We calculate the category of $D$-modules on the loop space of the affine line
in coherent terms. Specifically, we find that this category is derived equivalent
to the category of
ind-coherent sheaves on the moduli space of rank one de Rham local systems
with a flat section. Our result establishes a conjecture coming out of the
$3d$ mirror symmetry program, which obtains new compatibilities
for the geometric Langlands program from rich dualities of QFTs that
are themselves obtained from string theory conjectures.

Tate's thesis in the de Rham setting

Justin Hilburn

Perimeter Institute,
31 Caroline St N, Waterloo, ON N2L 2Y5, Canada

Sam Raskin

The University of Texas at Austin,
Department of Mathematics,
RLM 8.100, 2515 Speedway Stop C1200,
Austin, TX 78712

§ INTRODUCTION

§.§ Statement of the main results

We work over a field $k$ of characteristic zero.

Let $\sY$ be the moduli space of rank $1$ de Rham local systems on
the punctured disc equipped with a flat section, and
let $\fL \bA^1$ denote the algebraic loop space of $\bA^1$.

Our main result asserts:

There is a canonical equivalence of DG categories:

\[
\Delta:D^!(\fL \bA^1) \simeq \IndCoh^*(\sY).
\]

Moreover, this equivalence is compatible with the local geometric
class field theory of Beilinson-Drinfeld.

Here the left hand side is the category of $D$-modules on $\fL \bA^1$,
as defined in <cit.>, <cit.>. The right hand side
is the category of ind-coherent sheaves on $\sY$, which we construct
in <ref> (following <cit.>). In both cases,
there are infinite-dimensional subtleties in the definitions;
these are far more severe on the right hand side.

The assertions of the theorem are made more precise in the body of the paper.
For now, we content ourselves with the following remark on local class field theory.

Beilinson-Drinfeld construct an equivalence:

\begin{equation}\label{eq:intro-lcft}
D^*(\fL\bG_m) \simeq \QCoh(\LocSys_{\bG_m})
\end{equation}

for $\LocSys_{\bG_m}$ the moduli space of rank $1$ de Rham local systems on the
punctured disc. The left hand side here naturally acts on
the left hand side of Theorem <ref>, while the right hand side
naturally acts on the right hand side of Theorem <ref>. The compatibility
asserts that the equivalence of Theorem <ref> is compatible with
these actions.

Both sides of the above equivalence have natural $t$-structures.
However, this equivalence is not $t$-exact; it is necessarily
an equivalence of derived categories. This is in contrast to (<ref>),
which largely amounts to an equivalence of abelian categories.

For $K$ a local field, Tate's thesis <cit.>
(see also <cit.>)
considers the decomposition of the space
$\sD(K)$ of tempered distributions on $K$
as a representation for $K^{\times}$. We consider Theorem <ref>
as an analogue of the arithmetic situation. We observe that
our result applies not only at the level of eigenspaces,
but describes a categorical analogue of the whole space $\sD(K)$.

We plan to return to global aspects of the subject in future work.

As far as we are aware, our work is the first one in geometric representation
theory to prove a theorem about coherent sheaves on a
space of the form $\Maps(\o{\sD}_{dR},Y)$ for $Y$ an Artin stack
that is not a scheme and is not a classifying stack (for us: $Y = \bA^1/\bG_m$).
As we will see in <ref> and <ref>, there are substantial
technical difficulties that arise in this setting.
Roughly speaking, the singularities of the space $\sY$ are genuinely of infinite type.
We can only overcome these difficulties in our specific (abelian) setting.

As discussed later in <ref>, the main piece of our
construction is a realization of $D$-modules on $\bA^n$ in spectral terms;
this is the content of <ref>.
Already in the $n = 1$ case, our realization of $D$-modules on
$\bA^1$ as coherent sheaves on some space is novel; in this case,
we highlight that the expression
is as coherent sheaves on the subspace $\sY^{\leq 1} \subset \sY$
discussed in detail in  <ref>.

In the physicist's language, we are
comparing line operators for the $A$-twist of a pure hypermultiplet
with the $B$-twist of a $U(1)$-gauged hypermultiplet. Moreover, we
consider these $3d$ $\sN = 4$ theories as boundary theories for
(suitably twisted) pure $U(1)$-gauge theory (with electro-magnetic duality
exchanging its $A$ and $B$-twists).

§.§ Connections to $3d$ mirror symmetry

The equivalence of Theorem <ref> is a first[Specifically,
as an equivalence of categories of line operators, with both matter and
a non-trivial gauge group on the $B$-side.]
instance of what is expected to be a broad family of equivalences,
which goes under the heading
$3d$ mirror symmetry.
This subject is closely tied to the geometric Langlands program.

As context for our results, we
provide an informal introduction to these ideas below.
Our objective is to connect our equivalence as closely as possible with the
mathematical physics literature, and to promote those ideas to mathematicians
interested in the circle of ideas around geometric Langlands.

We hope the reader will forgive the
redundancy of our discussion given
the numerous other great sources in the literature.

We emphasize that we do not claim originality for any of the ideas
appearing below. We withhold attributions and leave much contact
with the existing literature until <ref>.

For another exposition of this subject also targeted at mathematicians,
see <cit.>. For a recent physics-oriented exposition, we refer
to <cit.> 2.

The physics of the last thirty years has highlighted the role of dualities:
quantum field theories that are equivalent by non-obvious means.
We emphasize that
physicists regard these dualities as conjectural: they do not claim
to know how to match the QFTs, only (at best) parts of them.

Examples abound. For instance,
Montonen–Olive's (conjectural) $S$-duality for
$4$-dimensional Yang-Mills theory with $\sN = 4$ supersymmetry for two
Langlands dual gauge groups $G$ and $\check{G}$ is of keen interest.

The dualities physicists consider fit into a sophisticated logical hierarchy.
But briefly, most are subsumed in the existence of Witten's (conjectural) $\sM$-theory,
which is supposed to recover other (known) QFTs by various constructions.

A general problem is to extract mathematical structures
from quantum field theories.[Often, this is by a sort of analogy. Physicists
are drawn to differential geometry: their bread and butter are moduli spaces
of solutions to non-linear PDEs (namely, Euler-Lagrange equations).

But for a mathematician,
it may be preferable to consider algebraic varieties as analogous to Kähler
manifolds, or symplectic varieties as analogous to hyper-Kähler manifolds.
Often, rich algebraic geometry results.]
In this case, physical dualities relate to mathematical

For instance, in <cit.>,
Kapustin and Witten found such a relationship
between Montonen–Olive's (conjectural) duality for
$4$-dimensional Yang-Mills with $\sN = 4$ supersymmetry and geometric Langlands
conjectures in mathematics. Their perspective was developed in
<cit.> and
<cit.>, which exactly clarified some means of extracting
algebro-geometric invariants from Lagrangian field theories.

§.§.§ $3d$ $\sigma$-models

In one setting, so-called[We remind what these parameters
encode in the subsequent discussion.]
$3d$ $\sN = 4$ quantum field theories, there are
rich relations with algebraic geometry.

First, for every algebraic stack $\sX$, we suppose
there is (in some algebraic sense)
a $3d$ (Lagrangian) quantum field theory
$T_{\sX}$ with $\sN = 4$ supersymmetry;
we spell out some more precise expectations below.

At the quantum level, physicists agree our supposition is problematic:
cf. the discussion at the
end of <cit.> 2.3. (The
classical field theory is fine, but may be unrenormalizable, and even if it is
renormalizable, there may not be a distinguished scheme.)
So it is better
to regard $T_{\sX}$ as a heuristic:
its twists (see below) are all that is defined
(at the quantum level).

In fact, physicists
say that $T_{\sX}$ depends only on the symplectic stack
$T^* \sX$. At the classical level, this theory
is the $3d$ $\sigma$-model with target $T^*\sX$.
The $\sN = 4$ supersymmetry comes from the
symplectic structure on $T^*\sX$.

I.e., more generally, there is a classical $3d$ $\sN = 4$ theory
corresponding to a $\sigma$-model with target any symplectic stack $\sS$.
One hopes for a quantization associated with a Lagrangian
foliation $\lambda:\sS \to \sX$;
at the moment, this quantization (or rather, its $A/B$ twists)
is only understood for $\sS$ a twisted cotangent bundle over some $\sX$
(and $\lambda$ the canonical projection).

This perspective is important, but sometimes
inconvenient (especially at the quantum level),
so we emphasize the role of $\sX$ e.g. in our notation.

§.§.§ Algebraic QFTs

Next, we describe how we think about a $3d$ QFT $T$ in algebraic geometry.
We include this discussion to make precise the connection between the mathematical
physics we wish to discuss and the algebraic geometry we wish to study.

Recall from <cit.> that an oriented $3$-dimensional fully-extended TFT
$T$ would
attach a number $T(M)$ to a closed 3-manifold $M$, a vector space
$T(M)$ to a closed 2-manifold $M$, a DG category $T(M)$ to a closed
1-manifold $M$, and a DG 2-category (i.e., $\DGCat_{cont}$-module
category) to a 0-manifold $M$. Cobordisms define morphisms. Disjoint unions
go to tensor products.

A variant: given an ambient $3$-manifold $N$, a fully-extended TFT on $N$ should
assign such data to manifolds $M$ equipped with embeddings into $N$.
(We are not aware of a reference.)

The algebraic situation is similar, but only some data is defined.
We heuristically describe the main structures, without giving formal definitions.

Fix a smooth, projective, geometrically connected algebraic curve $X$,
which we regard as analogous to a $2$-manifold. (The curve $X$ should not be confused
with the $\sX$ that sometimes occurs when considering a $\sigma$-model with
target $T^*\sX$.)

A[As the input to the discussion is an algebraic curve $X$,
we sometimes refer to this formalism as algebraic QFT.
We emphasize that we are not referring to some kind of
“algebraic twist" (analogous to a “holomorphic twist")
of a supersymmetric Lagrangian field theory —  the
formalism is insensitive to such considerations. Rather, this
terminology refers to the fact that this formalism is defined without
analysis, and makes sense in the general setting of algebraic geometry (of
curves here, with evident adaptations in the terminology otherwise)
over fields (perhaps of characteristic zero).]
$3d$ field theory $T$ on[It might be better
to say the theory lives on $X \times \bR$.] $X$ includes the
data of a vector space $T(X) \in \Vect$ and a category
$T(\o{\sD}_x) \in \DGCat_{cont}$ for every point $x \in X$,
regarding $\o{\sD}_x$ as analogous to a circle.

We regard the formal disc $\sD_x$ as a cobordism
$\emptyset \to \o{\sD}_x$. There is an associated functor:

\[
\Vect = T(\emptyset) \to T(\o{\sD}_x) \in \DGCat_{cont}
\]

i.e., there is a preferred object of $T(\o{\sD}_x)$,
the so-called vacuum (or unit) object.

We regard $X \setminus x$ as a cobordism
$\o{\sD}_x \to \emptyset$. There is an associated functor:

\[
T(\o{\sD}_x) \to T(\emptyset) = \Vect \in \DGCat_{cont}.
\]

This functional evaluated on the vacuum is
$T(X) \in \Vect$.

Unlike in the topological situation, the above is not symmetric.
I.e., we only allow cobordisms in the directions specified above.

In this $3d$ setting, the category $T(\o{\sD}_x)$ is often
called the category of line operators for the theory. (Physicists would
draw the line $x \times \bR \subset X \times \bR$ passing through
the interior of our circle $\o{\sD}_x \times 0 \subset X \times \bR$.)

The assignment $x \rightsquigarrow T(\o{\sD}_x)$ should extend
to a factorization category on $X$ in the sense of
<cit.>. The vacuum objects should correspond to a unital
structure on this factorization category. The functionals
above should extend to a well-defined functional on the
independent category of this unital factorization category;
see <cit.> for the definition.

In <cit.>, Beilinson and Drinfeld defined such a structure
for any chiral algebra $\sA$ on $X$, i.e., they (in effect)
defined a $3d$ field theory $T_{\sA}$.
The vector space $T_{\sA}(X)$ is the
space of conformal blocks of $\sA$. The category
$T_{\sA}(\o{\sD}_x)$ is $\sA\mod_x$, the category of (unital) chiral
modules for $\sA$ supported at $x$. The vacuum object is the
vacuum representation of $\sA$, and the functional
$\sA\mod_x \to \Vect$ is the functor $C^{ch}(X,\sA,-)$ from
loc. cit. 4.2.19.

(This theory is analogous to the $3d$ TFT associated
with a $\sE_2$-algebra $\sA$ in
<cit.> Theorem 4.1.24.[We
note that loc. cit. regards this theory as a fully-extended
$2d$ theory with more categorical complexity than we outlined
before. We instead regard the theory as a not-fully-extended
$3d$ theory of the type outlined above.
This compares to Remark 4.1.27 from <cit.>.])

Informally, it is good to think of $3d$ theories on $X$ as
the natural home for the Morita theory of chiral algebras,
just as $\DGCat_{cont}$ is the natural home for Morita theory of
associative algebras.

The relationship between chiral algebras and $3d$ $\sN = 4$ (and
sometimes $\sN = 2$) theories is the starting point of the series of
works <cit.>, <cit.>, and <cit.>, which aim to use
chiral algebras to
study mirror dual theories in this way. To describe this work in more detail,
we use ideas reviewed later in this introduction.

Specifically, a $3d$ $\sN = 4$
theory $T$ with a suitably deformable
(cf. <cit.> 2.3; there is an implicit
choice of $A$ or $B$ twists fixed in our discussion here)
$\sN = (0,4)$ boundary condition $\sB$ should yield an
algebraic $3d$ field theory $T^{alg}$
and a boundary condition $\sB^{alg}$ for it; here
we regard $\sB^{alg}$ as an interface $T^{alg} \to \on{triv}$
to the trivial theory.
This data yields a factorization algebra $\sA_{T,\sB}$ on $X$: its fiber at $x$
is by evaluating $\sB^{alg}(\o{\sD}_x):T(\o{\sD}_x) \to \on{triv}(\o{\sD}_x) = \Vect$
at the vacuum object for $T$. In this case, there is a canonical
interface $T^{alg} \to T_{\sA_{T,\sB}}$. Sometimes, it is even an equivalence,
meaning that one can study the full theory $T^{alg}$ via the factorization
algebra $\sA_{T,\sB}$.

As David Ben-Zvi emphasized to us, unital factorization categories
with a functional on their independent categories
do seem to provide a robust
definition of $[1,2]$-extended $3d$ algebraic QFT (while allowing for
some quite degenerate theories). He also informs us that
these ideas were developed in collaboration with Sakellaridis and Venkatesh,
and will be developed in their forthcoming work.

Is there some sense in which the theories of Example <ref>
are $[0,2]$-extended, as for their $E_2$-counterparts?
And what general constructions of morphisms
(alias: interfaces) between such theories exist?[This question
is of practical importance to us. Our
main theorem amounts to a construction of a non-trivial interface
between two $3d$ theories. It is desirable to understand this construction on
better conceptual grounds.]

§.§.§ Supersymmetry

We now recall the main practical
application of supersymmetry: supersymmetric
QFTs (typically) come with canonical
deformations,[Sometimes, we think of
deformations as quantizations. For us, the difference
is as follows.

Briefly, in $0$-dimensions, $\sP_0$-algebras can be quantized
by $\sE_0$-algebras,
i.e., pointed vector spaces. On the other hand, pointed vector
spaces $V$ may be further deformed by
finding a filtered vector space $V^{def}$ with $\gr_{\dot} V^{def} = V$.
(In homological settings, such a structure is equivalent to equipping $V$ with
a differential in a suitable sense, or one might say, deforming
the differential on $V$.)

In higher dimensions, one can say essentially the same words, following
<cit.>. Lagrangian
densities give rise to factorization $\sP_0$-algebras,
which govern the corresponding
classical field theory. These may be quantized by factorization
$\sE_0$-algebras, i.e., factorization algebras; this amounts to
quantizing the classical field theory (or more specifically, constructing
the OPE for local operators). These factorization algebras
may be further deformed to other factorization algebras.]
called twists.[For instance,
for a manifold $X$, its de Rham complex deforms
its complex of global differential forms (considered
as equipped with zero differential). Similarly,
for $X$ a smooth
algebraic variety, its de Rham complex
deforms its Hodge cohomology (or rather, the cochain
analogue). These examples appear in supersymmetric
quantum mechanics.] Twisting preserves dimensions
but lowers the amount of supersymmetry; in our examples,
there is no residual supersymmetry.

See e.g. <cit.> and <cit.>
for an introduction to the twisting procedure. See
<cit.> and <cit.>
for a detailed classification of the twists of supersymmetric QFTs. For the specific
twists considered in $3d$ mirror symmetry, see <cit.> 2.2.

In the setting of $3d$ QFTs $T$ with $\sN = 4$ supersymmetry,
there are two twists of interest for us: the $A$-twist $T_A$ and
the $B$-twist $T_B$. In practice, these are algebraic theories.

For our theories $T_{\sX}$ of interest,
we let $T_{\sX,A}$ and $T_{\sX,B}$ denote the corresponding
$3d$ QFTs.

The category $T_{\sX,A}(\o{\sD}_x)$ of line operators for the $A$-twisted theory
should be the category[At this point,
we are operating heuristically and do not want to be overly prescriptive, so
we do not consider finer points such as $D^!$ vs. $D^*$. Similarly in
Example <ref>.

With that said, in our example, $D^!$ is what appears.]
$D(\fL\sX)$ of $D$-modules on the algebraic loop space:

\[
\fL \fX = \Maps(\o{\sD}_x,\sX)
\]

of $\sX$.

The category $T_{\sX,B}(\o{\sD}_x)$ of line operators for the $B$-twisted
theory should be:

\[
\IndCoh\big(\Maps(\o{\sD}_{x,dR},\sX)\big).
\]

If $\sX$ is smooth affine, each of the categories above essentially
come from chiral algebras, as in Example <ref>.
Indeed, up to mild corrections,
the $A$-twist is essentially governed by a CDO for $\sX$, while the
$B$-twist is governed by the commutative chiral algebra of functions
on[As in <cit.>, it is convenient to think of
$\sX$ as $\Maps(\sD_{dR},\sX)$ here.] $\sX$.

The algebraic QFTs appearing above are of a special nature. First, they
are defined functorially on every smooth curve $X$. Moreover, for
$X = \bA^1$, the resulting sheaves of categories are strongly
$\bG_a$-equivariant (cf. <cit.> Chapter 2).
These observations are a sort of de Rham analogue of the
fact that the $A$ and $B$-twists are topological twists.

Below, we speak of both algebraic and non-algebraic (or analytic)
theories. Typically, we speak of supersymmetry for the non-algebraic theories,
and obtain algebraic theories by twisting.

§.§.§ $3d$ mirror symmetry (first pass)

Given a $3d$ $\sN = 4$ theory $T$, there is another $3d$ $\sN = 4$
theory $T^{\star}$; this theory has the same underlying QFT as $T$, but the
embedding of the supersymmetry algebra is modified by an automorphism
of that algebra. The salient property for us is that the $A$-twist
$T_A^{\star}$ of $T^{\star}$ is the $B$-twist $T_B$ of the original theory.
Moreover, $(T^{\star})^{\star} = T$. We refer to $T^{\star}$ as the
abstract mirror dual to the theory $T$.

We now describe a simplified version of $3d$ mirror symmetry conjectures.

