$U\hookrightarrow\operatorname{Spec}(\pi_{0}A)_{\textup{cl}}$ of an affine
scheme. Let $(\operatorname{Spec}(\pi_{0}A_{f_{i}})_{\textup{cl}}\rightarrow
U)_{i\in I}$ be an affine open cover by basis elements. Certainly, we can lift
this open cover to an open subscheme $V\coloneqq\mathop{\rm Im}(\coprod_{i\in
I}\operatorname{Spec}(A[f_{i}^{-1}]))$, where the image is taken in
$\operatorname{Spec}(A)$. Let $B$ be a animated $A$-algebra with structure
morphism $w\colon\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$.
Then $w$ factors through an $u\colon\operatorname{Spec}(B)\rightarrow V$, i.e.
$u\in V(B)$, if and only if there is an étale cover $(B\rightarrow
B_{j})_{j\in J}$ such that for every $j$ there is an $i$ with
$\pi_{0}w_{j}(f_{i})$ invertible, where $w_{j}$ is the composition of $w$ with
the natural map $B\rightarrow B_{j}$.

To see this, assume we have a map $u\colon\operatorname{Spec}(B)\rightarrow V$
of derived $A$-schemes. Then base change with the affine open cover of $V$
gives an affine open $\coprod_{i\in I}\operatorname{Spec}(B_{i})$ cover of
$\operatorname{Spec}(B)$ that maps to $\coprod_{i\in
I}\operatorname{Spec}(A[f_{i}^{-1}])$ via projection (note that $V$ is an open
subscheme of an affine scheme and thus separated, so in particular the
diagonal of $V$ is affine). The projection $\coprod_{i\in
I}\operatorname{Spec}(B_{i})\rightarrow\coprod_{i\in
I}\operatorname{Spec}(A[f_{i}^{-1}])$ is induced by the termwise projections
(note that coproducts in $\infty$-topoi are universal). Thus, by the universal
property of localization, we see that $\pi_{0}w_{i}(f_{i})$ is invertible in
$\pi_{0}B_{i}$.

Now assume there is an étale cover
$(\operatorname{Spec}(B_{j})\rightarrow\operatorname{Spec}(B))_{j\in J}$ such
that for every $j$ there is an $i$ with
$\operatorname{Spec}(B_{j})\rightarrow\operatorname{Spec}(A[f_{i}^{-1}])$. In
particular, we get a map $\operatorname{Spec}(B_{j})\rightarrow\coprod_{i\in
I}\operatorname{Spec}(A[f_{i}^{-1}])$ and by taking coproducts and fiber
products, we get a map

$\operatorname{Spec}(B)\simeq\mathop{\textup{colim}}_{\Delta}(\check{C}(\coprod_{j\in
J}\operatorname{Spec}(B_{j})/B)_{\bullet})\rightarrow\coprod_{i\in
I}\operatorname{Spec}(A[f_{i}^{-1}])\rightarrow V.$

###  3.3. Quasi-coherent modules over derived stacks

In this section we will shortly look at quasi-coherent modules over derived
stacks. We show that they behave as “expected”. Namely, quasi-coherent modules
over derived stacks still satisfy descent181818We will make this explicit
later on, as we did not define Grothendieck topologies on derived stacks.. We
also have pullback and pushforward functors that are adjoint to another.
Further, we show that the $\infty$-category of quasi-coherent modules (in the
derived sense) over a (classical) scheme $X$ is equivalent to
${\mathcal{D}}_{\textup{qc}}(X)$.

We will closely follow [GR17, §I.3], [Lur04] and [Kha18] and generalize some
results following their ideas.

######  Definition 3.43.

Let $X$ be a presheaf on $\textup{AR}_{R}^{\operatorname{op}}$, we define the
$\infty$-category of quasi-coherent modules over $X$ to be

$\operatorname{QCoh}(X)\coloneqq\lim_{\operatorname{Spec}(A)\rightarrow
X}\operatorname{Mod}_{A}.$

An element ${\mathcal{F}}\in\operatorname{QCoh}(X)$ is called quasi-coherent
module over $X$ or ${\mathcal{O}}_{X}$-module. For any affine derived scheme
$\operatorname{Spec}(A)$ and any morphism
$f\colon\operatorname{Spec}(A)\rightarrow X$, we denote the image of a quasi-
coherent module ${\mathcal{F}}$ under the projection
$\operatorname{QCoh}(X)\rightarrow\operatorname{Mod}_{A}$, with
$f^{*}{\mathcal{F}}$.

We define the $\infty$-category of perfect quasi-coherent modules over $X$ to
be

$\operatorname{QCoh}_{\operatorname{perf}}(X)\coloneqq\lim_{\operatorname{Spec}(A)\rightarrow
X}\operatorname{Mod}^{\operatorname{perf}}_{A}.$

We say that a perfect quasi-coherent module ${\mathcal{F}}$ over $X$ has Tor-
amplitude in $[a,b]$ if for every derived affine scheme
$\operatorname{Spec}(A)$ and any morphism
$f\colon\operatorname{Spec}(A)\rightarrow X$ the $A$-module
$f^{*}{\mathcal{F}}$ has Tor-amplitude in $[a,b]$.