These conjectures state that for certain 3d mirror dual pairs
$(\sX,\sX^{\star})$ of algebraic stacks, there is
an equivalence:

\[
T_{\sX}^{\star} \simeq T_{\sX^{\star}}
\]

of $3d$ field theories.

For instance, passing to $B$-twists and categories of line operators on both sides,
such a conjecture predicts an equivalence:

\[
D(\fL\sX) \simeq \IndCoh\big(\Maps(\o{\sD}_{x,dR},\sX^{\star})\big).
\]

The pair $(\sX,\sX^{\star}) = (\bA^1,\bA^1/\bG_m)$ is supposed to be a $3d$ mirror
dual pair of stacks. In this case, Theorem <ref> amounts to
an equivalence of line operators for the corresponding twisted[This
may seem incomplete, given that $3d$ mirror symmetry is stated more symmetrically
in terms of untwisted theories. However, as in Remark <ref>,
the untwisted QFTs are on unsteady ground.]

§.§.§ $S$-duality

The theory of $3d$ mirror symmetry as presented above admits a generalization
in which there is an auxiliary pair $(G,\check{G})$ of Langlands dual
reductive groups; the previous setting amounts to a pair of trivial groups.

The theory becomes much richer in this setting. There are connections
to $S$-duality in $4d$ gauge theory and the geometric Langlands correspondence.
Moreover, many of the examples in conventional (i.e., trivial group) $3d$ mirror
symmetry can be better understood as being built from more primitive
examples in the general setting.

The major cost is that even the formulation of the conjectures
becomes conditional on forms of $S$-duality conjectures.

We discuss these ideas in more detail below.

First, we need to discuss $4d$ algebraic QFTs, which we denote $\mathsf{T}$.

This is easy: the formalism is the same as in <ref>,
but one categorical level higher. For instance, there should
be a DG category attached to $X$, and a DG 2-category $\mathsf{T}(X)$
(of surface defects) attached to

We can speak of interfaces $\mathsf{T}_1 \to \mathsf{T}_2$ between
theories, which are morphisms in the natural sense. An interface
$\mathsf{triv} \to \mathsf{T}$ from
the trivial[This theory attaches $\Vect$ to $X$ and
$\DGCat_{cont}$ to $\o{\sD}_x$.]
theory is a boundary condition for $\mathsf{T}$.
This amounts to objects:

\[
\begin{gathered}
\sB_X \in \mathsf{T}(X) (\in \DGCat_{cont})  \\
\sB_x \in \mathsf{T}(\o{\sD}_x) (\in \mathsf{2DGCat}).
\end{gathered}
\]

A boundary condition $\mathsf{triv} \to \mathsf{triv}$ is the same
as a $3d$ theory.

Note that theories can be tensored. The $4d$ theories we consider are naturally
self-dual. So we can (and do) consider interfaces and boundary conditions as
operating in both directions. In such a situation, given boundary conditions
$\sB_1,\sB_2$ for $\mathsf{T}$, we let $\pair{\sB_1}{\sB_2}$ denote
the resulting boundary condition $\mathsf{triv} \to \mathsf{triv}$
obtained by composing $\mathsf{triv} \xar{\sB_1} \mathsf{T} \xar{\sB_2} \mathsf{triv}$
and applying Remark <ref>.

We now discuss twists.

For $4d$ $\sN = 4$ (non-algebraic) theory $\mathsf{T}$, there are supposed
to be $A$ and $B$ twists $\mathsf{T}_A$ and $\mathsf{T}_B$.
Again, there is an involutive operation of
abstract mirror dual $\mathsf{T}^{\star}$, exchanging $A$ and $B$ twists.

We now recall that in $4d$, for a reductive group $G$,
there is Yang-Mills $\YM_G$ theory with $\sN = 4$
supersymmetry. Again, there are $A$ and $B$-twists with algebraic

The $A$-twisted theory $\YM_{G,A}$ attaches to $X$ the DG category
$D(\Bun_G(X))$ of $D$-modules on $\Bun_G(X)$, and to $\o{\sD}_x$
the DG 2-category $\fL G\mod$ of DG categories with a (strong)
action of the loop group $\fL G$ to $\o{\sD}_x$.
For the latter, the vacuum object is $D(\Gr_G)$ and the (“chiral homology")
functional $\fL G\mod \to \Vect$ is given by tensoring over $D^*(\fL G)$ with

The $B$-twisted theory $\YM_{G,B}$ attaches to $X$ the DG category
$\QCoh(\LocSys_G(X))$ of quasi-coherent sheaves on
$\LocSys_G(X)$, and to $\o{\sD}_x$ attaches
the DG 2-category $\ShvCat_{/\LocSys_{G}(\o{\sD}_x)}$.

Given a stack $\sX$ with a $G$-action, there is a corresponding
boundary condition $\sB_{\sX}$ for $\YM_G$.

For $\sX = G$, this boundary condition is called the Dirichlet
boundary condition; we denote it by $\Dir_G$.

For $\sX = \Spec(k)$, this boundary condition is called the Neumann
boundary condition; we denote it by $\Neu_G$.

For any boundary condition $\sB$ of $\YM_G$, we let $T_{\sB}$ denote
the $3d$ theory $\pair{\sB}{\Dir_G}$. By standard compatibility of $3d$ and $4d$
supersymmetry algebras, any such $T_{\sB}$ has $\sN = 4$ supersymmetry.

For a $3d$ $\sN = 4$ theory $T$, the data of $G$-flavor symmetry
is the data of a boundary condition $\sB$ for $\YM_G$ and an identification
$T \simeq T_{\sB}$. Similarly, $G$-gauge symmetry for $T$
is the data of a boundary condition $\sB$ and an identification
$T \simeq \pair{\sB}{\Neu_G}$.

The above is compatible with twists; $\sB$ gives a boundary condition
$\sB_A$ for the $A$-twist $\YM_{G,A}$, and similarly for $B$-twists.
The resulting $3d$ (algebraic) theories are $T_{\sB,A}$ and $T_{\sB,B}$.

For $\sX$ as above, the boundary condition $\sB_{\sX,A}$ should define
the object $D(\fL \fX) \in \fL G\mod$, and the boundary condition
$\sB_{\sX,B}$ should define $\IndCoh(\Maps(\o{\sD}_{x,dR},\sX/G)) \in
\ShvCat_{/\LocSys_G(\o{\sD})}$. Globally, there should be objects
of $D(\Bun_G)$ and $\QCoh(\LocSys_G)$ as well. These objects
are considered by Ben-Zvi, Sakellaridis, and Venkatesh, who
term them period/$L$-sheaves, interpreting
them as sheaf-theoretic analogues of period integrals/$L$-function values
from harmonic analysis and number theory. They were also considered
(away from singular loci) in <cit.> 9-10.

Parallel to Remark <ref>, at least at the classical (non-quantum) level,
the boundary condition $\sB_{\sX}$ can be defined more generally for
Hamiltonian $G$-spaces, with $\mu:T^*\sX \to \fg^{\vee}$ inducing
$\sB_{\sX}$ (and again, something weaker than
a suitable Lagrangian foliation should be needed to quantize).

In the above setting, $S$-duality is supposed to be an equivalence:

\[
\YM_G^{\star} \simeq \YM_G
\]

of $4d$ theories. Passing to $B$-twists, we obtain:

\[
\YM_{G,A} \simeq \YM_{\check{G},B}.
\]

Up to smearing[Throughout this exposition of the general ideas,
we give ourselves the freedom of ignoring these subtleties.
In the body of this paper (specifically, <ref> and <ref>),
we treat these technical points in detail in our context. The non-abelian future of the
subject will of course similarly need to confront these points more seriously.]
away homological algebra subtleties
(cf. <cit.>), this is a form of the geometric Langlands
conjectures, encoding local and global theories and compatibilities
between them (cf. <cit.>).

The subsequent discussion is predicated on the existence of $S$-duality, so
we assume it in what follows.

Given a boundary condition $\sB$ for $\YM_G$, we let
$\sB^{\star}$ denote the resulting abstract mirror dual
boundary condition for $\YM_G^{\star}$, and then let
$\check{\sB}$ denote the resulting $S$-dual boundary condition for

§.§.§ $3d$ mirror symmetry redux

Now fix a reductive group $G$.

In general, a[Traditionally, 3d mirror symmetry refers to
the case where $G = \check{G} = \on{triv}$. We find the present terminology
convenient, although it departs somewhat from established conventions.]
$3d$ mirror dual pair (for $(G,\check{G})$)
consists of a pair $(\sX,\sX^{\star})$ where $G \actson \sX$,
$\check{G} \actson \sX^{\star}$, and there is an equivalence:

\[
\check{\sB}_{\sX} \simeq \sB_{\sX^{\star}}
\]

of boundary conditions for $\YM_{\check{G}}$. I.e., the
boundary condition $S$-dual to $\sB_{\sX}$ should be $\sB_{\sX^{\star}}$.
For $G$ trivial, this recovers our earlier notion of $3d$ mirror dual pairs.

Suppose $(\sX,\sX^{\star})$ as above. Passing to $B$-twists, we find that the objects:

\[
D(\fL\sX) \in \fL G\mod \simeq
\ShvCat_{/\LocSys_{\check{G}}(\o{\sD})}
\ni
\IndCoh(\Maps(\o{\sD}_{x,dR},\sX^{\star}/\check{G}))
\]

are supposed to match, where the middle isomorphism is local geometric
Langlands (considered modulo homological subtleties).

We remark that this does not amount to an equivalence of categories
between the categories on the left and on the right here. Rather, this
statement is inherently conditional on local geometric Langlands.
(And we intend it to be a little fuzzy regarding homological subtleties.)

With that said, in the case that $G$ is abelian, the above does
amount to an equivalence of categories compatible with certain symmetries.
(In general, the Whittaker category for the LHS above should be close to the
RHS above.)

§.§.§ Some examples of mirror dual pairs

To illustrate the above, we briefly include some examples.
For many other examples, we refer to <cit.>, which is
a table of examples maintained by Jonathan Wang. (The terminology in the
first two examples is taken from Ben-Zvi, Sakellaridis, and Venkatesh,
who are appealing to the analogy with harmonic analysis.)

For $G = \check{G} = GL(V) \times GL(V)$, the pair
$(GL(V) \times V,\End(V))$ should be $3d$ mirror dual.

In particular, the objects:

\[
D(\fL GL(V) \times \fL V) \in \fL GL(V) \times \fL GL(V) \mod
\]

should correspond to:

\[
\IndCoh(\Maps(\o{\sD}_{dR},GL(V) \backslash\End(V)/GL(V)) \in
\ShvCat_{/\LocSys_{GL(V)\times GL(V)}(\o{\sD})}
\]

under local geometric Langlands.

For $V = \bA^1$,
the above asserts that:

\[
D(\fL\bG_m \times \fL \bA^1) \in \fL \bG_m \times \fL \bG_m \mod
\]

corresponds to:

\[
\IndCoh(\sY \times \LocSys_{\bG_m}).
\]

As our group is abelian, we expect an honest equivalence
of categories. Passing to $\fL\bG_m$-invariants $\leftrightarrow$
$\{$fiber at $\on{triv} \in \LocSys_{\bG_m}\}$, we obtain a conjecture:

\[
D(\fL \bA^1) \simeq \IndCoh(\sY).
\]

This is our main result.

[Gaiotto-Witten <cit.>, 3.3.1]

Let $G = \check{G} = GL(V)$ for $\dim(V) = n$.
Let ${\sL au}$ be the (Laumon) moduli space
of data $(V_1,V_2,\ldots,V_{n-1},f_1,\ldots,f_{n-1})$ where
$V_i$ is a vector space of dimension $i$ and $f_i:V_i \to V_{i+1}$ is a
morphism, where by definition, $V_n = V$.

Then the pair:

\[
(\sX,\sX^{\star}) = (GL(V),{\sL au})
\]

is expected to be $3d$ mirror dual (with respect to our $(G,\check{G})$).

Unwinding, this in effect gives a sort of formula (or kernel) for
geometric Langlands for $GL_n$; unfortunately, the necessary symmetries on the
resulting object are not apparent.

§.§.§ Where do mirror dual pairs come from?

We briefly address the stated question. There are two styles of answer.

First, one can try to reverse engineer examples. If one believes a mirror
dual $\sX^{\star}$ to some $\sX$ should exist,
one can sometimes calculate the ring of
functions on $T^*{\sX^{\star}}$ as the Coulomb
branch[I.e., total cohomology of endomorphisms of the unit object
in $T_{\sX^{\star},B} = \IndCoh(\Maps(\o{\sD}_{dR},\sX))$.] of $\sX^{\star}$,
which must correspond to the Higgs branch[I.e., total cohomology
of endomorphisms of the unit object
in $T_{\sX,A} = D(\fL \sX)$.] of $\sX$.

Alternatively, as in the work of Ben-Zvi, Sakellaridis, and Venkatesh,
one can reverse engineer examples by fitting phenomena from number theory
into the above framework via the analogy between automorphic forms
and automorphic $D$-modules.

But sometimes (always?) there is a better answer than either.
Many $3d$ mirror symmetry examples can be derived from (super)string
theory dualities and the existence of $\sM$-theory. (For example, see
<cit.> 3.1.2 for the derivation in our example.)
We are not aware of a mathematical counterpart to this idea, i.e., a simple conjectural
framework that subsumes all examples in $3d$ mirror symmetry. Clearly this
would be highly desirable.

Forthcoming work of the first author and Philsang Yoo will develop in detail the
mathematical derivation of mirror pairs from $S$-duality in string theory.

§.§.§ The role of this paper

Our objective in writing this paper was to test the above ideas in the
simplest[Not covered by geometric Langlands/$S$-duality for $\YM$,
and in which the full weight of the infinite dimensional geometry appears.]
case of interest, which is the Tate case discussed above.
Here $(G,\check{G}) = (\bG_m,\bG_m)$ and[The discrepancy here with the
ungauged version considered in Example <ref> is
explained by the fact that $\Dir$ and $\Neu$ are $S$-dual for tori.]
$(\sX,\sX^{\star}) = (\bA^1,\bA^1)$.

As we consider abelian gauge groups,
for which geometric Langlands is unconditional, this example can be
studied in complete detail.
We perform this study at the level of categories of line operators. We will
return to global considerations (compatibility with chiral homology) in
future work.

Given that we obtain complete positive results in this case, and
because many of the technical difficulties inherent in the $3d$ mirror symmetry
project occur in our example, we believe that our results provide strong
support for the general conjectures.

§.§.§ Some references

Many of the ideas discussed above were developed in
collaborative work, often unpublished, of a number of mathematical physicists.
We are grateful to many people who have shared these ideas
with us over the years, and find their intellectual generosity inspiring.
The downside of this situation is that we find some difficulty in accurately
attributing priority for the ideas. We do our best here, but apologize in
advance for any omissions or inaccuracies, which are not intended as slights.

The first instances of $3d$ mirror symmetry were considered in
<cit.>. The connection with string theory dualities
was made in <cit.>. These ideas were developed further
in <cit.>, which considered interactions between the
<cit.> constructions and $S$-duality for super Yang-Mills.
In turn, <cit.> translated those ideas into conjectures regarding
geometric Langlands via the Kapustin-Witten dictionary <cit.>
between Langlands and $S$-duality. The physics literature on BPS line operators
is too vast to survey here, but we refer the reader to <cit.> for a
review of the literature in the $3d$ $\sN = 4$ context.

The algebro-geometric interpretation of the $A$-side discussed
above is suggested by work of Braverman, Finkelberg and Nakajima
in <cit.>, <cit.>, <cit.>,
which emphasized the role of Coulomb branches.
The description of categories of line operators in twisted $3d$ $\sN = 4$
theories was given in unpublished work of Kevin Costello,
Tudor Dimofte, Davide Gaiotto, Philsang Yoo, and the first
author.[These ideas were recorded in 1.1 of
<cit.> and in <cit.> 7.]
The earliest derivation
was based on applying the method of Elliott-Yoo <cit.>
in the $3d$ $\sN = 4$ context,
i.e., calculating (derived) moduli spaces of solutions to Euler-Lagrange equations and
quantizing (in the shifted symplectic sense).[Because
we are not aware of a publicly available
account of this derivation, we direct the reader to <cit.>,
which is a series of talks Kevin Costello gave in 2014 that discuss some of
the main ingredients. It is the earliest talk we know of that contains
these components.]
By applying the yoga of $3d$ mirror symmetry, this description of
categories of line operators also
led to a conjectural form of Theorem <ref>
(and other related examples).[For a physics-oriented discussion of this
work, see <cit.>.]
Philsang Yoo and the first author considered these line operator categories
in the framework of local geometric Langlands as a mathematical interpretation
of <cit.> (inspired in part by <cit.>).[See
<cit.> and <cit.> for discussions of this work.]
On the $A$-side, an alternative derivation of the category of
line operators is suggested by
<cit.> 4; here the method is as described in Remark <ref>.

Connections between $3d$ mirror symmetry
and number theory (e.g., duality for spherical varieties and period
integrals <cit.>) were developed by Ben-Zvi, Sakellaridis,
and Venkatesh, in forthcoming[The work is also highly publicized –
there are recorded lectures about the work readily available online
(see e.g. <cit.>).]
work, which has led to profound new insights and many new examples of
dual pairs.

Finally, in addition to the above works,
we have drawn inspiration particularly from <cit.>,
<cit.>, <cit.>
<cit.>, <cit.>,
<cit.>, <cit.>, and <cit.> in our thinking about
mirror symmetry.

§.§.§ Some related works

We also highlight some works that are closely related to ours.

First, <cit.> studies a version of our problem
near the formal completion of $0 \in \sY$ (i.e., Theorem <ref>
in perturbation theory) using VOAs; in their picture (based on the
method of Remark <ref>), the
Lie algebra controlling the deformation theory is a
$\mathfrak{psu}(1|1)$ Kac-Moody algebra. We understand that these ideas
are currently being developed further by Andrew Ballin, Thomas Creutzig,
Tudor Dimofte, and Wenjun Niu.

Second, the works <cit.> and <cit.>
of Braverman-Finkelberg-Ginzburg-Travkin also obtained geometric results from
mirror symmetry conjectures, but in the non-abelian setting.
There it is not possible at present
to prove that boundary conditions match under $S$-duality, since
local geometric Langlands is only conjectural for non-abelian $G$.
Therefore, the cited authors instead consider the categories obtained by
pairing the vacuum boundary condition(s) for
$\YM_G$/$\YM_{\check{G}}$, and
establish the resulting conjectures for certain examples of
$3d$ mirror dual pairs $(G,\check{G})$ and $(\sX,\sX^{\star})$.

§.§ Outline of the paper

We now describe the contents of the present work.

First, we describe the general strategy.

By definition, there are fully faithful functors:

\[
D(\fL_n^+\bA^1) \into D^!(\fL^+ \bA^1) \into D^!(\fL \bA^1).
\]

Here $\fL_n^+\bA^1$ is the (scheme corresponding to)
the vector space $k[[t]]/t^nk[[t]]$; $\fL^+ \bA^1$ is the
(scheme corresponding to the pro-finite dimensional) vector
space $k[[t]]$, and $\fL\bA^1$ is the (indscheme corresponding to
the Tate) vector space $k((t))$. The first functor is a $!$-pullback
and the second functor is a (suitably normalized) $*$-pushforward functor.

The main step in our construction is the construction of corresponding

\[
\Delta_n:D(\fL_n^+\bA^1) \to \IndCoh^*(\sY).
\]

As the left hand side is modules over a Weyl algebra, this amounts
to constructing an object of the right hand side with an action of a Weyl
algebra. We construct this object explicitly, via generators and relations.

From there, we bootstrap up to construct a functor
$\Delta$ as in Theorem <ref>. We show it is an equivalence using
the geometry of $\sY$.

There is an adage in geometric Langlands that most of the work occurs on the
geometric (i.e., automorphic) side. In our work it is the opposite: the
geometric side is easy, and most of our work is on the spectral side.

We introduce $\sY$ (and related spaces) in <ref>. Here we
give the main geometric tools in our study of $\sY$. We study
ind-coherent sheaves on $\sY$ in <ref>, after some preliminary
remarks about Abel-Jacobi maps in <ref>. We construct the
functors $\Delta_n$ in <ref>; as indicated above, this is our main
construction. We check the corresponding compatibility of this construction
with class field theory
in <ref>. We then show fully faithfulness of $\Delta_n$ in
<ref>. We then prove Theorem <ref> in <ref>.

§.§ Categorical conventions

At various points, we use homotopical algebra and derived
algebraic geometry in Lurie's higher categorical form,
cf. <cit.>, <cit.>, <cit.>, though our notation more
closely follows the conventions of <cit.>.

In general, our terminology should be assumed to be derived.
We refer to $(\infty,1)$-categories as categories and
$\infty$-groupoids as groupoids. We let
$\Gpd$ denote the category of groupoids.

We let $\DGCat_{cont}$ denote the category of cocomplete[Rather, presentable.]
DG categories (over $k$) and continuous DG functors, cf. <cit.> I.1.10.
We let $\otimes$ denote its standard (Lurie) tensor product.
We let $\Vect \in \DGCat_{cont}$ denote the DG category of (chain complexes
of) vector spaces. For a DG algebra $A/k$, we let $A\mod$ denote the
corresponding DG category.
For a DG category $\sC$ and $\sF,\sG \in \sC$, we let
$\ul{\Hom}_{\sC}(\sF,\sG) \in \Vect$ denote the complex of
maps from $\sF$ to $\sG$.

For a DG category $\sC$ with a $t$-structure, we use cohomological
indexing conventions and let $\sC^{\leq 0}$ denote the connective
objects, $\sC^{\geq 0}$ denote the coconnective objects, and
let $\sC^{\heart} \coloneqq \sC^{\leq 0} \cap \sC^{\geq 0}$ denote
the heart of the $t$-structure. Similarly, we let
$\sC^+$ denote the eventually coconnective objects.

By default, schemes are DG[It will turn out a posteriori
that the primary objects
we consider are classical schemes (or stacks). But because we are studying
derived categories of coherent sheaves, the machinery and perspective of derived
algebraic geometry is quite convenient.]
schemes over $k$. We let
$\AffSch$ denote the category of affine schemes (over $k$),
and $\Sch$ the category of schemes (over $k$).

When we wish to refer to consider various non-derived objects as
derived objects, we use the
term classical. So we may speak of classical schemes, classical rings,
classical vector spaces, classical $A$-modules, etc., all of which
are full subcategories of the corresponding (suitably) derived categories.

Finally, we sometimes use standard ideas and notation from the theory of
categories with $G$-actions. We refer to <cit.> for an introduction
to these ideas.

§.§ Acknowledgements

We thank Sasha Beilinson, David Ben-Zvi, Dario Beraldo,
Mat Bullimore, Dylan Butson, Sasha Braverman,
Justin Campbell, Kevin Costello, Gurbir Dhillon, Tudor Dimofte, Chris Elliott,
Tony Feng, Davide Gaiotto, Nik Garner, Dennis Gaitsgory, Sam Gunningham,
Joel Kamnitzer, Ivan Mirkovic, Tom Nevins, Wenjun Niu, Pavel Safronov,
Yiannis Sakellaridis, Ben Webster, Jonathan Wang, Alex Weekes, and
Philsang Yoo for many helpful and inspiring conversations related
to this work. The first author particularly wishes to acknowledge and thank
Philsang Yoo for joint work conjecturing Theorem <ref>.

J.H. is part of the Simons Collaboration on Homological Mirror Symmetry supported by
Simons Grant 390287. S.R. was supported by NSF grant DMS-2101984.
This research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Research, Innovation and Science.

§ GEOMETRY OF THE SPECTRAL SIDE

§.§ Overview

In this section, we define $\sY$ and establish our main algebro-geometric tools
for studying its coherent sheaves.

Here is a more detailed overview of this section.

In <ref>, we give background on jet and loop spaces,
essentially setting up notation for later use.
In <ref>, we give background on $\LocSys_{\bG_m}$.
In <ref>, we define $\sY$. In <ref>,
we introduce variants of $\sY$ that are more traditional objects
of algebraic geometry. In <ref>, we prove a series
of flatness results; these are the key tools for studying $\sY$ and
its coherent sheaves.

The remainder of the section consists of a series of codas, essentially
developing material and notation for later reference. These may be skipped
at first pass and referred to as needed. In <ref>, we introduce
additional notation for later reference. In <ref>, we deduce
some exact sequences from our flatness results; these will be used
later in some inductive arguments. In <ref>, we describe
certain well-behaved opens in $\sZ^{\leq n}$.
In <ref>, we explicitly describe $\sY$ using
generators and relations. Finally, in <ref>, we draw pictures
explicitly describing the geometry in the regular singular situation.

§.§ Notation for jet and loop spaces

For a commutative DG algebra $A \in \ComAlg$,
$A[[t]] \in \ComAlg$ is defined as
$(\lim_n A \otimes k[[t]]/t^n)$ and $A((t))$ is defined
as $A[[t]] \otimes_{k[[t]]} k((t))$. We also let
$A[[t]]/t^n \coloneqq A \otimes_k k[[t]]/t^n$.

For a prestack $Z$,
$\fL Z \in \PreStk \coloneqq \TwoHom(\AffSch^{op},\Gpd)$ denote the functor:

\[
\Spec(A) \mapsto \Hom_{\AffSch}(\Spec(A((t))),Z).
\]

Similarly, we let $\fL^+ Z$ denote the functor:

\[
\Spec(A) \mapsto \Hom_{\AffSch}(\Spec(A[[t]]),Z).
\]

For $n \geq 0$, we let $\fL_n^+$ denote the functor:

\[
\Spec(A) \mapsto \Hom_{\AffSch}(\Spec(A[[t]]/t^n),Z).
\]

There is an evident comparison map:

\[
\vareps_Z:\fL^+ Z \to \underset{n}{\lim} \, \fL_n^+ Z.
\]

We often use the following lemma without mention.

For $Z$ an algebraic stack of
the form $Y/G$ with $Y$ affine and $G$ an affine
algebraic group, the map $\kappa_Z$ is an isomorphism.

The result is obvious when $Z = Y$ is affine.
For $Z = \bB G$, we refer e.g. to <cit.> Lemma 2.12.1.
The argument from loc. cit. immediately
extends to the general form of $Y$ considered here.

In fact, this result is true much more
generally for Noetherian algebraic
stacks with affine diagonals: see <cit.> Corollary 1.5.

We often let $\ev:\fL^+ Z \to \fL_1^+ Z = Z$
denote the evaluation map.

We have the following basic representability result.

Suppose $Z$ is an affine scheme almost of finite type.
Then $\fL^+ Z$ is an affine scheme and $\fL Z$ is an

We have the following variant of the above.

Suppose $Z$ is equipped with a $\bG_m$-action.
We let $\fL Zdt \in \PreStk$ denote the functor:

\[
\fL Z dt \coloneqq \fL(Z/\bG_m) \underset{\fL \bB \bG_m}{\times} \Spec(k)
\]

where $\Spec(k) \to \fL \bB \bG_m$ corresponds to the
(continuous) tangent sheaf $T_{\o{\sD}}$ on $\o{\sD}$.
We define $\fL^+Z dt$ similarly. A choice of trivialization
of $T_{\o{\sD}}$ defines an isomorphism $\fL Z dt \simeq \fL Z$
identifying $\fL^+ Z dt$ and $\fL^+ Z$; in particular,
Proposition <ref> applies in this setting.

Informally, $\fL Z dt$ parametrizes sections of
the fiber bundle $\o{\Theta}(\Omega_{\o{\sD}}^1) \overset{\bG_m}{\times} Z \to
\o{\sD}$, where $\Omega_{\o{\sD}}^1$ is the line bundle
of (continuous) 1-forms on $\o{\sD}$ and $\Theta(-)$
(resp. $\o{\Theta}$) denotes the
(resp. punctured) total space of a line bundle.

$\fL \bA^1$ is the algebro-geometric version of the space of Laurent
series, while $\fL \bA^1 dt$ is the algebro-geometric verison of the
space of 1-forms on the punctured disc (for the standard $\bG_m$-action
on $\bA^1$ by homotheties). Similarly, $\fL \bG_m$ is the algebro-geometric
version of the space of invertible Laurent series.
One easily finds that these indschemes are formally smooth and classical.

§.§ Rank $1$ local systems

We now give a detailed study of $\LocSys_{\bG_m}$, the moduli space of
rank $1$ local systems on the punctured disc. This material is well-known,
but it is convenient to review to introduce notation and constructions
we will need in the more complicated setting of our space $\sY$.

We define $\LocSys_{\bG_m}$ as follows.

We have a map $\fL \bG_m \to \fL \bA^1 dt$ sending
$f \in \fL \bG_m$ to $d\log(f)$, cf. <cit.> 1.12.
This map is a map of (classical) group indschemes, where
$\fL \bA^1 dt$ is given its natural additive structure.

We define $\LocSys_{\bG_m}$ as the stack[I.e., we sheafify
for the fppf topology. It is equivalent to sheafify for the
Zariski topology as $\Ker(\fL^+ \bG_m \to \bG_m)$ is pro-unipotent.
For the same reason, the resulting prestack is a sheaf for the
fpqc topology. In other words, there is no room for ambiguity in
quotient $\fL \bA^1 dt/\fL \bG_m$.

For normalization purposes, we remark that for
a point $\omega \in \fL \bA^1 dt$, we consider
$\nabla = d - \omega$ as the corresponding rank $1$ connection
(on the trivial line bundle).

We need variants of $\LocSys_{\bG_m}$ as well.

\[
\LocSys_{\bG_m,\log} \coloneqq
\LocSys_{\bG_m} \underset{\fL \bB\bG_m}{\times} \fL^+ \bB \bG_m =
\fL \bA^1 dt/\fL^+ \bG_m.
\]

This is the moduli space of line bundles on the
disc with a connection on the punctured disc.

There is an
evident action of $\Gr_{\bG_m} \coloneqq \fL \bG_m/\fL^+ \bG_m$
on $\LocSys_{\bG_m,\log}$ such that
$\LocSys_{\bG_m} = \LocSys_{\bG_m,\log}/\Gr_{\bG_m}$.

Define $\fL^{pol} \bA^1 dt$ as the quotient
$\fL \bA^1 dt / \fL^+\bA^1 dt$, the quotient being with respect
to the additive structure. In other words, $\fL^{pol} \bA^1 dt$
parametrizes polar parts of differential forms on the disc.
Note that $\fL^{pol} \bA^1 dt$ is an indscheme of ind-finite type.

As $d\log$ maps $\fL^+ \bG_m$ into $\fL^+ \bA^1 dt$, the
projection map:

\[
\fL \bA^1 dt \to \fL^{pol} \bA^1 dt
\]

intertwines the gauge action of $\fL^+ \bG_m$ on
the left hand side with its trivial action on the right hand
side. Therefore, we obtain a canonical map:

\[
\Pol: \LocSys_{\bG_m,\log} \to \fL^{pol} \bA^1 dt.
\]

This map takes the polar part of a connection.

By construction, there is a map:

\[
\LocSys_{\bG_m,\log} \to \bB \fL^+ \bG_m \xar{\ev} \bB \bG_m.
\]

This map sends a pair $(\sL,\nabla)$ (with $\sL$ a line
bundle on the disc and $\nabla$ its connection on the punctured
disc) to the fiber of $\sL$ at the origin.

The following result is well-known.

The map:

\[
\LocSys_{\bG_m,\log} \to \fL^{pol} \bA^1 dt \times \bB \bG_m
\]

is an isomorphism.

The map $\fL^+ \bG_m \xar{d\log,\ev} \fL^+ \bA^1 dt \times \bG_m$
is easily seen to be an isomorphism. The result is immediate
from here.

We now deduce a similar description of $\LocSys_{\bG_m}$
from Proposition <ref>.

There is a canonical residue map
$\Res:\fL^{pol} \bA^1 dt \to \bA^1$. There is a canonical projection
$\fL^{pol} \bA^1 dt \to \Ker(\Res)$ as the latter identifies
with $\fL \bA^1 dt/t^{-1} \cdot \fL^+ \bA^1 dt$,
so we obtain a product decomposition:

\[
\fL^{pol} \bA^1 dt = \bA^1 \times \Ker(\Res).
\]

There is an isomorphism $\LocSys_{\bG_m} \simeq
\bA^1/\bZ \times \Ker(\Res)_{dR} \times \bB \bG_m$ fitting
into a commutative diagram:

\[
\xymatrix{
\LocSys_{\bG_m,\log} \ar[rr]^{\text{Prop. \ref{p:locsys-log}}}
\ar[d] & & \fL^{pol} \bA^1 dt \times \bB \bG_m \ar[d] \\
\LocSys_{\bG_m} \ar[rr]^{\simeq}
& &
\bA^1/\bZ \times \Ker(\Res)_{dR} \times \bB \bG_m
\]

There is a canonical homomorphism
$\Gr_{\bG_m} \to \bZ$ given by minus[This sign is included to match
with standard normalizations.] the valuation map. This map splits
canonically as well: $\bZ = \Gr_{\bG_m}^{red}$.
Therefore, we obtain a canonical product decomposition:

\[
\Gr_{\bG_m}^{red} = \bZ \times \Gr_{\bG_m}^{\circ}
\]

for $\Gr_{\bG_m}^{\circ}$ the connected component
of the identity. Here we consider $\bZ$ as a discrete indscheme
over $k$ in the natural way, i.e., as $\coprod_{n \in \bZ} \Spec(k)$.

We have a commutative diagram:

\[
\xymatrix{
\Gr_{\bG_m} \ar[r]^{d\log} \ar@{=}[d] &
\fL^{pol} \bA^1 dt \ar@{=}[d] \\
\bZ \times \Gr_{\bG_m}^{\circ} \ar[r]
& \bA^1 \times \Ker(\Res)
\]

where the bottom map is obtained as the product
of the homomorphisms $\bZ \into \bA^1$ and
$d\log:\Gr_{\bG_m}^{\circ} \to \Ker(\Res)$. The latter
is easily seen to induce an isomorphism between
$\Gr_{\bG_m}^{\circ}$ and the formal group
$\Ker(\Res)_0^{\wedge}$ of $\Ker(\Res)$. This gives the claim.

This isomorphism actually depends mildly on the choice of uniformizer $t$;
this is needed to trivialize the action of
$\Gr_{\bG_m}$ on the $\bB \bG_m$-factor.

We will also use truncated versions of the above spaces in
which we bound the irregularity of our local systems.

We fix $n \in \bZ^{\geq 0}$ in what follows.

Define $\fL^{\leq n} \bA^1 dt \subset \fL \bA^1 dt$ as the
classical closed subscheme whose points are differential
forms with poles of order at most $n$. Therefore,
$\fL^{\leq n} \bA^1 dt$ is the image of $\fL^+ \bA^1 dt$ under the
map $t^{-n} \cdot -:\fL \bA^1 dt \to \fL \bA^1 dt$.

We similarly define $\fL^{pol,\leq n} \bA^1 dt$ as
$\fL^{\leq n} \bA^1 dt/\fL^+ \bA^1 dt$. For $n > 0$, we define
$\Ker(\Res)^{\leq n}$ as $\Ker(\fL^{\leq n} \bA^1 dt \to \bA^1)$.

For $n>0$, define $\fL^{\leq n} \bG_m$ as the fiber product:

\[
\fL^{\leq n} \bG_m \underset{\fL \bA^1 dt}{\times} \fL^{\leq n} \bA^1 dt
\]

where we are using $d\log:\fL \bG_m \to \fL \bA^1 dt$.

For[We separate the cases to be derivedly correct.]
$n = 0$, define $\fL^{\leq 0} \bG_m$ as $\fL^+ \bG_m$.

Finally, define $\Gr_{\bG_m}^{\leq n}$ as $\fL^{\leq n}\bG_m/\fL^+\bG_m$.

$\fL^{\leq n} \bG_m$ is a formally smooth classical indscheme.

It suffices to show the same for $\Gr_{\bG_m}^{\leq n}$.

As in the proof of Proposition <ref>, the map
$\Gr_{\bG_m} \xar{d\log} \fL \bA^1 dt \to \Ker(\Res)$ induces an

\[
\Gr_{\bG_m}^{\leq n} \isom \bZ \times (\Ker(\Res)^{\leq n})_0^{\wedge}.
\]

Here $\Ker(\Res)^{\leq n}$
is defined as the kernel of the
residue on polar forms with poles of order $\leq n$.
As the latter is an affine space, its formal completion at the origin
is formally smooth and classical. This gives the claim.

Next, define $\LocSys_{\bG_m}^{\leq n}$ as the quotient
$\fL^{\leq n} \bA^1 dt/\fL^{\leq n} \bG_m$ under the gauge action.
We similarly define $\LocSys_{\bG_m,\log}^{\leq n}$ as
$\fL^{\leq n} \bA^1 dt/\fL^+ \bG_m$. By definition,
we have a Cartesian diagram:

The isomorphisms from Propositions <ref> and <ref>
induce further isomorphisms:

\[
\xymatrix{
\LocSys_{\bG_m,\log}^{\leq n} \ar[rr]^{\simeq}
\ar[d] & & \fL^{pol,\leq n} \bA^1 dt \times \bB \bG_m \ar[d] \\
\LocSys_{\bG_m,\log} \ar[rr]^{\simeq}
& & \fL^{pol} \bA^1 dt \times \bB \bG_m
\]

\[
\xymatrix{
\LocSys_{\bG_m}^{\leq n} \ar[rr]^{\simeq} \ar[d]
& &
\bA^1/\bZ \times \Ker(\Res)_{dR}^{\leq n} \times \bB \bG_m \ar[d] \\
\LocSys_{\bG_m} \ar[rr]^{\simeq}
& &
\bA^1/\bZ \times \Ker(\Res)_{dR} \times \bB \bG_m.
\]

This result is clear from our earlier analysis.

§.§ Definition of $\sY$

We define $\sY$, the moduli of rank $1$ de Rham local systems on $\o{\sD}$
with a flat section, as follows.

First, observe that we have a map $d:\fL \bA^1 \to \fL \bA^1 dt$
defined by the exterior derivative on $\o{\sD}$.
We also have a product map $\fL \bA^1 \times \fL \bA^1 dt \to \fL \bA^1 dt$
coming from the product $\bA^1 \times \bA^1 \to \bA^1$.

Let $\sY^{\prime}$ denote the equalizer:

\[
\sY^{\prime} \coloneqq \on{Eq}
\big(\fL \bA^1 \times \fL \bA^1 dt
\underset{(g,\omega) \mapsto g\omega}{\overset{d\circ \pi_1}{\rightrightarrows}}
\fL \bA^1 dt\big) \in \PreStk
\]

We emphasize that this is a derived equalizer;
although the terms appearing are classical indschemes, this
equalizer is a priori a DG indscheme.