######  Remark 3.44.

We see that that by definition $\operatorname{QCoh}(-)$ (resp.
$\operatorname{QCoh}_{\operatorname{perf}}(-)$) is a right Kan extension of
$\operatorname{Mod}_{-}\colon\textup{AR}_{R}\rightarrow\textup{Cat}_{\infty}$
(resp.
$\operatorname{Mod}_{-}^{\operatorname{perf}}\colon\textup{AR}_{R}\rightarrow\textup{Cat}_{\infty}$)
onto ${\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}$
along the Yoneda emebdding.

A priori $\operatorname{Mod}_{-}$ is a functor from animated rings to the
$\infty$-category of not necessarily small $\infty$-categories. But for the
purpose of this article if we talk about the right Kan extension along the
Yoneda embedding to presheaves on $\textup{AR}_{\mathbb{Z}}$, we assume
smallness of the module categories.

######  Remark 3.45.

Note that limits preserve monomorphisms in $\textup{Cat}_{\infty}$ (as this
$\infty$-category has limits). Therefore, if
$F,G\colon\textup{AR}_{R}\rightarrow\textup{Cat}_{\infty}$ are functors and
$\alpha\colon F\rightarrow G$ is a natural transformation such that
$\alpha(A)$ is a monomorphism for all $A\in\textup{AR}_{R}$, we see that for
the induced morphism $R\alpha$ on the right Kan extensions $RF$ resp. $RG$ of
$F$ resp. $G$ under the Yoneda embedding
$\textup{AR}_{R}\hookrightarrow{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}$
the evaluation on some
$X\in{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}$
yields a monomorphism $R\alpha(X)\colon RF(X)\hookrightarrow RG(X)$.

This can be applied to see that for example for all
$\in{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}$,
we have that the natural morphism
$\operatorname{QCoh}_{\operatorname{perf}}(X)\rightarrow\operatorname{QCoh}(X)$
is fully faithful and we can see
$\operatorname{QCoh}_{\operatorname{perf}}(X)$ as a full subcategory of
$\operatorname{QCoh}(X)$.

######  Lemma 3.46.

Let ${\mathcal{C}}$ be the limit of stable $\infty$-categories
${\mathcal{C}}_{k}$ indexed by some simplicial set $K$ with finite limit
preserving transition maps, then ${\mathcal{C}}$ is stable.

###### Proof.

Since ${\mathcal{C}}_{k}$ have finite limits, we know that ${\mathcal{C}}$ has
finite limits. Then the spectrum of ${\mathcal{C}}$ is stable by [Lur17, Cor.
1.4.2.17], but by [Lur17, Rem. 1.4.2.25], we know that
$\operatorname{Sp}({\mathcal{C}})$ itself is a limit of the tower
$\dots\rightarrow{\mathcal{C}}_{*}\xrightarrow{\Omega}{\mathcal{C}}_{*}$. In
particular, we have

$\operatorname{Sp}({\mathcal{C}})\simeq\operatorname{Sp}(\lim_{K}{\mathcal{C}}_{\bullet})\simeq\lim_{K}\operatorname{Sp}({\mathcal{C}}_{\bullet})\simeq\lim_{K}{\mathcal{C}}_{\bullet}\simeq{\mathcal{C}},$

where we use [Lur17, Prop. 1.4.2.21] for the second to last equivalence. ∎

######  Remark 3.47.

By Lemma 3.46, we know that for any
$X\in{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})$ the $\infty$-category
$\operatorname{QCoh}(X)$ is stable, since it is the limit of stable
$\infty$-categories and the transition maps are given by base change (the base
change functor preserves fiber sequences, as they are equivalently cofiber
sequences, and finite products, that are equivalent to finite coproducts).

The following proposition is a generalization of [GR17, §I.3 Cor. 1.3.11] but
we can follow the idea of the proof.

######  Proposition 3.48.

Let ${\mathcal{C}}$ be a presentable $\infty$-category and let
$F\colon\textup{AR}_{R}\rightarrow{\mathcal{C}}$ be a (hypercomplete) sheaf
with respect to the Grothendieck topology $\tau\in\\{\textup{fpqc,
\'{e}tale}\\}$ on $\textup{AR}_{R}$. Let $RF$ denote the right Kan extension
of $F$ along the Yoneda embedding
$\textup{AR}_{R}\hookrightarrow{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}$.
Further let us denote the corresponding $\infty$-topos of (hypercomplete)
$\tau$-sheaves on $\textup{AR}_{R}$ with $\textup{Shv}_{\tau}$. Then for any
diagram $p\colon
K\rightarrow{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})$, where $K$ is
a simplicial set, and morphism $\mathop{\textup{colim}}_{K}X_{k}\rightarrow Y$
that becomes an equivalence in $\textup{Shv}_{\tau}$ after
sheafification191919Recall${}^{\ref{foot.sheafi}}$ that we can describe
$\textup{Shv}_{\tau}$ as a localization of
${\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})$ (as seen in the proof),
so we get a functor
$L\colon{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})\rightarrow\textup{Shv}_{\tau}$
left adjoint to the inclusion, which we call sheafification., we have that the
natural map $RF(Y)\rightarrow\lim_{K}RF(X_{k})$ is an equivalence.