There is a canonical $\fL \bG_m$-action on $\sY^{\prime}$,
heuristically given by the formula:

\[
(f \in \fL \bG_m,(g,\omega) \in \sY^{\prime}) \mapsto
\]

We will define this action more rigorously below,
but in the meantime, we define:

\[
\sY \coloneqq \sY^{\prime}/\fL \bG_m.
\]

Let us explain the above formulae.

Note that $\sY^{\prime}$ by definition parametrizes pairs
$(g \in \fL\bA^1, \omega \in \fL \bA^1 dt) $ with
$dg = g\omega$, which we can rewrite as
$\nabla g = 0$ for $\nabla \coloneqq d - \omega$.
This data amounts to a connection on the trivial bundle on $\o{\sD}$
(corresponding to $\omega$) and a flat section (corresponding to $g$).

Quotienting by $\fL \bG_m$ amounts to modding out by gauge transformation,
i.e., not fixing a trivialization on our line bundle on $\o{\sD}$.

Above, we did not define the action of $\fL \bG_m$ on $\sY^{\prime}$
completely rigorously: as $\sY^{\prime}$ is a priori DG, such
formulae are not sufficient. (In fact, using Theorem <ref>, $\sY^{\prime}$
is classical, and the implicit anxiety here is not needed.)

Here are two approaches.

First, in <cit.>,
Gaitsgory defines a prestack $\Maps(\o{\sD}_{dR},Y)$ for any
prestack $Y$. Taking $Y = \bA^1/\bG_m$ then immediately gives the definition
of $\sY$ as $\Maps(\o{\sD}_{dR},\bA^1/\bG_m)$. (One readily
finds $\Maps(\o{\sD}_{dR},\bB \bG_m) = \LocSys_{\bG_m}$ and
$\sY^{\prime} = \Maps(\o{\sD}_{dR},\bA^1/\bG_m)
\times_{\Maps(\o{\sD}_{dR},\bB \bG_m)} \Omega_{\bG_m}^1$, consistent
with our constructions.)

We prefer an alternative approach that we find more explicit, and that we
describe in full detail here. The reader who is not concerned about
homotopical details (which are not serious anyway by Theorem <ref>)
may skip this material.

Below, we make various constructions with $\fL \bA^1 \times \fL \bA^1 dt$. Here we may
just work with formulae as this indscheme is classical.

We consider two monoid structures on $\fL \bA^1 \times \fL \bA^1 dt$.
The first has product:

\[
(g,\omega) \dot (\tilde{g},\tilde{\omega}) \coloneqq
\]

while the second has product:

\[
(g,\omega) \star (\tilde{g},\tilde{\omega}) \coloneqq
\]

Moreover, the map:

\[
\begin{gathered}
\mu:\fL \bA^1 \times \fL \bA^1 dt \to \fL \bA^1 \times \fL \bA^1 dt \\
(g,\omega) \mapsto (g,dg-g\omega)
\end{gathered}
\]

is a map of monoids $(\fL \bA^1 \times \fL \bA^1 dt,\dot) \to
(\fL \bA^1 \times \fL \bA^1 dt,\star)$.

We obtain a commutative diagram of maps of monoids:

\[
\xymatrix{
\fL \bG_m \ar[rrr]^(.4){f \mapsto (f,d\log(f))}
\ar[d] &&& (\fL \bA^1 \times \fL \bA^1 dt,\dot) \ar[d]^{\mu} \\
\on{*} \ar[rrr] &&& (\fL \bA^1 \times \fL \bA^1 dt,\star)
\]

Therefore, we obtain a canonical map of monoids:

\[
\fL \bG_m \to (\fL \bA^1 \times \fL \bA^1 dt,\dot)
\underset{(\fL \bA^1 \times \fL \bA^1 dt,\star)}{\times}
\fL \bA^1
\]

where the previously unconsidered map is this
fiber product is $\fL \bA^1 \xar{(\id,0)} \fL \bA^1 \times \fL \bA^1 dt$; the source $\fL \bA^1$ is given its natural
product structure.

The right hand side above evidently
identifies with $\sY^{\prime}$, so we obtain a monoid
structure on $\sY^{\prime}$ and a map of monoids
$\fL \bG_m \to \sY^{\prime}$. In particular, we obtain
an action of $\fL \bG_m$ on $\sY^{\prime}$; this is
our desired action.

We now formulate the following result, whose proof will
be given in <ref>.

$\sY$ is a classical prestack.

In particular, $\sY$ is completely determined by its values
on usual commutative rings, i.e., not commutative DG rings.
(We formally deduce the same for $\sY^{\prime}$.)

§.§ Intermediate spaces

As for $\LocSys_{\bG_m}$, there are several variants of
$\sY$ that will be crucial to our

First, we define:

\[
\sY_{\log} = \sY \times_{\fL \bB\bG_m} \fL^+ \bB \bG_m =
\sY^{\prime}/\fL^+\bG_m.
\]

Note that we have a canonical action of $\Gr_{\bG_m}$ on
$\sY_{\log}$ with $\sY_{\log}/\Gr_{\bG_m} = \sY$.

By construction, $\sY_{\log}$ parametrizes the data
$(\sL,\nabla,s)$ where $\sL$ is a line bundle
on the disc $\sD$,
a connection $\nabla$ on $\sL|_{\o{\sD}}$, and
$s \in \Gamma(\o{\sD},\sL)$ a flat section.

We have a natural ind-closed $\sZ \subset \sY_{\log}$ parametrizing
similar data, but with $s \in \Gamma(\sD,\sL)$ flat as a
section on the punctured disc.

Formally, we define:

\[
\sZ \coloneqq \on{Eq}
\big(\fL^+ \bA^1 \times \fL \bA^1 dt
\underset{(g,\omega) \mapsto g\omega}{\overset{d\circ \pi_1}{\rightrightarrows}}
\fL \bA^1 dt\big)/\fL^+ \bG_m \in \PreStk.
\]

Here the $\fL^+ \bG_m$-action is constructed as in

The subspace $\sZ \subset \sY_{\log}$ plays a key role in our
main construction in <ref>.

In formulae, we have:

\[
\sZ = \Maps(\o{\sD}_{dR},\bA^1/\bG_m)
\underset{\Maps(\o{\sD},\bA^1/\bG_m)}{\times}
\Maps(\sD,\bA^1/\bG_m)
\]

for suitable meaning of $\o{\sD}_{dR}$
(cf. <cit.>).

A little informally, $\sZ$ is the moduli of $(\sL,\nabla,s)$ with $\sL$
a line bundle on $\sD$, $\nabla$ a connection on the punctured disc,
and $s \in \Gamma(\sD,\sL)$ with $\nabla(s) = 0 \in
\Gamma(\o{\sD},\sL \otimes \Omega^1)$.

Next, we define truncated versions of the above spaces.
Fix $n \geq 0$.

We then define:

\[
\sY_{\log}^{\leq n} \coloneqq \sY_{\log}
\underset{\LocSys_{\bG_m,\log}}{\times}
\LocSys_{\bG_m,\log}^{\leq n}.
\]

We define:

\[
\sZ^{\leq n} \coloneqq \on{Eq}
\big(\fL^+ \bA^1 \times \fL^{\leq n} \bA^1 dt
\underset{(g,\omega) \mapsto g\omega}{\overset{d\circ \pi_1}{\rightrightarrows}}
\fL^{\leq n} \bA^1 dt\big)/\fL^+ \bG_m
\]

similarly to $\sZ$; again, the construction
of <ref> applies and provides rigorous
meaning to the $\fL^+ \bG_m$-action in this formula.

The prestack $\sZ^{\leq n}$ is a (DG) algebraic[We refer to
<cit.> for the definition.] stack.

By definition, we have:

\[
\on{Eq}\big(\fL^+ \bA^1 \times \fL^{\leq n} \bA^1 dt
\underset{(g,\omega) \mapsto g\omega}{\overset{d\circ \pi_1}{\rightrightarrows}}
\fL^{\leq n} \bA^1 dt\big) =
\fL^{\leq n} \bA^1 dt \underset{\LocSys_{\bG_m,\log}^{\leq n}}{\times}
\sZ^{\leq n}.
\]

Therefore, $\sZ^{\leq n} \to \LocSys_{\bG_m,\log}^{\leq n}$ is
an affine morphism. As $\LocSys_{\bG_m,\log}^{\leq n}$ is an algebraic stack
by Proposition <ref>, we obtain the result.

We remind that $\bB \fL^+ \bG_m$ is not an algebraic stack
while $\bB \bG_m$ is: by definition, algebraic stacks are required
to admit fppf covers, not merely fpqc covers.

§.§ Flatness results

We now establish some technical results, showing that certain morphisms
are flat. Ultimately, these results are the technical backbone of
our study of $\sY$ and its coherent sheaves.

We begin with the following result.

The map:

\[
\begin{gathered}
\mu_n:\bA^n \times \bA^n = \bA^{2n} \to \bA^n \\
(a_0,\ldots,a_{n-1},b_0,\ldots,b_{n-1}) \mapsto
(a_0b_0,a_0b_1+a_1b_0,\ldots,\sum_{i = 0}^{n-1} a_ib_{n-1-i})
\end{gathered}
\]

is flat.

This map is defined by a family of homogeneous polynomials
(of degree $2$). Therefore,[As is standard: there exists
an open $U \subset \bA^n$ such that geometric fibers over
points in $U$ are equidimensional with
the expected dimension. By homogeneity, $U$
is closed under the $\bG_m$-action. As $U$ contains $0$,
it must be all of $\bA^n$. Then recall that a morphism
of smooth schemes is flat if and only if its geometric fibers are
equidimensional of the expected dimension.]
it suffices to show that the
fiber $Z \coloneqq \mu_n^{-1}(0)$
over $0$ is equidimensional with the expected
dimension $2n - n = n$.

First, for $0 \leq m \leq n$, let $Z_m \subset Z$ be the locally
closed subscheme where $a_0 = a_1 = \ldots = a_{m-1} = 0$
and $a_m \neq 0$, the last condition being considered as vacuous
for $m = n$. We claim that $\dim(Z_m) = n$ for all $m$. (We
will also see that $Z_m$ is smooth and connected, so we deduce
that $Z$ has the $n+1$ irreducible components $\overline{Z_m}$.)

Clearly $Z_n = \bA^n$, so
suppose $m \neq n$. Then:

\[
Z_m \subset (\bA^1 \setminus 0) \times \bA^{n-m-1} \times \bA^n
\]

is closed, where the coordinates on the latter affine
space are $a_m,a_{m+1},\ldots,a_{n-1},b_0,\ldots,b_{n-1}$;
the equations defining $Z_m$ here are:

\[
\sum_{i=0}^j a_{m+i} b_{j-i} = 0, \ldots j = 0,\ldots,n-m-1.
\]

In particular,
$b_j = -\frac{\sum_{i=1}^j a_{m+i} b_{j-i}}{a_0}$ for
$j = 0,\ldots,n-m-1$. It follows
that the morphism:

\[
\begin{gathered}
Z_m \subset
(\bA^1 \setminus 0) \times \bA^{n-m-1} \times  \bA^{m} \\
(a_m,\ldots,a_{n-1},b_0,\ldots,b_{n-1}) \mapsto
\end{gathered}
\]

is an isomorphism, giving the claim.

The above lemma is standard. See for example
<cit.> Theorem 2.2, especially the remarks
following its proof.

We now have the following variant.

Fix a linear map $T:\bA^{2n} \to \bA^n$.
Then $\mu_n+T:\bA^{2n} \to \bA^n$ is flat.

Introduce an auxiliary parameter $\lambda$ and
consider the map $\bA^{2n} \times \bA_{\lambda}^1 \to
\bA^n \times \bA_{\lambda}^1$ given by
$(\mu_n+\lambda T,\lambda)$. This map is defined by
homogeneous polynomials (all but one are degree $2$).
Its fiber over $0$ coincides with $\mu_n^{-1}(0)$, so this
map is flat.
Restricting to
$\bA^n \overset{x\mapsto (x,1)}{\subset} \bA^n \times \bA_{\lambda}^1$
gives the claim.

We now consider variants of the above in which
we pass to a limit.

Let $\bA^{\infty} \in \AffSch$ denote the affine scheme
$\Spec(\Sym(k^{\oplus \bZ^{\geq 0}}))$.

The map:

\[
\begin{gathered}
\mu_{\infty}:\bA^{\infty} \times \bA^{\infty} \to \bA^{\infty}
\\
\big((a_0,a_1,\ldots),(b_0,b_1,\ldots)\big) \mapsto
(a_0b_0,a_0b_1+a_1b_0,\ldots,\sum_{i = 0}^{j} a_ib_{j-i},\ldots)
\end{gathered}
\]

is flat.

More generally, suppose we are given linear maps
$T_n:\bA^{2n} \to \bA^n$ fitting into commutative diagrams:

\[
\xymatrix{
\bA^{2n+2} = \bA^{n+1} \times \bA^{n+1} \ar[d] \ar[rr]^(.65){T_{n+1}} & &
\bA^{n+1} \ar[d] \\
\bA^{2n} \ar[rr]^{T_n} & & \bA^n
\]

with vertical maps induced by the projection
$\bA^{n+1} \xar{(a_0,\ldots,a_n) \mapsto (a_0,\ldots,a_{n-1})} \bA^n$.
Let $T$ denote the induced map
$\bA^{\infty} \times \bA^{\infty} \to \bA^{\infty}$.

Then $\mu_{\infty}+T$ is flat.

As $\mu_{\infty}+T$ is obtained from the morphisms
$\mu_n+T_n$ by passing to the inverse limit in $n$, the
result follows from Lemma <ref>.

We also need a refinement of the above. The reader may skip this
material and return to it as needed.

Let $C = \Spec(k[x,y]/xy)$. Geometrically, $C$ is a the union of the
$x$ and $y$-axes in the plane. For a scheme $S$ and a morphism
$\vph:S \to C$, we say that $\vph$ is flat along the $x$-axis
if the derived fiber product $S \times_C \bA_x^1$ is a classical
scheme. I.e., for $S$ classical, this means no Tors are formed in
forming this fiber product.

We say $\vph$ is flat along the $y$-axis if
the parallel condition holds for the $y$-axis.
We say $\vph$ is flat along the axes if it
is flat along both the $x$ and $y$-axis.

Fix $n>0$ and let $Z = \mu_n^{-1}(0) \subset \bA^{2n}$ as in
the proof of Lemma <ref>.
is an evident map:

\[
Z \xar{(a_0,\ldots,a_{n-1},b_0,\ldots,b_{n-1}) \mapsto (a_0,b_0)} C
\subset \bA^2.
\]

The above map is flat along the axes.
Moreover, there is a canonical isomorphism:

\[
\mu_{n-1}^{-1}(0) \times \bA^1 \simeq Z \underset{C}{\times} \bA_x^1
\]

given by:

\[
\big((a_0,a_1,\ldots,a_{n-2},b_0,\ldots,b_{n-2}),\lambda\big) \in
\mu_{n-1}^{-1}(0) \times \bA^1 \mapsto
(a_0,a_1,\ldots,a_{n-2},\lambda,0,b_0,b_1,\ldots,b_{n-2}) \in Z.
\]

Recall that in the proof Lemma <ref> that we calculated
the irreducible components of $Z$. In <cit.>, the multiplicities
of these components were also calculated.[In the notation from
the proof of Lemma <ref>, the multiplicity of
$Z_m$ is $\binom{n}{m}$.]
This latter result follows from Proposition <ref>:
given $S \xar{(f,g)} C$ flat along axes, for an irreducible
component $S_m$ of $S$, $\on{mult}_S(S_m) =
\on{mult}_{\{f = 0\}}(S_m \cap \{f = 0\}) + \on{mult}_{\{g = 0\}}(S_m \cap \{g = 0\})$;
one can then calculate the multiplicities by induction on $n$.
We remark that this argument is quite similar to the one given in loc. cit.

We will deduce the above result from the following general lemma.

Suppose $T$ is a scheme equipped with a map $(f,g):T \to \bA^2$.
Let $S = T \times_{\bA^2} C = \{fg = 0\} \subset T$, where this fiber
product is understood as a derived fiber product.

If $g:T \to \bA^1$ is flat, then $S \to C$ is flat along the $x$-axis.

This is tautological:

\[
S \underset{C}{\times} \bA_x^1 =
T \underset{\bA^2}{\times} C \underset{C}{\times} \bA_x^1 =
T \underset{\bA^2}{\times} \bA_x^1 =
T \underset{\bA_y^1}{\times} 0.
\]

As $g$ is assumed flat,
the latter scheme is classical by definition.

The morphism:

\[
\begin{gathered}
\mu_n^{\prime}:\bA^n \times \bA^n = \bA^{2n} \to \bA^n \\
(a_0,\ldots,a_{n-1},b_0,\ldots,b_{n-1}) \mapsto
(b_0,a_0b_1+a_1b_0,\ldots,\sum_{i = 0}^{n-1} a_ib_{n-1-i})
\end{gathered}
\]

is flat.

We highlight the (only) difference between $\mu_n^{\prime}$ and
in the former, the first coordinate entry is $b_0$, not $a_0b_0$.

As in the proof of Lemma <ref>,
as each coordinate of $\mu_n^{\prime}$ is homogeneous, we
reduce to showing that $(\mu_n^{\prime})^{-1}(0)$ has the expected
dimension $n$. But at the classical level,
this fiber clearly identifies with
$\mu_{n-1}^{-1}(0) \times \bA_{b_{n-1}}^1$ (via the map in the statement
of Proposition <ref>),
which has dimension $n$ by Lemma <ref>.

Consider the map $\bA^{2n} \xar{\mu_n} \bA^n \to \bA^{n-1}$ where
the last map projects onto the last $n-1$-coordinates.
Let $T$ denote the inverse image of $0$ under this map.

By Lemma <ref>, the map $b_0:T \to \bA^1$ is flat.
Therefore, Lemma <ref> gives the flatness along
the $x$-axis. Flatness along the $y$-axis obviously
follows by symmetry.
The resulting description of the fiber product is

We now have the following generalizations.

\[
T:\bA^{2n} = \bA^{n} \times \bA^{n} \to 0 \times \bA^{n-1} \subset \bA^n
\]

is a linear map.

Let $Z^q = (\mu_n+T)^{-1}(0)$. Then the natural map
$Z^q \xar{(a_0,b_0)} C$ is flat along the axes.

Moreover, in the notation of Corollary <ref>, we may take
$n = \infty$ in this result.

For $1\leq n<\infty$, the map:

\[
\mu_n^{\prime}+T:\bA^{2n} \to \bA^n
\]

is flat by the argument for Lemma <ref>,
using Corollary <ref> instead of Lemma <ref>.
The proof of Proposition <ref> then applies to see that
$Z^q \to C$ is flat along the $x$-axis.

The proof of Corollary <ref> allows us to deduce the
$n = \infty$ case.

We now apply the above results to $\sZ$ and $\sY$.

For every $n \geq 0$, $\sZ^{\leq n}$ is a
classical algebraic stack.

In coordinates, the morphism:

\[
\mu:\fL^+ \bA^1 \times \fL^{\leq n} \bA^1 dt
\xar{(g,\omega) \mapsto g\omega - dg}
\fL^{\leq n} \bA^1 dt
\]

is given by:[As always, the meaning of this
formula is on $A$-points for any commutative ring $A$. That is,
the $a_i$ and $b_i$'s are regarded as elements of $A$.]

\[
\begin{gathered}
(g = \sum_{i=0}^{\infty} a_i t^i,
\omega = \sum_{i = 0}^{\infty} b_i t^{i-n} dt) \mapsto
g \omega - dg =
\sum_{i \geq 0}
\big(\sum_{j = 0}^i a_j b_{i-j}\big) t^{-n+i} dt
+ \sum_{i \geq 0} i b_i t^{i-1} dt.
\end{gathered}
\]

By Corollary <ref>, this is a flat morphism of affine schemes.
Therefore, the inverse image $\mu^{-1}(0)$ is a classical affine scheme.
As this inverse image is the universal $\fL^+ \bG_m$-torsor
over $\sZ^{\leq n}$, it follows that $\sZ^{\leq n}$ is classical as well.
We now obtain the result from Proposition <ref>.

Let $C = \Spec(k[x,y]/xy)$ as in <ref>.
Consider the $\bG_m$-action on $C$ of horizontal
homothety. I.e., for the corresponding grading on
$k[x,y]/xy$, $\deg(x) = 1$ and $\deg(y) = 0$.

Fix a coordinate $t$ on the formal disc.
We have a corresponding
map $\sZ^{\leq n} \to C/\bG_m$ defined as follows.

First, we we have a natural map
$\sZ \to \fL^+\bA^1/\bG_m \xar{\ev} \bA^1/\bG_m$.
This map takes $(\sL,\nabla,s) \in \sZ$ to $s|_0 \in \sL|_0$
for $0 \in \sD$ the base-point.

Next, we have a map:

\[
\sZ^{\leq n} \to \LocSys_{\bG_m,\log}^{\leq n}
\xar{\Pol} \fL^{pol,\leq n} \bA^1 dt
\simeq \prod_{i=1}^n \bA^1 \frac{dt}{t^i}
\to \bA^1 \frac{dt}{t^n} = \bA^1.
\]

This map records the leading (i.e.,
degree $-n$) coefficient of the connection.

The corresponding map
$\sZ^{\leq n} \to \bA^1 \times \bA^1/\bG_m$ evidently
maps into $C/\bG_m \subset \bA^1 \times \bA^1/\bG_m$.

The above is somewhat non-canonical as the second
map above depends
on the choice of coordinate $t$.
The more canonical statement would be
to replace the $\bA^1$ by the (scheme corresponding
to the) line $t^{-n} k[[t]]dt/t^{-n+1}k[[t]] dt$.

The following result plays a key technical role in our work.

Suppose $n>0$.

The map $\sZ^{\leq n} \to C/\bG_m$ is flat
along the axes.[By this, we mean that the
corresponding map $\sZ^{\leq n} \underset{\bB\bG_m}\times \Spec(k) \to C$ is flat along the axes in the
sense of <ref>.]

We have a canonical isomorphism:

\[
\sZ^{\leq n} \underset{C/\bG_m}{\times} \bA^1/\bG_m
\simeq \sZ^{\leq n-1}.
\]

Let $\iota:\sZ^{\leq n} \to \sZ^{\leq n}$
denote the map:

\[
(\sL,\nabla,s) \mapsto (\sL(1),\nabla,s)
\]

where $\sL(1)$ is the line bundle on the disc
whose sections on the disc are allowed
to have a pole of order $1$ at the base-point $0 \in \sD$.[Another way to say this:
$\Gr_{\bG_m}$ acts on $\sY_{\log}$, and the corresponding
action of $1 \in \bZ = \Gr_{\bG_m}(k)$ preserves
$\sZ^{\leq n}$ for all $n$; the induced map
$\sZ^{\leq n} \to \sZ^{\leq n}$ is our $\iota$.]

Then $\iota$ fits into a (derived) Cartesian diagram:

\[
\xymatrix{
\sZ^{\leq n} \ar[r]^{\iota} \ar[d] & \sZ^{\leq n} \ar[d] \\
\bA^1 \times \bB \bG_m \ar[r]^{(\id,0)} & C/\bG_m.
\]

For (<ref>), it suffices to check
flatness along axes after passing to the fpqc cover
$\sZ^{\leq n} \times_{\bB \fL^+ \bG_m} \Spec(k)$.
The corresponding map to $C/\bG_m$ evidently lifts
to $C$, and it suffices to check that the corresponding
map to $C$ is flat along axes. Using coordinates
as in the proof of
Proposition <ref>, we deduce the claim
from Corollary <ref>.

The other assertions are immediate from the constructions.
For instance, the diagram in
(<ref>) is obviously classically
Cartesian, so derived Cartesian by (<ref>).

The above morphism
$\iota:\sZ^{\leq n} \into \sZ^{\leq n}$ is an
almost finitely presented[See
<cit.> Definition 7.2.4.26
for the definition in the affine case.
In the following remark in loc. cit., it is shown
that this notion is preserved under base-change.
By <cit.> Proposition 4.1.4.3, this condition
can be checked flat locally. Therefore, there
is an evident notion of a representable
morphism of prestacks being locally almost of finite
presentation, and we are using the term in this

We emphasize that in this setting,
passing to an affine cover of $\sZ^{\leq n}$,
the corresponding map of classical affine
schemes is a finitely presented morphism in the sense
of classical algebraic geometry,
but being almost finitely presented is a
stronger notion (in spite of the terminology):
in the more classical reference <cit.>
Exposé III Définition 1.2, this property
of a morphism is called pseudo-coherence.]
closed embedding.

Almost finite presentation is preserved under
(derived) base-change, and any morphism of
schemes almost of finite presentation
is itself almost of finite presentation.
So the claim follows from
Proposition <ref> (<ref>).

$\sY_{\log}^{\leq n}$ is a classical ind-algebraic stack.
More precisely, the total space of the canonical $\bG_m$-
torsor on
$\sY_{\log}^{\leq n}$ is classical and a reasonable indscheme
in the sense of <cit.> 6.8.

We clearly have:

\begin{equation}\label{eq:y-log-colim-iota}
\colim \big(\sZ^{\leq n} \xar{\iota} \sZ^{\leq n} \xar{\iota } \ldots \big)
= \sY_{\log}^{\leq n}.
\end{equation}

Each of these morphisms is an
almost finitely presented closed
Therefore, $\sY^{\leq n} \times_{\bB \bG_m} \Spec(k)$
is a filtered colimit of the classical affine schemes
$\sZ^{\leq n} \times_{\bB \bG_m} \Spec(k)$ under
almost finitely presented closed

We now remind the general definition of reasonable
indscheme from <cit.>:
it is a (DG) indscheme that can be written
as a colimit of eventually coconnective
quasi-compact quasi-separated schemes
under closed embeddings almost of finite presentation.
So clearly this property is verified here.

One clearly obtains similar results for the Higgs analogue
of $\sY$. A posteriori, one sees that
$\Maps(\o{\sD},C)$ is a reasonable ind-affine indscheme.
For a weaker (classical) notion of reasonable indscheme, a similar
result with $C$ replaced by any finite type affine scheme
is well-known: see <cit.> Lemma 2.4.8. It is natural to ask: in what
generality is such a result true for the stronger
(derived) notion used here?

By Corollary <ref>,
$\sY^{\leq n} = \sY_{\log}^{\leq n}/\Gr_{\bG_m}^{\leq n}$
is a classical (non-algebraic) prestack.
We deduce the same for $\sY = \colim_n \sY^{\leq n}$.

We need one mild improvement of Proposition <ref>.
Roughly, the statement says that the use of the base-point $0 \in \widehat{\sD}$
is not essential in loc. cit.

First, we have a map:

\begin{equation}\label{eq:axis-1}
\widehat{\sD} \times \sZ \to \widehat{\sD} \times \fL^+\bA^1/\bG_m \to
\bA^1/\bG_m
\end{equation}

where the latter map is evaluation. Explicitly, this map
sends $(\tau,(\sL,\nabla,s)) \in \widehat{\sD} \times \sZ$ to
$s|_{\tau} \in \sL|_{\tau}$.

Now fix $n$. We let $\sD^{\leq n} \coloneqq \Spec(k[[t]]/t^n)$.

We have a second map:

\begin{equation}\label{eq:axis-2}
\sD^{\leq n} \times \sZ^{\leq n} \to \bA^1/\bG_m
\end{equation}

defined as follows. By definition, this map will
factor as:

\[
\sD^{\leq n} \times \sZ^{\leq n} \to
\sD^{\leq n} \times \LocSys_{\bG_m,\log}^{\leq n} \xar{\id \times \Pol}
\sD^{\leq n} \times \fL^{pol,\leq n} \bA^1 dt \to
\bA^1/\bG_m
\]

where the last map remains to be defined.
First, note that the dualizing complex
$\omega_{\sD^{\leq n}} \in \IndCoh(\sD^{\leq n})$ lies in degree $0$
and is a line bundle; it corresponds to the module $(k[[t]]/t^n)^{\vee} =
t^{-n} k[[t]]dt/k[[t]]dt$. Therefore, given $\tau \in \sD^{\leq n}$,
we obtain a line $\tau^*(\omega_{\sD^{\leq n}})$. Moreover,
a point $\omega \in \fL^{pol,\leq n} \bA^1 dt$ defines an evident section
of the above line bundle $\omega_{\sD^{\leq n}}$. Restricting to the
point $\tau$, we obtain the desired construction.

A choice of coordinate $t$ trivializes the above line bundle
on $\sD^{\leq n}$: the basis element is $\frac{dt}{t^n}$.
Such a trivialization lifts the map (<ref>) to
a map to $\bA^1$. This map is explicitly given at
$(\tau,(\sL,\nabla,s))$ by evaluating the function
$\Pol(\nabla) \cdot \frac{t^n}{dt}$ on $\sD^{\leq n}$
at $\tau$.

The map
$\sZ^{\leq n} \to \bA^1/\bG_m \times \bA^1/\bG_m$ factors
through $C/(\bG_m\times\bG_m)$. For $n>0$,
the resulting map
$\sZ^{\leq n} \to C/(\bG_m\times\bG_m)$ is flat along

For the first assertion, observe that
for $(\sL,\nabla,s) \in \sZ$,
$s\Pol(\nabla) = 0$ as a polar section
of the line bundle $\sL dt$ on the disc.
This implies the claim.

For flatness, note that we
have a canonical evaluation morphism:

\[
\widetilde{\ev}:
\sD^{\leq n} \times \fL_n^+ C \to C.
\]

It is immediate from
Proposition <ref> that this map
is flat along axes. Indeed, we need to show
that $\widetilde{\ev}^*(i_*(\sO_{\bA_x^1}))$
is concentrated in cohomological degree $0$
(where $i:\bA_x^1 \to C$ is the embedding).
It suffices to check this after further restriction
to $\fL_n^+ C$, where the assertion is
exactly loc. cit.

From here, the claim proceeds as in the proof of
Proposition <ref>.

§.§ Some notation

We now collect a bit of notation related
to $\sY$.

We begin by introducing notation for various structural maps.

For $n>0$, recall that we have the morphism:

\[
\iota:\sZ^{\leq n} \to \sZ^{\leq n}.
\]

For $r \geq 0$, we sometimes let $\iota^r$ denote the $r$-fold
composition of $\iota$.

We let:

\[
\begin{gathered}
\delta_n:\sZ^{\leq n-1} \to \sZ^{\leq n} \\
i_n:\sZ^{\leq n} \to \sY_{\log}^{\leq n} \\
\pi_n:\sZ^{\leq n} \to \sY^{\leq n} \\
\lambda_n:\sY_n \to \sY
\end{gathered}
\]

denote the canonical maps.

We also have the maps $\zeta$ and $\widetilde{\zeta}$ fitting into
a diagram:

\[
\xymatrix{
\LocSys_{\bG_m,\log}^{\leq n}
\ar[rr]^{\widetilde{\zeta}} \ar[d]^{\widetilde{\pi}_n}
&& \sZ^{\leq n}
\ar[d]^{\pi_n} \\
\LocSys_{\bG_m}^{\leq n} \ar[rr]^{\zeta} && \sY^{\leq n}.
\]

Here the horizontal arrows send
$(\sL,\nabla)$ to $(\sL,\nabla,0)$; i.e., we
take $0$ as the flat section of our local system.

We introduce the following line bundle on $\sZ^{\leq n}$.

Using our base-point $0 \in \sD$, we obtain a canonical line bundle
$\sO_{\sZ^{\leq n}}(-1)$ on $\sZ^{\leq n}$; its fiber at
$(\sL,\nabla,s)$ is the fiber $\sL|_0$ of $\sL$ at $0$.

For $r \in \bZ$, we let:

\[
\sO_{\sZ^{\leq n}}(r) \coloneqq \sO_{\sZ^{\leq n}}(-1)^{\otimes -r}.
\]

denote its (suitably normalized) tensor powers.
For $\sF \in \QCoh(\sZ^{\leq n})$, we let
$\sF(r)$ denote its tensor with $\sO_{\sZ^{\leq n}}(r)$.

We let $\widetilde{\sZ}^{\leq n}$ denote the total space of the $\bG_m$-bundle
defined by $\sO_{\sZ^{\leq n}}(-1)$, i.e.,
$\widetilde{\sZ}^{\leq n} \coloneqq \sZ^{\leq n} \times_{\bB \bG_m} \Spec(k)$.
By Proposition <ref> (or its proof), we remark that
$\widetilde{\sZ}^{\leq n}$ is an affine scheme, and by Theorem <ref>,
it is a classical affine scheme. We use notation $\iota$ and $\delta_n$
sometimes for the corresponding maps for $\widetilde{\sZ}^{\leq n}$.

Clearly the above line bundle extends canonically to $\sY_{\log}$ and
$\sY_{\log}^{\leq n}$; we use similar notation in those settings.

§.§ Fundamental exact sequences

We now record two exact sequences that we will use for some inductive arguments.
The results here amount to restatements of Proposition <ref>.
Suppose $n > 0$ in what follows.

We will presently construct a short exact sequence:

\begin{equation}\label{eq:fund-ses}
0 \to \delta_{n,*}\sO_{\sZ^{\leq n-1}}(1)
\to \sO_{\sZ^{\leq n}} \to \iota_*\sO_{\sZ^{\leq n}} \to 0
\end{equation}

in $\QCoh(\sZ^{\leq n})^{\heart}$.

The map $\sO_{\sZ^{\leq n}} \to \iota_*\sO_{\sZ^{\leq n}}$ is just the
canonical adjunction morphism.

Now observe that there is a canonical map:

\begin{equation}\label{eq:o(1)}
\sO_{\sZ^{\leq n}}(1) \to \sO_{\sZ^{\leq n}};
\end{equation}

its fiber at $(\sL,\nabla,s)$ is the map:

\[
\sL^{\vee}|_0 \to \sO|_0
\]

given by pairing with the section $s|_0 \in \sL|_0$.
We claim that this map factors uniquely as:

\[
\sO_{\sZ^{\leq n}}(1) \onto \delta_{n,*}\sO_{\sZ^{\leq n-1}}(1)
\dashrightarrow \sO_{\sZ^{\leq n}}
\]

and that the resulting map defines a short exact sequence
as in (<ref>).

In fact, this is implicit in work we have done already: these claims are
obtained by pullback from the short exact sequence:

\[
0 \to k[x,y]/(xy,x) \xar{x \cdot -} k[x,y]/xy \to k[x,y]/(xy,y) \to 0
\]

of bi-graded $k[x,y]/xy$ modules along the morphism
from Proposition <ref>. We emphasize that we are using the
flatness asserted in Proposition <ref> (<ref>).

For later use, we record the following observation. Fix
a coordinate $t$ on the disc and use the notation of <ref> below.

Then we also have a short exact sequence:

\begin{equation}\label{eq:fund-ses-2}
0 \to \iota_*\sO_{\sZ^{\leq n}} \to \sO_{\sZ^{\leq n}} \to
\delta_{n,*}\sO_{\sZ^{\leq n-1}} \to 0
\end{equation}

in $\QCoh(\sZ^{\leq n})^{\heart}$ in which the right
map is the canonical morphism. The left arrow is the unique map fitting
into a commutative diagram:

\[
\xymatrix{
\sO_{\sZ^{\leq n}} \ar[d] \ar[dr]^{b_{-n} \cdot -} \\
\iota_*\sO_{\sZ^{\leq n}}<EMAIL_ADDRESS>& \sO_{\sZ^{\leq n}}.
\]

Here $b_{-n}$ is as in <ref> below.
Again, the existence of the dotted arrow and the short exact sequence
follow from Proposition <ref>.

§.§ Nice opens

At some points, it is convenient to refer to the following geometric observations
about $\sZ^{\leq n}$. Roughly speaking, the idea is that
$\sZ^{\leq n}$ is “nice" away from $\sZ^{\leq n-1}$, which we sometimes use
for inductive statements.

We proceed separately in the cases where $n = 1$ and $n > 1$.

§.§.§ $n>1$ case

Define $\sU_n \coloneqq \sZ^{\leq n}$ as the open substack:

\[
\sU_n \coloneqq \sZ^{\leq n} \setminus \sZ^{\leq n-1}.
\]

The natural map:

\[
\LocSys_{\bG_m,\log}^{\leq n} \setminus \LocSys_{\bG_m,\log}^{\leq n-1}
\to
\sU_n
\]

(induced by $\widetilde{\zeta}$ from <ref>)
is an isomorphism.

By Proposition <ref>, we deduce that $\sU_n \simeq
(\bA^1 \setminus 0 ) \times \bA^{n-1} \times \bB \bG_m$. In particular,
$\sU_n$ is a smooth stack of finite type.

It is convenient to use the notation of <ref>, which is introduced
In that notation, we have $\sU_n = \Spec(A_n[b_{-n}^{-1}])/\bG_m$.

In the (classical) commutative ring $A_n[b_{-n}^{-1}]$, we have relations:

\[
\begin{gathered}
0 = b_{-n} a_0 \Rightarrow a_0 = 0 \\
0 = b_{-n} a_1 + b_{-n+1} a_0 = b_{-n} a_1 \Rightarrow a_1 = 0 \\
\ldots \Rightarrow a_i = 0 \text{ for all i}
\end{gathered}
\]

(using $n>1$).
It follows that we have an isomorphism:

\[
A_n[b_{-n}^{-1}] = k[b_{-1},\ldots,b_{-n},b_{-n}^{-1}].
\]

This amounts to the claim.

§.§.§ $n = 1$ case

This case is slightly more technical.

We define $\sU_1$ as the pro-(Zariski open substack of $\sZ^{\leq 1}$):

\[
\sU_1 \coloneqq \underset{r}{\lim} \, {\sZ^{\leq 1} \setminus
\big(\sZ^{\leq 0} \cup \ldots \cup \iota^r(\sZ^{\leq 0})\big)}.
\]

Here each of the structural maps in the limit is affine,
so the limit exists and is well-behaved.

The natural map:

\[
(\bA^1 \setminus \bZ^{\geq 0}) \times \bB \bG_m \to
\LocSys_{\bG_m,\log}^{\leq 1} \xar{\widetilde{\zeta}}
\sZ^{\leq 1}
\]

maps through $\sU_1$ and induces an isomorphism:

\[
(\bA^1 \setminus \bZ^{\geq 0}) \times \bB \bG_m \isom \sU_1.
\]

We again use the coordinates of <ref>.
In this notation, $\sU_1$ the quotient stack:

\[
\Spec(A_1[b_{-1}^{-1},(b_{-1}-1)^{-1},(b_{-1}-2)^{-1},\ldots])/\bG_m
\]

obtained by inverting the elements $\{b_{-1}-i\}$ for each $i \geq 0$.
In this ring, we have the relations:

\[
(b_{-1}-i)a_i = 0
\]

for each $i \geq 0$, which means that $a_i = 0$ in this localization
for each $i$. The claim then follows.