###### Proof.

First, let us note that since ${\mathcal{C}}$ is presentable, we can find a
small subcategory ${\mathcal{C}}^{\prime}\subseteq{\mathcal{C}}$ such that
${\mathcal{C}}$ is a localization of ${\mathcal{P}}({\mathcal{C}}^{\prime})$
(see [Lur09, Thm. 5.5.1.1]). In particular, the elements $RF(Y)$ and
$\lim_{K}RF(X_{k})$ may be regarded as functors from ${\mathcal{C}}^{\prime}$
to $\SS$ and the natural morphism is an equivalence if and only if it is an
equivalence after composing with the evaluation for every
$c\in{\mathcal{C}}^{\prime}$ (see [Lur21, 01DK]). We note that the inclusion
of ${\mathcal{C}}$ into ${\mathcal{P}}({\mathcal{C}}^{\prime})$ preserves
limits and since the evaluation of a functor
$G\in{\mathcal{P}}({\mathcal{C}}^{\prime})$ at $c$ is equivalent to
$\operatorname{Hom}_{{\mathcal{P}}({\mathcal{C}}^{\prime})}(j(c),G)$, where
$j\colon{\mathcal{C}}^{\prime}\hookrightarrow{\mathcal{P}}({\mathcal{C}}^{\prime})$
denotes the Yoneda embedding (see [Lur09, Lem. 5.5.2.1]), we see that also the
evaluation preserves limits. So it is enough to check that for every
$c\in{\mathcal{C}}^{\prime}$, the morphism
$RF(Y)(c)\rightarrow\lim_{\Delta}(RF(X_{k})(c))$ is an equivalence. In
particular, we may replace $F$ by
$\operatorname{Hom}_{{\mathcal{P}}({\mathcal{C}}^{\prime})}(j(c),-)\circ F$
for any $c\in{\mathcal{C}}^{\prime}$ and so without loss of generality, we may
assume that ${\mathcal{C}}\simeq\SS$.

We will first discuss the case of $\tau$-sheaves. By definition of the
$\infty$-category $\textup{Shv}_{\tau}$, we know that all $\tau$-sheaves are
$S$-local, where $S$ is the collection of those monomorphism
$U\hookrightarrow\operatorname{Spec}(A)$, where $A\in\textup{AR}_{R}$, such
that it defines a $\tau$-covering sieve (see [Lur09, §6.2.2] for details).
This in particular defines a localization functor
$L\colon{\mathcal{P}}(\textup{AR}_{R})\rightarrow{\mathcal{P}}(\textup{AR}_{R})$
with essential image given by $\textup{Shv}_{\tau}$. Using [Lur09, Prop.
5.5.4.2], we see that any equivalence in $\textup{Shv}_{\tau}$ is local, i.e.
any morphism $f\colon U\rightarrow V$ in ${\mathcal{P}}(\textup{AR}_{R})$ such
that $Lf$ is an equivalence and any $Q\in\textup{Shv}_{\tau}$ we have that the
natural morphism

$\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(V,Q)\rightarrow\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(U,Q)$

is an equivalence. In particular, in our situation, we have that

$\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(Y,Q)\rightarrow\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(\mathop{\textup{colim}}_{K}X_{k},Q)\simeq\lim_{K}\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(X_{k},Q)$

is an equivalence (note that the colimit in the second $\operatorname{Hom}$ is
taken in the $\infty$-category ${\mathcal{P}}(\textup{AR}_{R})$, whereas for
$Y\simeq\mathop{\textup{colim}}_{K}X_{k}$ colimit is taken in
$\textup{Shv}_{\tau}$ which do not agree in general since the inclusion
$\textup{Shv}_{\tau}\hookrightarrow{\mathcal{P}}(\textup{AR}_{R})$ does not
preserve colimits in general).

Since $F$ is a sheaf with respect to the topology $\tau$, we therefore have

$\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(Y,F)\simeq\lim_{K}\operatorname{Hom}_{{\mathcal{P}}(\textup{AR}_{R})}(X_{k},F).$

Since we can write any presheaf on $\textup{AR}_{R}$ as a colimit of
representable ones (see [Lur09, Lem. 5.1.5.3]) and the Yoneda lemma [Lur09,
Lem. 5.5.2.1], we finally have the equivalence

$RF(Y)\xrightarrow{\sim}\lim_{K}RF(X_{k}).$

The case of hypercomplete $\tau$-sheaves is completely analogous, noting that
the $\infty$-topos of hypercomplete $\tau$-sheaves can be realized as a
localization of ${\mathcal{P}}(\textup{AR}_{R})$ with respect to hypercovers
(see [Lur09, Cor. 6.5.3.13]). ∎

######  Definition 3.49.

Let ${\mathcal{C}}$ be a presentable $\infty$-category and let $\tau$ be the
fpqc or étale topology on $\textup{AR}_{R}$. A functor
$F\colon{\mathcal{P}}(\textup{AR}_{R}^{\operatorname{op}})^{\operatorname{op}}\rightarrow{\mathcal{C}}$
is a (hypercomplete) sheaf or satisfies $\tau$-descent if for any effective
epimorphism $X\rightarrow Y$ (resp. a hypercover $X^{\bullet}\rightarrow Y$),
we have

$RF(Y)\simeq\lim_{\Delta}RF(\check{C}(X/Y)_{\bullet})\textup{ (resp.
}RF(Y)\simeq\lim_{\Delta_{s}}RF(X^{\bullet})).$

######  Remark 3.50.

In the setting of Proposition 3.48, we see that if $F$ is a (hypercomplete)
sheaf, then so is its right Kan extension $RF$.

######  Remark 3.51.

An important example of a presentable $\infty$-category is the
$\infty$-category $\textup{Cat}_{\infty}$ of small $\infty$-categories.
Presentability of this $\infty$-category follows from the fact that it is the
$\infty$-category of a combinatorial simplicial model category (marked
simplicial sets with the model structure of [Lur09, Prop. 3.1.5.2]), which are
precisely the presentable $\infty$-categories ( [Lur09, Prop. A.3.7.6]).

######  Proposition 3.52.

Let ${\mathcal{C}}$ be a presentable $\infty$-category and let
$F\colon\textup{AR}_{R}\rightarrow{\mathcal{C}}$ be an étale sheaf. Let $RF$
denote a right Kan extension of $F$ along the Yoneda embedding
$\textup{AR}_{R}\hookrightarrow{\mathcal{P}}(\textup{AR}_{R})^{\operatorname{op}}$.
Then for any derived scheme $X$ over $R$, the natural morphism

$RF(X)\rightarrow\lim_{\begin{subarray}{c}U\hookrightarrow X\\\ \textup{affine
open}\end{subarray}}F(U)$

is an equivalence.

###### Proof.

This lemma is a generalization of [Kha18, Lec. 1 Prop. 3.5] but can be proven
the same. For the convenience of the reader, we give a proof.

Let $X$ be derived scheme and $Y\coloneqq\coprod_{i\in
I}\operatorname{Spec}(A_{i})\rightarrow X$ be a Zariski atlas. By Remark 3.50
we have

$RF(X)\simeq\lim_{\Delta}RF(\check{C}(Y/X)_{\bullet}).$

For any affine open $U\hookrightarrow X$ let $Y_{U}\coloneqq\coprod_{i\in
I}U\times_{X}\operatorname{Spec}(A_{i})\rightarrow U$ be the induced cover on
$U$. Thus the question reduces to showing the equivalence

$RF(\check{C}(Y/X)_{n})\rightarrow\lim_{\begin{subarray}{c}U\hookrightarrow
X\\\ \textup{ affine open}\end{subarray}}RF(\check{C}(Y_{U}/U)_{n}),$

for all $[n]\in\Delta$. By cofinality, we may replace $X$ in the limit
argument by $\check{C}(Y/X)_{n}$ for any $n$.

To see this, note that for every affine open
$U\hookrightarrow\check{C}(X/Y)_{n}$, we get a morphism $U\rightarrow
U\times_{X}U\simeq\check{C}(Y/X)_{n}\times_{X}U\times_{\check{C}(Y/X)_{n}}U\simeq\check{C}(Y_{U}/U)_{n}\times_{\check{C}(Y/X)_{n}}U\rightarrow\check{C}(Y_{U}/U)$.
Thus

$\lim_{\begin{subarray}{c}U\hookrightarrow\check{C}(Y/X)_{n}\\\ \textup{affine
open}\end{subarray}}RF(U)\simeq\lim_{\begin{subarray}{c}U\hookrightarrow X\\\
\textup{affine open}\end{subarray}}RF(\check{C}(Y_{U}/U)_{n}).$

Assume the pairwise intersection of the $\operatorname{Spec}(A_{i})$ is
affines, then $X$ is affine and the question is trivial. Now assume the
pairwise intersection is not affine then it is open in an affine scheme and
thus separated. Thus they admit Zariski covers, where each of the pairwise
intersection is affine. Repeating the whole process concludes the proof. ∎

######  Remark 3.53.

Let us remark that the functors $A\mapsto\operatorname{Mod}_{A}$ and
$A\mapsto\operatorname{Mod}_{A}^{\textup{perf}}$ are hypercomplete sheaves for
the fpqc topology. The first assertion follows by [Lur18, Cor. D.6.3.3]. The
second assertion is clear since modules satisfy flat hyperdescent and since
perfect modules are precisely the dualizable ones (see [Lur17, Prop.
7.2.2.4]), we can construct a dual fpqc locally (see [Lur17, Prop. 4.6.1.11])
- see the proof [AG14, Lem. 5.4] for a more detailed explanation.