In both cases above, we observe that $\widetilde{\sU}_n \coloneqq
\widetilde{\sZ}^{\leq n} \times_{\sZ^{\leq n}} \sU_n$ is a regular Noetherian
affine scheme.

§.§.§ A remark

In spite of the above observations, we warn that the natural map:

\[
\zeta: \LocSys_{\bG_m} \setminus \LocSys_{\bG_m}^{\leq 0} \to
\sY|_{\LocSys_{\bG_m} \setminus \LocSys_{\bG_m}^{\leq 0}}
\]

is not an isomorphism. However, it is an isomorphism
if one truncates to irregularity of order $n \leq 1$,
or if one only evaluates on Noetherian
test rings (the left hand side is locally of finite type,
but the right hand side is not).

§.§ Coordinates

To make the above completely explicit, we provide explicit

Define a classical commutative ring $A_n$ by taking
(infinitely many) generators
$a_0,a_1,a_2,\ldots$ and $b_{-1},\ldots,b_{-n}$
and (infinitely) relations coming from equating Taylor series coefficients
of the formal Laurent series:

\[
\sum_{i=0}^{\infty} ia_it^{i-1} =
\sum_{i=0}^{\infty} a_it^i \cdot
\sum_{i=-n}^{-1} b_{-i}t^{-i}.
\]

We consider $A_n$ as graded with $\deg(a_i) = 1$
for all $i$ and $\deg(b_i) = 0$.
This grading on $A_n$ corresponds geometrically
to a $\bG_m$-action on $\Spec(A_n)$.

Define a map $\Spec(A_n) \to \sZ^{\leq n}$
by sending a point $(a_0,a_1,a_2,\ldots,b_{-1},\ldots,b_{-n})
\in \Spec(A_n)$ to the trivial line bundle
$\sO$ on the disc and equipping it with the connection
$\nabla = d-\sum_{i = 1}^n b_{-i} t^{-i} dt$ on the punctured
disc and the section $s = \sum_{i \geq 0} a_i t^i$,
which is annihilated by the connection by design.

This map actually factors through $\Spec(A_n)/\bG_m$; the
latter parametrizes the data of a line $\ell$ and
points as above, except that the $a_i$ are sections
of $\ell$. We then take our line bundle to be
$\sO \otimes \ell$, take $\nabla$ given by the same formula,
and similarly for our section $s$.

From Propositions <ref> and <ref>,
we obtain the next result.

The above map
$\Spec(A_n)/\bG_m \to \sZ^{\leq n}$ is an isomorphism.
I.e., we have a $\bG_m$-equivariant isomorphism
$\Spec(A_n) \simeq \widetilde{\sZ}^{\leq n}$.

§.§ Regular singular sketches

For the sake of explicitness, we draw pictures of $\sZ^{\leq 1}$,
$\sY_{\log}^{\leq 1}$, and $\sY^{\leq 1}$ using
the presentation of <ref>.

By the above, $A_1$ is the (classical) algebra with generators
$b_{-1},a_0,a_1,\ldots$ and relations:

\[
(b_{-1}-i)a_i = 0.
\]

For an integer $m \geq 0$, let $A_{1,\leq m}$ denote the
subalgebra generated by $b_{-1},a_0,\ldots,a_m$, and let
$\widetilde{\sZ}_{\leq m}^{\leq 1} \coloneqq \Spec(A_{1,\leq m})$.
We obtain the following picture for $\widetilde{\sZ}_{\leq m}^{\leq 1}$:

\[
\begin{tikzpicture}
\draw[gray, thick] (-3,0) -- (3,0);
\draw[gray, thick] (0,-1) -- (0,1);
\draw[gray, thick] (.5,-1) -- (.5,1);
\draw[gray, thick] (1,-1) -- (1,1);
\draw[gray, thick] (1.5,-1) -- (1.5,1);
\node at (-4, 0)   (a) {$\widetilde{\sZ}_{\leq 3}^{\leq 1} = $} ;
\end{tikzpicture}
\]

where the horizontal axis is the $b_{-1}$-axis, and the fiber
over $b_{-1} = i$ is given the coordinate $a_i$.

The structural maps $\widetilde{\sZ}_{\leq m+1}^{\leq 1} \to
\widetilde{\sZ}_{\leq m}^{\leq 1}$ coming from the
embedding $A_{1,\leq m} \into A_{1,\leq m+1}$ correspond to contracting
the rightmost line in the picture. Therefore, as:

\[
\widetilde{\sZ}^{\leq 1} = \underset{m}{\lim} \, \widetilde{\sZ}_{\leq m}^{\leq 1}
\]

we obtain the picture:

\[
\begin{tikzpicture}
\draw[gray, thick] (-3,0) -- (3,0);
\draw[gray, thick] (0,-1) -- (0,1);
\draw[gray, thick] (.5,-1) -- (.5,1);
\draw[gray!90!red, thick] (1,-1) -- (1,1);
\draw[gray!60!red, thick] (1.5,-1) -- (1.5,1);
\draw[gray!30!red, thick] (2,-1) -- (2,1);
\draw[gray!15!red, thick] (2.5,-1) -- (2.5,1);
\node at (-4, 0)   (a) {$\widetilde{\sZ}^{\leq 1} = $} ;
\node at (3.5, 0)   (b) {$\ldots$} ;
\end{tikzpicture}
\]

where the reddening indicates that interpret the
picture in the pro-sense rather than the ind-sense.

For $n > 1$, the stack $\sZ^{\leq n}$ is non-reduced, so there
is additional complexity in attempting to draw pictures.
(In addition, its Krull dimension grows with $n$.)

The map $\iota:\widetilde{\sZ}^{\leq 1} \to \widetilde{\sZ}^{\leq 1}$
corresponds to a rightward
shift in the above picture. Therefore,
if we let $\widetilde{\sY}_{\log}^{\leq 1}$ similarly be the total space
of the $\bG_m$-torsor over $\sY_{\log}^{\leq 1}$,
from (<ref>) we obtain the picture:

\[
\begin{tikzpicture}
\draw[gray, thick] (-3,0) -- (3,0);
\draw[gray, thick] (0,-1) -- (0,1);
\draw[gray, thick] (.5,-1) -- (.5,1);
\draw[gray!90!red, thick] (1,-1) -- (1,1);
\draw[gray!60!red, thick] (1.5,-1) -- (1.5,1);
\draw[gray!30!red, thick] (2,-1) -- (2,1);
\draw[gray!15!red, thick] (2.5,-1) -- (2.5,1);
\draw[gray, thick] (-.5,-1) -- (-.5,1);
\draw[gray!90!blue, thick] (-1,-1) -- (-1,1);
\draw[gray!60!blue, thick] (-1.5,-1) -- (-1.5,1);
\draw[gray!30!blue, thick] (-2,-1) -- (-2,1);
\draw[gray!15!blue, thick] (-2.5,-1) -- (-2.5,1);
\node at (-4.5, 0)   (a) {$\widetilde{\sY}_{\log}^{\leq 1} = $} ;
\node at (3.5, 0)   (b) {$\ldots$} ;
\node at (-3.5, 0)   (b) {$\ldots$} ;
\end{tikzpicture}
\]

where the bluing in the picture indicates that we interpret
the limit in the ind-sense (and the reddening is as before).

Informally, $\widetilde{\sY}_{\log}^{\leq 1}$ is a semi-infinite comb. It has
infinitely many bristles, which are attached to the handle at integer
points; in the positive direction, these bristles have
pro-nature, and in the negative direction they have ind-nature.[We
are unfortunately unable to produce non-synesthetic
pictures adequately distinguishing between projective and inductive limits.
If asked to better explain the above pictures,
we would resort to the formal descriptions
$\sZ^{\leq 1} = \lim_m \sZ_{\leq m}^{\leq 1}$ and $\sY_{\log}^{\leq 1} =
\colim_{\iota} \sZ^{\leq 1}$.]

In each of these pictures, $\bG_m$ acts by scaling in the vertical direction, i.e.,
scaling the bristles. So forming stacky quotients for this action,
we obtain the promised sketches of
$\sZ^{\leq 1}$ and $\sY_{\log}^{\leq 1}$.

Finally, the action of $\bZ = \Gr_{\bG_m}^{\leq 1}$ on $\sY_{\log}$ is generated
by rightward shift in the above pictures. Quotienting by this action gives a
picture[Roughly speaking, this picture looks like $\bA^1/\bZ$ with a
single bristle attached at $\bZ/\bZ$. But we must not forget the semi-infinite nature
of that bristle, which we find difficult to visually express in this setting.]
for $\sY^{\leq 1}$.

§ LOCAL ABEL-JACOBI MORPHISMS

In this section, we collect some standard results
about local Abel-Jacobi maps and (simplified)
Contou-Carrère pairings for use in <ref>.
This material is standard and included for the reader's

Let $\widehat{\sD} \in \IndSch$ denote the formal disc
$\Spf(k[[t]]) \coloneqq \colim_n \Spec(k[[t]]/t^n)$.

Let $\widehat{\Delta}:\widehat{\sD} \to \widehat{\sD} \times \sD$ denote the
graph of the canonical map $\widehat{\sD} \to \sD$.
Clearly this map is a closed embedding (in particular, schematic).

We define:

\[
\sO(-\widehat{\Delta}) \coloneqq \Ker\big(
\sO_{\widehat{\sD} \times \sD} \to \widehat{\Delta}_*(\sO_{\widehat{\sD}})\big)
\in \QCoh(\widehat{\sD} \times \sD).
\]

One readily checks
that $\sO(-\widehat{\Delta})$ is a line bundle on
$\widehat{\sD} \times \sD$: in fact,
a choice of coordinate on the disc
gives a trivialization of it.
Moreover, as:

\[
\widehat{\sD}
\underset{\widehat{\sD} \times \sD }{\times}
(\widehat{\sD} \times \o{\sD} ) = \emptyset
\]

there is a canonical trivialization
$\sO(-\widehat{\Delta})|_{\widehat{\sD} \times \o{\sD}}$.

Thus, we obtain a map:

\[
\AJ^{-1}:\widehat{\sD} \to \Gr_{\bG_m}.
\]

define the local Abel-Jacobi map
$\AJ$ as $\AJ^{-1}$ composed
with the inversion map $\Gr_{\bG_m} \to \Gr_{\bG_m}$.

Explicitly, we have:

\[
\begin{gathered}
\AJ: \widehat{\sD} \to \Gr_{\bG_m} \\
\tau \mapsto (\sO_{\sD}(\tau),1)
\end{gathered}
\]

where $\sO_{\sD}(\tau)$ is the line bundle on
the disc with sections having at worst simple
poles at $\tau \in \widehat{\sD} \subset \sD$
and the evident trivialization “$1$" on $\o{\sD}$.

We now consider a closely related map to the Abel-Jacobi map.

We have an evident evaluation map:

\begin{equation}\label{eq:ev-a1}
\widehat{\sD} \times \fL^+ \bA^1 \to \bA^1
\end{equation}

that is linear in the second coordinate.
Using the non-degeneracy of the residue pairing,
we deduce that there is a unique map:

\begin{equation}\label{eq:dlog-aj}
\widehat{\sD} \to
\fL^{pol} \bA^1 dt.
\end{equation}

such that the composition:

\[
\widehat{\sD} \times \fL^+ \bA^1
\xar{\eqref{eq:dlog-aj} \times \id}
\fL^{pol} \bA^1 dt \times
\fL^+ \bA^1 \xar{(f,\omega) \mapsto \Res(f\omega)}
\bA^1
\]

recovers the evaluation map above.

Explicitly, (<ref>) is given in coordinates

\[
(\tau \in \widehat{\sD}) \mapsto \sum_{i = 0}^{\infty}
\frac{\tau^i}{t^{i+1}} dt.
\]

We have the following elementary calculation.

The map (<ref>) coincides with the composition:

\[
\widehat{\sD} \xar{\AJ} \Gr_{\bG_m}
\xar{-d\log} \fL^{pol} \bA^1 dt.
\]

Choose coordinates as in Remark <ref>.
The map $\AJ:\widehat{\sD} \to \Gr_{\bG_m}$ lifts to a
map to $\fL\bG_m$ via $\tau \mapsto \frac{1}{t-\tau}$.
We then have:

\[
d\log \frac{1}{t-\tau} =
\frac{d\frac{1}{t-\tau}}{\frac{1}{t-\tau}} =
-\frac{dt}{t-\tau} = -\frac{1}{t}\sum_{i = 0}^{\infty} \frac{\tau^i}{t^i} dt
\]

Comparing to Remark <ref>, we obtain the

Following the lemma, we denote the above map by
$-d\log \AJ$, and its additive inverse by $d\log \AJ$.

§.§ Compatibility with truncations

We use the following observation. We remind that
$\sD^{\leq n} \coloneqq \Spec(k[[t]]/t^n)$.

Clearly $d\log \AJ$ maps $\sD^{\leq n}$ maps
into $\fL^{pol,\leq n} \bA^1 dt$.
Therefore, by definition (see <ref>),
we deduce that $\AJ$ maps $\sD^{\leq n}$ into
$\Gr_{\bG_m}^{\leq n}$.

§.§ The positive Grassmannian

Next, we define:

\[
\fL^{pos} \bG_m \coloneqq
\fL \bG_m \underset{\fL \bA^1} {\times} \fL^+ \bA^1.
\]

There is an evident commutative monoid
structure on $\fL^{pos} \bG_m$ and homomorphism
$\fL^+ \bG_m \to \fL^{pos} \bG_m$.

We then define:

\[
\Gr_{\bG_m}^{pos} \coloneqq \fL^{pos} \bG_m/\fL^+ \bG_m =
\Ker(\fL^+\bA^1/\bG_m \to \fL \bA^1/\bG_m).
\]

Explicitly, $\Gr_{\bG_m}^{pos}$ parametrizes
the data of $\sL$ a line bundle on the disc and
$\sigma \in \Gamma(\sD,\sL)$ a (regular) section
such that $\sigma|_{\o{\sD}}$ trivializes $\sL$.

For $n \geq 0$, we define:

\[
\Sym^n \widehat{\sD} \coloneqq \Spf(k[[t_1,\ldots,t_n]])^{S_n}
\in \IndSch.
\]

for $S_n$ the symmetric group.
We then let:

\[
\Sym\widehat{\sD} \coloneqq
\coprod_n \Sym^n \widehat{\sD} \in \IndSch.
\]

As for usual smooth curves, $\Sym\widehat{\sD}$ parametrizes
effective Cartier divisors on $\sD$ supported on $\widehat{\sD}$.
Hence, we obtain an isomorphism:

\[
\Sym\widehat{\sD} \isom \Gr_{\bG_m}^{pos}
\]

such that the composition:

\[
\widehat{\sD} = \Sym^1 \widehat{\sD} \to
\Sym\widehat{\sD} \simeq \Gr_{\bG_m}^{pos}
\subset \Gr_{\bG_m}
\]

is $\AJ$.

This is an isomorphism of commutative monoids for the
evident product on $\Sym(\widehat{\sD})$.

§.§ Log Contou-Carrére

We now observe the following.

There is a unique pairing:

\[
\pair{-}{-}:\Gr_{\bG_m}^{pos} \times \fL^+ \bA^1 \to \bA^1
\]

that is bilinear for multiplicative monoid
structures on both sides and whose restriction
to $\widehat{\sD} \times \fL^+ \bA^1$ is

Explicitly, for $D \in \Gr_{\bG_m}^{pos}$ an effective
Cartier divisor
on $\sD$ supported on $\widehat{\sD}$ and
$f\in \fL^+ \bA^1$, we have:

\[
\pair{D}{f} = f(D) \coloneqq \Nm_D(f).
\]

There are some variants of the pairing above.

First, note that
$\pair{-}{-}|_{\Gr_{\bG_m}^{pos} \times \fL^+ \bG_m}$
maps into $\bG_m$. As $\Gr_{\bG_m}$ is the group completion
of $\Gr_{\bG_m}^{pos}$ (in the evident sense), we obtain a

\[
\pair{-}{-}:\Gr_{\bG_m} \times \fL^+ \bG_m \to
\bG_m.
\]

This pairing is compatible with Contou-Carrére's
pairing <cit.> in the evident sense.

Next, let $\Sym^m \sD^{\leq n} \coloneqq
\Spec((k[[t]]/t^n)^{\otimes m,S_m})$, which is an
affine scheme. We let $\Gr_{\bG_m}^{pos,\leq n}\coloneqq
\coprod_m \Sym^m \sD^{\leq n}$.

The evident map:

\[
\Gr_{\bG_m}^{pos,\leq n} \to \coprod_m \Sym^m \widehat{\sD} \to \Gr_{\bG_m}
\]

maps into $\Gr_{\bG_m}^{\leq n}$. Indeed,
this map is a map of monoids, so the same is
true of its composition with $d\log:\Gr_{\bG_m} \to
\fL^{pol} \bA^1 dt$. As $\sD^{\leq n}$ maps into
$\fL^{pol,\leq n} \bA^1 dt$ (cf. <ref>),
we obtain the claim by definition of $\Gr_{\bG_m}^{\leq n}$.

For similar reasons, the pairing:

\[
\pair{-}{-}:
\Gr_{\bG_m}^{pos,\leq n} \times \fL^+ \bA^1 \to \bA^1
\]

factors through a pairing:

\[
\Gr_{\bG_m}^{pos,\leq n} \times \fL_n^+ \bA^1 \to \bA^1
\]

that we also denote by $\pair{-}{-}$.

The base-point $0 \in \widehat{\sD}$ gives a point of
$\Gr_{\bG_m}^{pos,\leq n}$ (of degree 1), hence
a translation map $T:\Gr_{\bG_m}^{pos,\leq n} \to \Gr_{\bG_m}^{pos,\leq n}$.
There is a natural map:

\[
\colim \, (\Gr_{\bG_m}^{pos,\leq n} \xar{T} \Gr_{\bG_m}^{pos,\leq n} \xar{T} \ldots)
\to \Gr_{\bG_m}^{\leq n}
\]

that we claim is an isomorphism.
Indeed, the assertion is evident at the level of $k$-points (both sides
are $\bZ$), so it suffices to check the assertion on tangent complexes,
where it is straightforward (both sides naturally identify with

We will use the following constructions (in a quite simple setting).

Let $V$ be a vector space. We consider $V$ as a prestack[If $Y$ is
coconnective, this prestack is a (DG) indscheme.]
with functor of points $\Spec(A) \mapsto \Omega^{\infty}(A\otimes V) \in \Gpd$.

Now suppose $X$ is an ind-proper indscheme equipped with a morphism
$\vph:X \to V$. We obtain a canonical “integration" map:

\begin{equation}\label{eq:integration}
\Gamma^{\IndCoh}(X,\omega_X) \to V.
\end{equation}

Indeed, our map $\vph$ is (tautologically) equivalent to
a morphism $\sO_X \to V\otimes \sO_X \in \QCoh(X)$. Tensoring with the dualizing
sheaf on $X$, we obtain a morphism $\omega_X \to V \otimes \omega_X \in \IndCoh(X)$,
or what is the same, a map $p^!(k) \to p^!(V)$ for $p:X \to \Spec(k)$ the projection.
By ind-properness and adjunction,
we obtain a map $p_*^{\IndCoh}p^!(k) \to V$ as desired.