######  Remark 3.54.

Using the definition of the functors $\operatorname{QCoh}$ and
$\operatorname{QCoh}_{\operatorname{perf}}$, we see with Remark 3.53 and
Remark 3.50 that these functors satisfy descent in the sense that for any
effective epimorphism of derived stacks $X\twoheadrightarrow Y$, we have

$\operatorname{QCoh}(Y)\simeq\lim_{\Delta}\operatorname{QCoh}(\check{C}(X/Y)_{\bullet})\textup{
resp.
}\operatorname{QCoh}_{\operatorname{perf}}(Y)\simeq\lim_{\Delta}\operatorname{QCoh}_{\operatorname{perf}}(\check{C}(X/Y)_{\bullet}).$

######  Remark 3.55.

Using Remark 3.44 and Proposition 3.52, we see that a quasi-coherent module
over a derived scheme is given by a compatible family of modules
$({\mathcal{F}}_{A})_{A}$ for every affine open
$\operatorname{Spec}(A)\hookrightarrow X$.

######  Remark 3.56.

Let us recall the derived direct and inverse image. For a morphism of animated
rings $\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$, we get a
forgetful functor $\operatorname{Mod}_{B}\rightarrow\operatorname{Mod}_{A}$
(this follows from [Lur17, 4.6.2.17]). This functor is right adjoint to the
tensor product $B\otimes_{A}-$. We can globalize this to the case where we
replace the domain by an arbitrary derived stack $X$. Namely, any quasi-
coherent module ${\mathcal{F}}$ over $X$ is determined by its underlying
$C$-module $\iota^{*}{\mathcal{F}}$, for
$\iota\colon\operatorname{Spec}(C)\rightarrow X$. Since $C$ is naturally an
$A$-algebra, we can forget the $C$-structure and view ${\mathcal{F}}$ as a
limit in $\operatorname{Mod}_{A}$. The tensor product with each such $C$ also
induces a functor from $A$-modules to quasi-coherent $X$-modules. We can also
globalize this construction on the base for a geometric morphism of derived
stacks, i.e. if $f\colon X\rightarrow S$ is a geometric morphism of derived
stacks, we get an adjunction

${f^{\ast}\colon\operatorname{QCoh}(S)}$${\operatorname{QCoh}(X)\colon
f_{\ast},}$

note here that the right adjoint comes formally from the fact that the
pullback by construction commutes with colimits. If one adds assumptions to
$f$, then we can say more about the pushforward but we will not do this here
and refer to [Lur04, §5.5] since it is not of interest for us.

If we work with classical schemes, we will write $Lf^{*}$ and $Rf_{*}$ to
differentiate between the classical notions.

######  Proposition 3.57.

Let $X$ be a scheme. Then we have an equivalence of $\infty$-categories
${\mathcal{D}}_{\textup{qc}}(X)\simeq\operatorname{QCoh}(X)$, where
${\mathcal{D}}_{\textup{qc}}(X)$ denoted the derived $\infty$-category of
${\mathcal{O}}_{X}$-modules with quasi-coherent cohomologies.

###### Proof.

This is shown in the spectral setting in [Lur18] and can be followed in our
setting from Lurie’s PhD thesis [Lur04]. For convenience of the reader we will
show how to conclude this proposition as a consequence of both references.

A scheme $X$ is per definition a locally ringed space $(X,{\mathcal{O}}_{X})$.
We let $X^{\prime}\coloneqq\textup{Shv}_{\textup{(Sets)}}(X)$ denote the
Grothendieck topos associated to the small Zariski-site of $X$. The sheaf of
rings ${\mathcal{O}}_{X}$ can be viewed as a ring object of $X^{\prime}$. So
in particular, the tuple $(X^{\prime},{\mathcal{O}}_{X})$ defines a locally
ringed topos. We write ${\mathcal{X}}$ for the $1$-localic $\infty$-topos
assocaited to $X^{\prime}$ (which exists by [Lur09, Prop. 6.4.5.7]). As
explained in [Lur18, Rem. 1.4.1.5], we can view ${\mathcal{O}}_{X}$ as a sheaf
of connective $0$-truncated $E_{\infty}$-rings (which are just commutative
rings) on ${\mathcal{X}}$, which we denote by ${\mathcal{O}}_{{\mathcal{X}}}$
and hence get a spectrally ringed space
$({\mathcal{X}},{\mathcal{O}}_{{\mathcal{X}}})$, which is local (we refer to
[Lur18, §I.1.1] for the definitions).

Since ${\mathcal{O}}_{{\mathcal{X}}}$ takes values in commutative rings, we
can also view it as a sheaf on ${\mathcal{X}}$, with values in animated rings.
Therefore $({\mathcal{X}},{\mathcal{O}}_{{\mathcal{X}}})$ defines a spectral
scheme resp. a derived scheme in the sense of [Lur18, §I.1] resp. [Lur04].
Note that also the definition of an ${\mathcal{O}}_{{\mathcal{X}}}$-module
agrees in both references, i.e. in both references, we see an
${\mathcal{O}}_{{\mathcal{X}}}$-module as an
${\mathcal{O}}_{{\mathcal{X}}}$-module object in
$\textup{Shv}_{\operatorname{Sp}}({\mathcal{X}})$, where
${\mathcal{O}}_{{\mathcal{X}}}$ is naturally seen as a sheaf with values in
spectra.

By [Lur04, Thm. 4.6.5], we have an equivalence of $\operatorname{QCoh}(X)$ and
$\infty$-categories of sheaf ${\mathcal{O}}_{{\mathcal{X}}}$-modules $M$ on
${\mathcal{X}}$ such that

  1. (1)

$\pi_{i}M$ (which is defined as the sheafification of the presheaf
$V\mapsto\pi_{i}M(V)$) is a quasi-coherent sheaf on the underlying Deligne-
Mumford stack of $({\mathcal{X}},{\mathcal{O}}_{{\mathcal{X}}})$, and

  2. (2)

the underlying sheaf of spaces of $M$ is a hypersheaf (as explained above $M$
can be seen as a sheaf on ${\mathcal{X}}$ with values in $\operatorname{Sp}$,
i.e. a limit preserving functor
$M\colon{\mathcal{X}}^{\operatorname{op}}\rightarrow\operatorname{Sp}$, and
composing with $\Omega^{\infty}\colon\operatorname{Sp}\rightarrow\SS$ defines
the underlying sheaf of spaces of $M$).

Using [Lur18, Prop. 2.2.6.1], we see that $\operatorname{QCoh}(X)$ is
equivalent to the $\infty$-category of quasi-coherent sheaves on the spectral
Deligne-Mumford stack $({\mathcal{X}},{\mathcal{O}}_{{\mathcal{X}}})$. Now
[Lur18, Cor. 2.2.6.2] shows that indeed the derived $\infty$-category of
${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology is equivalent to
$\operatorname{QCoh}(X)$. ∎

###  3.4. Deformation theory of derived stacks

This section is derived from [TV08, §1.4],[AG14, §4.2], [Kha18, Lecture 5].

In this section, we will globalize the results of Section 2.3 and 2.4 to
geometric morphisms of derived stacks. For this, we will define a global
version of the cotangent complex and list properties. Most importantly, we
will show that any $n$-geometric morphism has a cotangent complex and that
smooth morphisms are characterized by the cotangent complex. Further, we will
use the results to show that geometric stacks are automatically hypercomplete
sheaves for the étale topology.

We let $R$ be a ring and assume every derived stack is a derived stack over
$R$.

Let $f\colon X\rightarrow Y$ be morphism of derived stacks. Let
$x\colon\operatorname{Spec}(A)\rightarrow X$ be an $A$-point, where $A$ is an
animated $R$-algebra. Let $M\in\operatorname{Mod}^{\textup{cn}}_{A}$ and let
us look at the commutative square

${X(A\oplus M)}$${X(A)}$${Y(A\oplus M)}$${Y(A),}$$\scriptstyle{f}$

where the vertical arrow are given by the canonical projection $A\oplus
M\rightarrow A$. We set the dervations at the point $x$ as

$\operatorname{Der}_{x}(X/Y,M)\coloneqq\operatorname{fib}_{x}(X(A\oplus
M)\rightarrow X(A)\times_{Y(A)}Y(A\oplus M)),$

where we see $x$ as a point in the target via the natural map induced by
$\operatorname{Spec}(A\oplus
M)\rightarrow\operatorname{Spec}(A)\xrightarrow{x}X\xrightarrow{f}Y.$

######  Definition 3.58 ([TV08, Def. 1.4.1.5]).

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks. We say
$L_{f,x}\in\operatorname{Mod}_{A}$ is a cotangent complex for $f$ at the point
$x\colon\operatorname{Spec}(A)\rightarrow X$, if it is $(-n)$-connective, for
some $n\geq 0$ and for all $M\in\operatorname{Mod}_{A}^{\textup{cn}}$ there is
a functorial equivalence

$\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{f,x},M)\simeq\operatorname{Der}_{x}(X/Y,M).$

When such $L_{f,x}$ exists, we say $f$ admits a cotangent complex at the point
$x$. If there is no possibility of confusion, we also write $L_{X/Y,x}$ for
$L_{f,x}$. We also write $L_{X}$ if $Y\simeq\operatorname{Spec}(R)$.