Using <cit.> II.3 Lemma 3.3.7, one finds that this is a bijection;
i.e., specifying a map (<ref>) is equivalent to giving
$X \to V$.

We also have the following variant.
Suppose we are also given $Y$ a prestack and
$\pi:X \times Y \to \bA^1$ a map.

The map $\pi$ can be encoded by a morphism $X \to \Gamma(Y,\sO_Y)$ (thinking of the target
as a prestack as above). Therefore,
we obtain a comparison map:

\[
c_{\pi}:\Gamma^{\IndCoh}(X,\omega_X) \to \Gamma(Y,\sO_Y) \in
\Vect
\]

by the above. (As in Remark <ref>,
one can actually recover $\pi$ from $c_{\pi}$.)

We now have the following non-degeneracy assertion for our

Each of the morphisms:

\[
\begin{gathered}
\Gamma^{\IndCoh}(\Gr_{\bG_m}^{pos},\omega_{\Gr_{\bG_m}^{pos}})
\to \Gamma(\fL^+ \bA^1,\sO_{\fL^+ \bA^1}) \\
\Gamma^{\IndCoh}(\Gr_{\bG_m},\omega_{\Gr_{\bG_m}})
\to \Gamma(\fL^+ \bG_m,\sO_{\fL^+ \bG_m}) \\
\Gamma^{\IndCoh}(\Gr_{\bG_m}^{pos,\leq n},\omega_{\Gr_{\bG_m}^{pos,\leq n}})
\to \Gamma(\fL_n^+ \bA^1,\sO_{\fL_n^+ \bA^1}) \\
\Gamma^{\IndCoh}(\Gr_{\bG_m}^{\leq n},\omega_{\Gr_{\bG_m}^{\leq n}})
\to \Gamma(\fL_n^+ \bG_m,\sO_{\fL_n^+ \bA^1})
\end{gathered}
\]

coming from our pairings and <ref>
is an isomorphism.

Moreover, each isomorphism is a map of commutative algebras,
where the left hand sides are commutative algebras using
the commutative monoid structures on the relevant spaces,
while the right hand sides are commutative algebras
as functions on schemes.

Each of these maps is a map of commutative algebras
because of the bilinearity of the pairings.

Next, we have:

\[
\begin{gathered}
\Gamma^{\IndCoh}(\Gr_{\bG_m}^{pos,\leq n},\omega_{\Gr_{\bG_m}^{pos,\leq n}}) =
\underset{m}{\oplus} \,
\Gamma^{\IndCoh}(\Sym^m\sD^{\leq n},\omega_{\Sym^m\sD^{\leq n}})
= \\
\underset{m}{\oplus} \, ((k[[t]]/t^n)^{\otimes m,S_m})^{\vee}
= \Sym((k[[t]]/t^n)^{\vee})
\end{gathered}
\]

where the second equality is by finiteness of
$\Sym^m \sD^{\leq n}$.

We similarly have
$\fL_n^+ \bA^1 = \Spec(\Sym((k[[t]]/t^n)^{\vee}))$ by
definition. Under these identifications, it suffices to
show that the map:

\[
\Sym((k[[t]]/t^n)^{\vee}) =
\Gamma^{\IndCoh}(\Gr_{\bG_m}^{pos,\leq n},\omega_{\Gr_{\bG_m}^{pos,\leq n}})
\to \Gamma(\fL_n^+ \bA^1,\sO_{\fL_n^+ \bA^1}) =
\Sym((k[[t]]/t^n)^{\vee})
\]

from the proposition is the identity:
it suffices to check this on
the generators of the left hand side (as we have a map
of commutative algebras), and there it follows
by construction of the pairing.

Passing to the colimit as $n \to \infty$, we recover
the first map under consideration.

The second and fourth cases follow from Remark <ref>.

The assertion $\Gamma^{\IndCoh}(\Gr_{\bG_m},\omega_{\Gr_{\bG_m}})
\simeq \Gamma(\fL^+ \bG_m,\sO_{\fL^+ \bG_m})$ via this
pairing is standard from Contou-Carrère <cit.>.
The other assertions
are apparently less well-known,[For example,
they concern Cartier duality for commutative monoids,
which is less often considered than the group case.]
though they easy results.

§ COHERENT SHEAVES ON $\SY$

§.§ Overview

In this section, we define $\IndCoh^*$ for $\sY$ and its relatives.
This section follows <cit.> 6, though we keep our exposition largely
independent of loc. cit.

We strive to keep our exposition as explicit as possible, so assume
whatever simplifying assumptions we like. We refer to loc. cit. for
a more thorough development of the subject.

§.§ Affine case

Suppose $S$ is an eventually coconnective (e.g. classical) affine scheme.

We define the (non-cocomplete, DG) subcategory $\Coh(S) \subset \QCoh(S)$
to consist of objects $\sF \in \QCoh(S)^+$ such that for each $n$, the object
$\tau^{\geq -n}(\sF) \in \QCoh(S)^{\geq -n}$ is compact.
In other words, $\sF$ should be eventually coconnective and
$\ul{\Hom}_{\QCoh(S)}(\sF,-)$ should commute with
filtered colimits in $\QCoh(S)^{\geq -n}$ for each $n$.
Clearly $\Perf(S) \subset \Coh(S)$.

For $S = \Spec(A)$, we also sometimes write $\Coh(A) \subset A\mod$ for
the subcategory $\Coh(S) \subset \QCoh(S) \simeq A\mod$.

We define $\IndCoh^*(S)$ as $\Ind(\Coh(S))$.
Observe that we have a
natural functor $\Psi:\IndCoh^*(S) \to \QCoh(S)$.

Define $\IndCoh^*(S)^{\leq 0}$ as the subcategory generated
under colimits by coherent objects that are connective (with respect to the
$t$-structure on $\QCoh(S)$). There is a corresponding
$t$-structure on $\IndCoh^*(S)$ with this as its connective subcategory.

The above functor $\Psi$ is $t$-exact and induces an equivalence
$\Psi:\IndCoh^*(S)^+ \isom \QCoh(S)^+$.

§.§.§ Pullbacks

Let $f:S \to T$ be an eventually coconnective morphism of affine, eventually
coconnective schemes.

§.§.§ Pushforwards

Now suppose $f:S \to T$ is a morphism of affine, eventually coconnective schemes.
In this case, there is a unique continuous DG functor:

\[
f_*^{\IndCoh}:\IndCoh^*(S) \to \IndCoh^*(T)
\]

fitting into a commutative diagram:

\[
\xymatrix{
\IndCoh^*(S) \ar[rr]^{f_*^{\IndCoh}} \ar[d]^{\Psi} &&  \IndCoh^*(T)\ar[d]^{\Psi} \\
\QCoh(S) \ar[rr]^{f_*} && \QCoh(T)
\]

and such that $f_*^{\IndCoh}(\Coh(S)) \subset \QCoh(T)^+$.
(Indeed, this functor is necessarily the ind-extension of
$\Coh(S) \xar{f_*} \QCoh(T)^+ \simeq \IndCoh^*(T)^+ \subset \IndCoh^*(T)$.

In general, the functor $f_*^{\IndCoh}$ can[Unfortunately, <cit.>, which
is currently being revised, erroneously claims otherwise.]be pathological, i.e., it may fail to map $\IndCoh^*(S)^+$ into $\IndCoh^*(T)^+$.
This issue complicates the theory, compared to the finite type setting.

Suppose $f:S \to T$ as above is eventually coconnective (i.e., of finite
$\Tor$-amplitude). In this case,
$f^*:\QCoh(T) \to \QCoh(S)$ maps $\Coh(T)$ to $\Coh(S)$. By ind-extension,
we obtain a functor $f^{*,\IndCoh}:\IndCoh^*(T) \to \IndCoh^*(S)$.
It is immediate to see that it is left adjoint to $f_*^{\IndCoh}$.

Moreover, $f^{*,\IndCoh}$ is manifestly right $t$-exact. Therefore,
$f_*^{\IndCoh}$ is left $t$-exact in this case.

Suppose $f:S \to T$ is proper (e.g., an almost finitely presented
closed embedding).

Then $f_*:\Coh(S) \to \QCoh(T)^+$ maps into $\Coh(T)$. Therefore,
the functor $f_*^{\IndCoh}$ admits a continuous right adjoint,
that we denote $f^!$ in this case.

§.§.§ Semi-coherence

We now provide a general hypothesis that ensures that $f_*^{\IndCoh}$ is
left $t$-exact, even when $f$ is not eventually coconnective.

A connective and eventually coconnective
commutative algebra $A$ is semi-coherent
if every $M \in A\mod^+$ can be written as a filtered colimit
of objects $M_i$ such that:

* $M_i \in \Coh(A)$
for every $i$.

* There exists an integer $r$ independent of $i$
such that every $M_i$ lies in $A\mod^{\geq -r}$.

We say $S = \Spec(A)$ is semi-coherent if $A$ is.

In the definition, if one restricts to $M \in A\mod^{\heart}$,
one finds that such an $r$ may be constructed (if it exists)
independent of $M$ (by considering direct sums of modules).
It follows that it suffices to check the hypothesis just for $M \in A\mod^{\heart}$.

Any Noetherian (and eventually coconnective)
$A$ is obviously semi-coherent. More generally, any
coherent (and eventually coconnective) $A$ is semi-coherent.

We now have the following simple observation.

Suppose $S$ is a semi-coherent eventually connective affine scheme.
Suppose $\sC \in \DGCat_{cont}$ is a DG category with a $t$-structure
compatible with filtered colimits.
Suppose $F:\IndCoh^*(S) \to \sC$ is a continuous left $t$-exact DG functor
such that $F(\Coh(S)) \subset \sC^+$.
Then $F(\IndCoh^*(S)^+) \subset \sC^+$.

Suppose $\sF \in \IndCoh^*(S)^+$. By assumption and $S$, we can write
$\sF$ as a filtered colimit $\colim_i \sF_i$ with
$\sF_i \in \Coh(S) \cap \QCoh(S)^{\geq -r}$ for
some $r$; equivalently, $\sF_i \in \Coh(S) \cap \IndCoh^*(S)^{\geq -r}$.
Then $F(\sF) = \colim_i F(\sF_i) \in \sC^{\geq -r}$ by assumption, giving
the claim.

Suppose $f:S \to T$ is a morphism of eventually coconnective affine schemes
with $S$ semi-coherent. Then $f_*^{\IndCoh}$ is $t$-exact.

By Lemma <ref>, $f_*^{\IndCoh}$ is left $t$-exact.
Therefore, we have a commutative diagram:

\[
\xymatrix{
\IndCoh^*(S)^+ \ar[rr]^{f_*^{\IndCoh}} \ar[d]^{\Psi} &&  \IndCoh^*(T)^+\ar[d]^{\Psi} \\
\QCoh(S)^+ \ar[rr]^{f_*} && \QCoh(T)^+
\]

in which the left arrows are $t$-exact equivalences and the
bottom arrow is $t$-exact (by affineness). This gives the claim.

We have the following important technical result.

The (classical) commutative algebra $A_n$ defined in <ref> is

In other words, $\widetilde{\sZ}^{\leq n}$ is semi-coherent.

We defer the proof of the proposition to <ref>,
at the end of this section. Until that point, we assume it.

§.§ Coherent sheaves with support

We now consider the following situation.

Let $Z$ be an affine, eventually coconnective scheme.
Let $f:Z \to \bA^1$ be a function. Let $U = \{f \neq 0\} \subset Z$,
and let $j:U \to Z$ be the embedding.
Let $i:Z_0 \into Z$ be a closed subscheme with
$Z \setminus Z_0 = U$.

The following type of result is standard in the finite type setting.

Suppose that:

* $i$ is almost finitely presented.

* The map $i_*^{\IndCoh}:\IndCoh^*(Z_0) \to \IndCoh^*(Z)$ is $t$-exact.

Then the functor:

\[
(j^{*,\IndCoh},i^!):\IndCoh^*(Z) \to \IndCoh^*(U) \times \IndCoh^*(Z_0)
\]

is conservative.

Let $\sF \in \IndCoh^*(Z)$ be given. We suppose $\sF$ is a non-zero
object with $j^{*,\IndCoh}(\sF) = 0$. Our goal is to construct
an object $\sH \in \IndCoh^*(Z_0)$ and a non-zero map
$i_*^{\IndCoh}(\sH) \to \sF$.

We have:[Indeed, $j_*^{\IndCoh}$ is automatically left $t$-exact
(hence $t$-exact) because $j$ is flat; therefore, this formula follows from
the quasi-coherent setting by ind-extension.]

\[
j_*^{\IndCoh}j^{*,\IndCoh}(\sF) = \colim \big(\sF \xar{f} \sF \xar{f} \ldots \big) =
\]

Let $\sG \in \Coh(Z)$ be given with a non-zero map $\alpha:\sG \to \sF$; such
exists because $\sF$ is assumed non-zero.

By compactness of $\sG$, there is an integer $n$ such that the composition:

\[
\sG \xar{\alpha} \sF \xar{f^n} \sF
\]

is null-homotopic. This map coincides with the composition:

\[
\sG \xar{f^n} \sG \xar{\alpha} \sF
\]

so this map must be null-homotopic as well.
It follows that $\alpha$ factors through a (necessarily non-zero) map:

\[
\sG/f^n\coloneqq \Coker(f^n:\sG \to \sG) \to \sF
\]

for some $n \gg 0$.

We have a standard co/fiber sequence:

\[
\sG/{f^{n-1}} \xar{f} \sG/f^n \to \sG/f.
\]

By descending induction, we see that there must exist
a non-zero map from $\sG/f \to \sF$. Replacing $\sG$ with $\sG/f \in \Coh(Z)$,
we can assume that $f$ acts by zero on our original object $\sG$.

We now digress to make the following simple commutative algebra
observation.[In our eventual application, $Z_0$ is the classical
scheme underlying $\{f=0\}$. This step is unnecessary under that
additional hypothesis.]
Suppose $Z = \Spec(A)$ below.

Let $I \subset H^0(A)$ be the ideal of
elements vanishing on $Z_0^{cl}$. Let $(f) \subset H^0(A)$
be the ideal generated by $f$. We claim that there are integers $r,s$
such that:

\[
I^r \subset (f), (f^s) \subset I.
\]

For any $g \in I$, we have $\{g \neq 0\} \subset \{f \neq 0\}$ by assumption.
Therefore, $f$ is invertible in $H^0(A)[g^{-1}]$, so $g^{r_0} \in (f)$ for some
$r_0$ (depending on $g$).
Because $i$ is almost finitely presented, $I$ is finitely generated,
so we obtain $I^r \subset (f)$ for $r \gg 0$.

The other inclusion (which we do not need) follows similarly: $\{f \neq 0\}$ is covered
by $\{g \neq 0\}$ for $g \in I$, so $1 \in I[f^{-1}] = H^0(A)[f^{-1}]$,
so $f^s \in I$ for some $s$.

We deduce the following: any module:

\[
M \in H^0(A/f)\mod^{\heart} \subset H^0(A)\mod^{\heart} =
\]

admits a finite filtration:

\[
0 = I^r M \subset I^{r-1} M \subset \ldots \subset IM \subset M
\]

with subquotients lying in $H^0(A)/I\mod^{\heart} \subset A\mod^{\heart}$.

We now return to earlier setting.

Recall that we have $\sG \in \Coh(Z)$ on which $f$ acts null-homotopically.
Because $\sG$ is bounded, it has a finite Postnikov filtration as an object
of $\QCoh(Z)$
with associated graded terms $H^i(\sG)[-i]$.
Each $H^i(\sG) \in
\QCoh(Z)^{\heart}$. Because $f$ acts by zero on them, they lie
in $\QCoh(\{f = 0\})^{\heart}$. By the previous step, each
of these terms is obtained by successively extending
objects in $\QCoh(Z_0)^{\heart}$.

Now the diagram:

\[
\xymatrix{
\IndCoh(Z_0)^+ \ar[r]^{i_*^{\IndCoh}} \ar[d]^{\Psi} & \IndCoh(Z)^+ \ar[d]^{\Psi} \\
\QCoh(Z_0)^+ \ar[r]^{i_*} & \QCoh(Z)^+}
\]

commutes because we assumed $i_*^{\IndCoh}$ $t$-exact;
moreover, the vertical arrows are equivalences. Therefore, $\sG$
admits a finite filtration as an object of $\IndCoh^*(Z)$
with associated graded terms $\IndCoh$-pushed forward from $Z_0$.
Therefore, one of these associated graded terms must have a non-zero map
to $\sF$, giving the claim.

§.§.§ Application to $\widetilde{\sZ}^{\leq n}$

We now apply the above to construct generators of $\IndCoh^*(\widetilde{\sZ}^{\leq n})$.

$n > 0$, the DG category $\IndCoh^*(\widetilde{\sZ}^{\leq n})$ is compactly
generated by the objects
$\{\iota_*^{r,\IndCoh}(\sO_{\widetilde{\sZ}^{\leq n}})\}_{r \geq 0}$.

For $n = 0$, the DG category $\IndCoh^*(\widetilde{\sZ}^{\leq 0})$ is compactly
generated by $\sO_{\widetilde{\sZ}^{\leq 0}})$.

We proceed by induction on $n$.

The $n = 0$ case is simple: $\widetilde{\sZ}^{\leq 0}$ is $\bA^1$ (with coordinate

We now fix $n > 0$, and that the proposition for $n - 1$.
We introduce the following notation.

Let $\sC \subset \IndCoh^*(\widetilde{\sZ}^{\leq n})$ be the subcategory generated
by the objects $\iota_*^{r,\IndCoh}\sO_{\widetilde{\sZ}^{\leq n}}$. We need
to show that $\sC$ is the full $\IndCoh^*$.

We observe that (<ref>) implies that
$\delta_{n,*}^{\IndCoh}(\sO_{\sZ}^{\leq n-1}) \in \sC$.
More generally, we find
$\iota_*^{r,\IndCoh}\delta_{n,*}^{\IndCoh}(\sO_{\sZ}^{\leq n-1}) \in \sC$ for
any $r\geq 0$.

Below, we will apply Lemma <ref> with
$i:Z_0 \to Z$ corresponding to
$\delta_n:\widetilde{\sZ}^{\leq n-1} \to \widetilde{\sZ}$. We remark
that $\delta_n$ is almost finitely presented by
Lemma <ref> (<ref>)
(cf. the proof of Corollary <ref>), and that $\delta_{n,*}^{\IndCoh}$
is $t$-exact by Proposition <ref>.

In other words, our application of the lemma is justified.

We now treat the $n = 1$ case.

Suppose $\sF \in \IndCoh^*(\widetilde{\sZ}^{\leq 1})$ lies in the
right orthogonal to $\sC$.

By Step <ref> and the $n = 0$ case, $\delta_1^!(\sF) = 0$, and
$\delta_1^!(\iota^r)^! \sF = 0$ more generally. Therefore,
by Lemma <ref>, $\sF$ is $*$-extended from:

\[
\widetilde{\sZ}^{\leq 1} \setminus
\big(\widetilde{\sZ}^{\leq 0} \cup \ldots \cup \iota^r(\widetilde{\sZ}^{\leq 0})\big)
\]

for each $r$. It follows that $\sF$ is $*$-extended from:

\[
\widetilde{\sU}_1 \coloneqq
\widetilde{\sZ}^{\leq 1} \times_{\sZ^{\leq 1}} \sU_1.
\]

But by Lemma <ref> (cf. <ref>),
$\IndCoh^*(\widetilde{\sU}_1) \simeq \QCoh(\widetilde{\sU}_1)$
is generated by its structure sheaf. Since this is the $*$-restriction
of the structure sheaf on $\sZ^{\leq 1}$, we obtain the claim.