######  Definition 3.59.

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks. We say that
$L_{f}\in\operatorname{QCoh}(X)$ is a cotangent complex for $f$ if for all
points $x\colon\operatorname{Spec}(A)\rightarrow X$ the $A$-module
$x^{*}L_{f}$ is a cotangent complex for $f$ at the point $x$.

If $L_{f}$ exists, we say that $f$ admits a cotangent complex. We will write
$L_{f,x}$ instead of $x^{*}L_{f}$ if $f$ admits a cotangent complex.

If $Y\simeq\operatorname{Spec}(R)$ and $L_{f}$ exists, we say that $X$ admits
an absolute cotangent complex.

######  Remark 3.60.

By Lemma 2.37, the cotangent complex, for any morphism of derived stacks, is
unique up to homotopy.

######  Remark 3.61.

Note that any morphism of affine derived schemes
$f\colon\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$ admits a
cotangent complex. For any point
$x\colon\operatorname{Spec}(C)\rightarrow\operatorname{Spec}(B)$, we have
$L_{\operatorname{Spec}(B)/\operatorname{Spec}(A),x}\coloneqq
L_{B/A}\otimes_{B}C$.

######  Lemma 3.62.

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks.

  1. (1)

If $X$ and $Y$ admit absolute cotangent complexes, then $f$ admits a cotangent
complex and we have the following cofiber sequence for any point
$x\colon\operatorname{Spec}(A)\rightarrow X$

$L_{Y,f\circ x}\rightarrow L_{X,x}\rightarrow L_{f,x}.$

  2. (2)

If $f$ admits a cotangent complex, then for any morphism of derived stacks
$Z\rightarrow Y$ and any point $x\colon\operatorname{Spec}(A)\rightarrow
X\times_{Y}Z$, we have

$L_{f,x}\simeq L_{X\times_{Y}Z/Z,x}.$

  3. (3)

If for any morphism $x\colon\operatorname{Spec}(A)\rightarrow X$ the
projection $\operatorname{pr}\colon X\times_{Y,f\circ
x}\operatorname{Spec}(A)\rightarrow\operatorname{Spec}(A)$ admits a cotangent
complex, then $f$ admits a cotangent complex and further we have

$L_{f,x}\simeq L_{\operatorname{pr},x}.$

  4. (4)

If for any point $x\colon\operatorname{Spec}(A)\rightarrow X$ the stack
$X\times_{Y,f\circ x}\operatorname{Spec}(A)$ admits a cotangent complex, then
$f$ has a cotangent complex and we have

$L_{\operatorname{Spec}(A),\operatorname{id}_{\operatorname{Spec}(A)}}\rightarrow
L_{X\times_{Y}A,x}\rightarrow L_{f,x}.$

###### Proof.

The proof in the model categorical case is given in [TV08, Lem. 1.4.1.16]. But
these properties are straightforward to check.

Part 1 and 2 follow from the definitions. Part 3 follows from 2 and part 4
follows from 1, 3 and Remark 3.61 ∎

######  Lemma 3.63.

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a morphism of derived stacks.
Assume $Y/Z$ admits a cotangent complex, then $X/Y$ admits a cotangent complex
if and only if $X/Z$ admits a cotangent complex. Further, we obtain a cofiber
sequence

$f^{*}L_{Y/Z}\rightarrow L_{X/Z}\rightarrow L_{X/Y}$

of quasi-coherent modules over $X$ if the cotangent complexes exist.

###### Proof.

This is stated in [Kha18, Lec. 5 Prop. 5.7]. But anyway we will give a proof.

Let us take a point $x\colon\operatorname{Spec}(A)\rightarrow X$. Then what we
impose is that we have a cofiber sequence

$L_{Y/Z,f\circ x}\rightarrow L_{X/Z,x}\rightarrow L_{X/Y,x}.$

By Lemma 2.37 it is enough to show that for any connective $A$-module $M$ the
fiber of $\operatorname{Der}_{x}(X/Z,M)\rightarrow\operatorname{Der}_{f\circ
x}(Y/Z,M)$ at the trivial derivation (given by $A\oplus M\rightarrow
A\rightarrow X\rightarrow Y$) is given by $\operatorname{Der}_{x}(X/Y,M)$. But
this is clear. ∎

######  Lemma 3.64.

Let $j\colon X\hookrightarrow Y$ be a monomorphism of derived stacks over $A$,
then $j$ admits a cotangent complex and $L_{j}\simeq 0$.

###### Proof.

The proof is the same as in [Kha18, Lec. 5 Prop. 5.9], but for the convenience
of the reader, we recall the proof.