We now assume $n>1$ case.

In this case, we have:

\[
\iota_*^{r,\IndCoh}\delta_{n,*}^{\IndCoh} \simeq
\delta_{n,*}^{\IndCoh}
\iota_*^{r,\IndCoh}.
\]

(Here we use $\iota$ to denote the map for both $\sZ^{\leq n}$
and $\sZ^{\leq n-1}$.)

By induction and Step <ref>, $\sC$ contains the subcategory
generated under colimits by the essential image of
$\delta_{n,*}^{\IndCoh}$. Therefore, by Lemma <ref>,
any $\sF$ in the right orthogonal to $\sC$ is $*$-extended
from $\widetilde{\sU}_n$.

But again, $\IndCoh^*(\widetilde{\sU}_n) \simeq \QCoh(\widetilde{\sU}_n)$ is generated
by its structure sheaf by Lemma <ref>; as
$\sO_{\widetilde{\sZ}^{\leq n}} \in \sC$, we obtain the claim.

§.§ Stacky case

Now suppose $S$ is a stack of the form $T/G$ for $G$ classical affine group scheme
and $T$ an eventually coconnective affine scheme.

In this case, $G$ acts weakly on $\IndCoh^*(T)$, i.e., $\QCoh(G)$
acts canonically on $\IndCoh^*(T)$ (by flatness of $G$ over a point).
We set $\IndCoh^*(S) \coloneqq \IndCoh^*(T)^{G,w}$.

There is a unique $t$-structure on $\IndCoh^*(S)$ such that the
forgetful functor:

\[
\IndCoh^*(S) \to \IndCoh^*(T)
\]

is $t$-exact.

In the above setting, given a map $f:S_1 \to S_2$ of stacks of the
above types, we obtain similar functoriality as in the non-stacky case;
we omit the details.

We now have:

The category $\IndCoh^*(\sZ^{\leq n})$ is compactly generated
by the objects $\iota_*^{r,\IndCoh}(\sO_{\sZ^{\leq n}})(m)$, where $r \geq 0$ and
$m \in \bZ$.

For $G$ an affine algebraic group (in particular, of finite type) and
$\sF \in \sC^{G,w}$, it is a general fact that $\sF$ is compact
if and only if its image $\Oblv(\sF) \in \sC$ is compact.[See
<cit.>, proof of Lemma 5.20.2.]
Moreover, the functor $\Oblv:\sC^{G,w} \to \sC$ always generates
the target under colimits.

Therefore, writing $\sZ^{\leq n} = \widetilde{\sZ}^{\leq n}/\bG_m$,
the above objects are indeed compact. The generation follows from
Proposition <ref>.

There is a natural functor:

\[
\Psi:\IndCoh^*(\sZ^{\leq n}) \to \QCoh(\sZ^{\leq n})
\]

obtained by passing to weak $\bG_m$-invariants for
the corresponding functor for $\widetilde{\sZ}^{\leq n}$. This functor
is an equivalence on coconnective (equivalently: eventually coconnective)
subcategories via the corresponding assertion for $\widetilde{\sZ}^{\leq n}$.

We let $\Coh(\sZ^{\leq n}) \subset \IndCoh^*(\sZ^{\leq n})$
denote the subcategory of compact objects. By the above,
the functor $\Coh(\sZ^{\leq n}) \to \QCoh(\sZ^{\leq n})$
induced by $\Psi$ is fully faithful; its essential image is
exactly the subcategory of
eventually coconnective almost compact objects.

We now observe that the functor:

\[
\iota_*^{\IndCoh}:\IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sZ^{\leq n})
\]

is $t$-exact.

Indeed, by definition, this assertion reduces to the analogous
statement for $\widetilde{\sZ}^{\leq n}$, which in turn follows
from Proposition <ref>.

The same analysis applies for pushforward functors:

\[
\delta_{n+1,*}^{\IndCoh}:\IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sZ^{\leq n+1}).
\]

§.§ Ind-algebraic stacks

We define:

\[
\IndCoh^*(\sY_{\log}^{\leq n}) \coloneqq \colim \, \big(
\IndCoh^*(\sZ^{\leq n}) \xar{\iota_*^{\IndCoh}}
\IndCoh^*(\sZ^{\leq n}) \xar{\iota_*^{\IndCoh}} \ldots \big) \in \DGCat_{cont}.
\]

We let:

\[
i_{n,*}^{\IndCoh}: \IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sY_{\log}^{\leq n})
\]

denote the structural functor.

Similarly, we define:

\[
\IndCoh^*(\sY_{\log}) \coloneqq \underset{n}{\colim} \, \IndCoh^*(\sY_{\log}^{\leq n}) =
\underset{\iota_*^{\IndCoh},\delta_{n,*}^{\IndCoh}}{\underset{n,m}{\colim}} \, \IndCoh^*(\sZ^{\leq n}) \in \DGCat_{cont}.
\]

By the above and <cit.> Lemma 5.4.3, there is a unique
$t$-structure on $\IndCoh^*(\sY_{\log}^{\leq n})$ such that
each structural functor:

\[
\IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sY_{\log}^{\leq n})
\]

is $t$-exact. Similarly, there is a unique $t$-structure on
$\IndCoh^*(\sY_{\log})$ such that each structural functor:

\[
\IndCoh^*(\sY_{\log}^{\leq n}) \to \IndCoh^*(\sY_{\log})
\]

is $t$-exact.

Each of these categories is compactly generated as all of our structural
functors preserve compact objects.

§.§ Grassmannian actions

Suppose $n>0$ in what follows.

Recall that $\Gr_{\bG_m}$ tautologically acts on $\sY_{\log}$.

Observe that $\sZ \subset \sY_{\log}$ is preserved
under the action of the submonoid
$\Gr_{\bG_m}^{pos} \subset \Gr_{\bG_m}$.

Indeed, suppose more generally that $Y$ is a prestack
mapping to $\fL \bA^1/\bG_m$. Then:

\[
\Gr_{\bG_m}^{pos} = \Ker(\fL^+ \bA^1/\bG_m \to \fL \bA^1/\bG_m)
\]

acts canonically on:

\[
Z \coloneqq Y \underset{\fL \bA^1/\bG_m}{\times} \fL^+ \bA^1/\bG_m
\]

compatibly with the canonical $\Gr_{\bG_m} =
\Ker(\fL^+\bB \bG_m \to \fL \bB \bG_m)$-action on:

\[
Y_{\log} \coloneqq Y \underset{\fL\bB\bG_m}{\times} \fL^+ \bB \bG_m.
\]

Taking $Y = \sY$, we recover the claim.

Next, observe that the action of
$\Gr_{\bG_m}^{pos,\leq n} \subset \Gr_{\bG_m}^{\leq n}$
on $\sY_{\log}^{\leq n}$ preserves $\sZ^{\leq n}$.

Indeed, this is an immediate consequence of the above.

By the above, there is a canonical action:

\[
\QCoh(\Gr_{\bG_m}^{pos,\leq n}) \actson \QCoh(\sZ^{\leq n})
\]

where the left hand side is given its convolution monoidal

For $\sF \in \Coh(\Gr_{\bG_m}^{pos,\leq n}) \subset \QCoh(\Gr_{\bG_m}^{\leq n})$,
the corresponding action functor:

\[
\QCoh(\sZ^{\leq n}) \to \QCoh(\sZ^{\leq n})
\]

clearly preserves $\Coh(\sZ^{\leq n})$. Therefore, we obtain
a canonical action:

\[
\IndCoh(\Gr_{\bG_m}^{pos,\leq n}) \actson \IndCoh^*(\sZ^{\leq n}).
\]

By Remark <ref>, we have:

\[
\IndCoh(\Gr_{\bG_m}^{\leq n}) \underset{\IndCoh(\Gr_{\bG_m}^{pos,\leq n})}{\otimes}
\simeq \big(
\IndCoh^*(\sZ^{\leq n}) \xar{\iota_*^{\IndCoh}}
\IndCoh^*(\sZ^{\leq n}) \xar{\iota_*^{\IndCoh}} \ldots \big) \simeq
\IndCoh^*(\sY_{\log}^{\leq n}).
\]

Thus, we obtain a natural action of $\IndCoh(\Gr_{\bG_m}^{\leq n})$
on $\IndCoh^*(\sY_{\log}^{\leq n})$ such that the functor
$i_{n,*}^{\IndCoh}$ is a morphism of $\IndCoh(\Gr_{\bG_m}^{pos,\leq n})$-module

The map $\delta_{n,*}^{\IndCoh}:\IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sZ^{\leq n+1})$
is naturally a map of $\IndCoh(\Gr_{\bG_m}^{pos,\leq n})$-module categories,
as is evident by again considering subcategories of compact objects.

Therefore, $\IndCoh(\Gr_{\bG_m})$ naturally acts on $\IndCoh(\sY_{\log})$,
compatibly with the above actions.

§.§ Ind-coherent sheaves on $\sY$

We now define:

\[
\IndCoh^*(\sY) \coloneqq \IndCoh^*(\sY_{\log})^{\Gr_{\bG_m},w}.
\]

By definition, the right hand side is:

\[
\TwoHom_{\IndCoh(\Gr_{\bG_m})\mod}(\Vect, \IndCoh^*(\sY_{\log})).
\]

We let:

\[
\Oblv:\IndCoh^*(\sY) \to \IndCoh^*(\sY_{\log})
\]

denote the (conservative) forgetful functor.
It admits a left adjoint:

\[
\Av_!^{\Gr_{\bG_m},w}.
\]

The resulting monad on $\IndCoh^*(\sY_{\log})$ is easily seen to be
$t$-exact.[This reduces to the assertion that for
$\sF \in \IndCoh(\Gr_{\bG_m}^{\leq n,pos})^{\heart}$, the
action functor $\IndCoh^*(\sZ^{\leq n}) \to \IndCoh^*(\sZ^{\leq n})$
is $t$-exact. Filtering $\sF$, we are reduced to the case that
it is the skyscraper sheaf at a $k$-point. In that case, the
relevant functor is a composition of functors $\iota_*^{\IndCoh}$,
so $t$-exact by Proposition <ref>.]
Therefore, we obtain:

There is a unique $t$-structure on $\IndCoh^*(\sY)$ such that the
functor $\Oblv:\IndCoh^*(\sY) \to \IndCoh^*(\sY_{\log})$ is $t$-exact.
Moreover, the functor $\Av_!^{\Gr_{\bG_m},w}$ is $t$-exact.

Indeed, this follows from:

Let $\sC \in \DGCat_{cont}$ be equipped with a $t$-structure compatible with
filtered colimits and a right $t$-exact monad $T:\sC \to \sC$.
Then $T\mod(\sC)$ admits a unique $t$-structure such that the
forgetful functor $T\mod(\sC) \to \sC$ is $t$-exact.

In the truncated setting, we similarly define:

\[
\IndCoh^*(\sY^{\leq n}) \coloneqq
\IndCoh^*(\sY_{\log}^{\leq n})^{\Gr_{\bG_m}^{\leq n},w}.
\]

We again have adjoint functors:

\[
\Oblv,\Av_!^{\Gr_{\bG_m}^{\leq n},w}
\]

and a $t$-structure on $\IndCoh^*(\sY^{\leq n})$.

By functoriality of the constructions, there is a natural functor:

\[
\lambda_{n,*}^{\IndCoh}:\IndCoh^*(\sY^{\leq n}) \to \IndCoh^*(\sY).
\]

This functor is $t$-exact and preserves compact objects. Moreover, we have:

For $n>0$, the functor $\lambda_{n,*}^{\IndCoh}$ is fully faithful.

Roughly speaking, this is true because the map $\sY^{\leq n} \to \sY$ is
formally étale for $n>0$, which follows from the corresponding
fact for $\LocSys_{\bG_m}^{\leq n} \to \LocSys_{\bG_m}$ (which
follows from Proposition <ref>). Unwinding the constructions
to convert this argument into a proof is straightforward.

§.§ Proof of Proposition <ref>

We now give the proof of Proposition <ref>, which we deferred earlier.

Our argument is by induction on $n$. We proceed in steps.

§.§.§ Step 1: $n = 0$ case

The $n = 0$ case is trivial, as $A_0 = k[a_0]$ is Noetherian.

§.§.§ Step 2: $n = 1$ case

The inductive step we give for $n>1$ below may be adapted to treat the
$n = 1$ case, but it requires somewhat more work; we indicate how this works
in <ref>. We prefer
to give a direct argument. Actually, we will show that $A_1$ is coherent.

Recall the Noetherian subalgebras $A_{1,\leq m} \subset A_1$ from <ref>.
It suffices to show that for any finitely presented $M \in A_1\mod^{\heart}$,
there is an integer $m \gg 0$ and finitely generated
$N \in A_{1,\leq m}\mod^{\heart}$ such that
there is an isomorphism $N \otimes_{A_{1,\leq m}} A_1 \isom M$, where
the tensor product is understood in the derived sense.

Choose some $m_0$ and $N_0 \in A_{1,\leq m_0}\mod^{\heart}$ with an
isomorphism $H^0(N_0 \otimes_{A_{1,\leq m_0}} A_1) \isom M$; this may be
done by choosing generators and relations for $M$ and $m_0$ such that
the relations all involve linear combinations with coefficients in $A_{1,\leq m_0}$.

Choose $m > m_0$ such that $N_0$ does not contain any $(b_{-1}-i)$-torsion
for any integer $i\geq m$; this can be done because the finitely generated
$A_{1,\leq m_0}$-module $N_0$ only admits finitely many associated primes.
Then it is easy to see that $N \coloneqq
H^0(A_{1,\leq m} \otimes_{A_{1,\leq m_0}} N_0)$ satisfies the hypotheses.

We see here that $A_1$ is actually coherent, not merely semi-coherent.
We do not consider this question for general $n$.

§.§.§ Step 3: filtered colimits, extensions, and effective bounds

We now make the following observations for dévissage.

For an integer $r$, we say that $M \in A_n\mod$ is $r$-good if $M$ can be
expressed as a filtered colimit of coherent objects concentrated in
degrees $\geq -r$. Clearly any such $M$ lies in $A_n\mod^{\geq -r}$.

In other words, $M$ is $r$-good if it lies in the full subcategory:

\[
\Ind(\Coh(A) \cap A\mod^{\geq -r}) \subset A\mod^{\geq -r}.
\]

Here the natural functor is fully faithful because
$\Coh(A) \cap A\mod^{\geq -r}$ consists of objects that are
compact in $A\mod^{\geq -r}$ by assumption.

It follows from the second description that $r$-good objects are closed
under filtered colimits (for fixed $r$). We also observe that $r$-good objects
are closed under extensions.

Finally, we say an object $M \in A_n\mod$ is good if it is
$r$-good for some $r$. So our task is to show that any $M \in A_n\mod^+$
is good, or equivalently, any $M \in A_n\mod^{\heart}$.

§.§.§ Step 4: reductions

We now begin our induction. Take $n>1$ and assume the result for $n-1$.

Suppose $M \in A_n\mod^+$.
It it suffices to show that:

\[
M[b_{-n}^{-1}] \coloneqq \underset{b_{-n}}{\colim} \, M \text{ and }
\Coker(M \to M[b_{-n}^{-1}])
\]

are good (where $\Coker$ indicates the homotopy cokernel,
i.e., the cone).

We check these assertions below. We remark that by Remark <ref>,
we are reduced to considering $M \in A_n\mod^{\heart}$.

§.§.§ Step 5: generic case

Note that $M[b_{-n}^{-1}] \in A_n[b_{-n}^{-1}]\mod^{\heart} \subset A_n\mod^{\heart}$.

As $n>1$, we have an isomorphism:

\[
A_n[b_{-n}^{-1}] = k[b_{-1},\ldots,b_{-n},b_{-n}^{-1}].
\]

by Lemma <ref>.

As this algebra is regular Noetherian, it follows that
$M[b_{-n}^{-1}]$ has bounded $\Tor$-amplitude as an
$A_n[b_{-n}^{-1}]$-module (simply because it is bounded from below),
so the same is true of $M[b_{-n}^{-1}]$ as an $A_n$-module.
By standard homological algebra, $M[b_{-n}^{-1}]$ can be represented
by a bounded complex of flat (classical) $A_n$-modules.
By Lazard, a flat $A_n$-module is good, so $M[b_{-n}^{-1}]$ is good.

§.§.§ Step 6: torsion case

We now show that $\widetilde{M} = \Coker(M \to M[b_{-n}^{-1}])$ is good.
More generally, we show that any bounded complex
$\widetilde{M} \in A_n\mod^+$ such that $b_{-n}$ acts locally nilpotently on
its cohomologies is good. As the complex $\widetilde{M}$ is assumed bounded,
it suffices to treat its cohomology groups one at a time, so
we can assume (up to shifting) that $\widetilde{M} \in A_n\mod^{\heart}$.

By induction, there exists an integer $r$ such that any module
$N \in A_{n-1}\mod^{\heart}$ is $r$-good as an $A_{n-1}$-module.
Because the projection $A_n \xar{b_{-n} \mapsto 0} A_{n-1}$ is
almost finitely presented by Proposition <ref> (<ref>),
so coherent $A_{n-1}$-complexes restrict to coherent $A_n$-complexes,
$N$ restricts to an $r$-good $A_n$-module.

Now let $\widetilde{M}_i \subset \widetilde{M}$ be the
non-derived kernel of the map $b_{-n}^i:\widetilde{M} \to \widetilde{M}$.
Clearly $\widetilde{M}_i/\widetilde{M}_{i-1} \in A_{n-1}\mod^{\heart} \subset
A_n\mod^{\heart}$, so is $r$-good for the above $r$ (which is independent
of everything in sight except $n$). The module
$\widetilde{M}_i$ is then $r$-good, as it is obtained by successively extending
$r$-good modules. Finally, $\widetilde{M} = \colim_i \widetilde{M}_i$ is
a filtered colimit of $r$-good modules, so is $r$-good. This concludes the argument.

§.§.§ Step 7: revisiting the $n = 1$ case

Finally, we make a remark that the above argument can be adapted to
treat the $n = 1$ case, if one so desires. As in Lemma <ref>,
we should consider the localization of
$A_1$ at $\{b_{-1},b_{-1}-1,b_{-1}-2,\ldots\}$
(instead of simply at $b_{-1}$); one again then obtains a Noetherian ring
of finite global dimension. The argument concludes by noting that
any classical $A_1/(b_{-1}-i) = k[a_i]$-module is $0$-good, so
any module for $A_1$ that is torsion for the above elements (thus
necessarily the direct sum of its torsion with respect to each) is
$0$-good (by an argument as above).

In other words, we need to localize at infinitely many elements before
obtaining a regular ring; the saving
grace is that the goodness for torsion modules at each is bounded independently
of the element.

§ SPECTRAL REALIZATION OF WEYL ALGEBRAS

§.§ Overview

Let $i_n:\sZ^{\leq n} \into \sY_{\log}^{\leq n}$ denote the
canonical embedding, as in <ref>.

We have $i_{n,*}^{\IndCoh}(\sO_{\sZ^{\leq n}}) \in \Coh(\sY_{\log}^{\leq n})$,
which lies in $\IndCoh^*(\sY_{\log}^{\leq n})^{\heart}$ by
Proposition <ref>.

Let $\sF_n \coloneqq