It suffices to show that at any point
$x\colon\operatorname{Spec}(B)\rightarrow X$ the cotangent complex is zero,
i.e. the space of derivations at $x$ is contractible. Since $j$ is a
monomorphism, i.e. its fibers are $(-1)$-truncated (so either contractible or
empty (see [Lur09, Def. 5.5.6.8])), we know that the canonical map

$X(B\oplus M)\rightarrow X(B)\times_{Y(B)}Y(B\oplus M)$

is also a monomorphism (note that any point in the fiber of the above morphism
defines a point in the fiber of $X(B\oplus M)\rightarrow Y(B\oplus M)$, which
per definition has either contractible or empty fibers). But the canonical map
$u\colon\operatorname{Spec}(B\oplus
M)\rightarrow\operatorname{Spec}(B)\rightarrow X$ defines a derivation, so the
space of derivations is nonempty and thus contractible. ∎

Now we can easily see that the homotopy groups of the localization with
respect to one element is given by the localization of the homotopy groups.

######  Lemma 3.65.

Let $A$ be an animated ring and
$f\in\pi_{0}A=\pi_{0}\operatorname{Hom}_{\operatorname{Mod}_{A}}(A,A)$. Then
we have $\pi_{i}(A[f^{-1}])\cong(\pi_{i}A)_{f}$ as $\pi_{0}A$-modules.

###### Proof.

From Proposition 2.13 and Lemma 2.19 it follows that the map
$\operatorname{Spec}(A[f^{-1}])\hookrightarrow\operatorname{Spec}(A)$ is
locally of finite presentation and a monomorphism. Since monomorphisms have a
vanishing relative cotangent complex (see Lemma 3.64), we conclude with
Proposition 2.56 that $A\rightarrow A[f^{-1}]$ is étale. Hence, we conclude
using the definition of étale morphisms. ∎

######  Lemma 3.66.

Let $(A\rightarrow A_{i})_{i\in I}$ be an étale covering in $\textup{AR}_{R}$
and let $M$ be an $A$-module. Then the family induced by base change $(A\oplus
M\rightarrow A_{i}\oplus(M\otimes_{A}A_{i}))_{i\in I}$ is an étale cover.

###### Proof.

We only need to show that $A_{i}\otimes_{A}(A\oplus M)\simeq
A_{i}\oplus(M\otimes_{A}A_{i})$, since étale covers are stable under base
change. But by construction the functors $A_{i}\otimes_{A}(-\oplus M)$ and
$(-\otimes_{A}A_{i})\oplus(M\otimes_{A}A_{i})$ from $\textup{AR-
Mod}^{\textup{cn}}_{R}$ to $\textup{AR}_{A_{i}}$ commute with sifted colimits
and thus we are reduced to classical commutative algebra, where it is clear. ∎

######  Lemma 3.67.

Let $f\colon X\rightarrow Y$ be a morphism of derived schemes. Then $f$ admits
a cotangent complex.

###### Proof.

A proof sketch is given in [Kha18, Lec. 5 Thm. 5.12], but since these are
lecture notes, we recall the proof.

We may assume that $Y=\operatorname{Spec}(R)$ using Lemma 3.63.

By Proposition 3.52, we know that

$\operatorname{QCoh}(X)\simeq\lim_{\begin{subarray}{c}\operatorname{Spec}(B)\hookrightarrow
X\\\ \textup{ open immersion }\end{subarray}}\operatorname{Mod}_{B}.$

For each of the open affines $\operatorname{Spec}(B)$ in $X$, we take
$L_{\operatorname{Spec}(B)}\coloneqq L_{B/\mathbb{Z}}$, viewed as a quasi-
coherent sheaf on $\operatorname{Spec}(B)$. Since the cotangent complex is
compatible with taking pullbacks, i.e. $L_{B}\otimes_{B}A\simeq L_{A}$ for a
triangle

${\operatorname{Spec}(A)}$${\operatorname{Spec}(B)}$${X,}$

where $\operatorname{Spec}(A),\operatorname{Spec}(B)$ are open in $X$, we see
that this indeed defines an object in the limit, which we denote with $L_{X}$
(for the compatibility, note that the relative cotangent complexes of
monomorphisms vanish by Lemma 3.64). This defines a cotangent complex on $X$.

We have to show that $L_{X}$ represents the space of derivations. Since we
have glued the cotangent complex for affine opens, we will use a descent
argument for arbitrary points. For this, we use that modules satisfy fpqc
descent and that for any point $x\colon\operatorname{Spec}(B)\rightarrow X$,
the base change with an affine open cover of $X$ gives an affine open cover
$(B_{i})$ of $B$. Hence, for a connective $B$-module $M$, it suffices to show
that

$\lim\operatorname{Hom}_{B_{i}}(L_{X,B_{i}},M_{i})\simeq\lim\operatorname{Der}_{B_{i}}(X/\mathbb{Z},M_{i}),$

where $M_{i}\coloneqq M\otimes_{B}B_{i}$, which is clear termwise, since each
$B_{i}$ factors through some affine open $A_{i}$ of $X$ by construction (to
write the derivations as a limit use the sheaf property of $X$,
$\operatorname{Spec}(\mathbb{Z})$ and Lemma 3.66). Note that for each affine
open the cotangent complex exists and we claim that $L_{X,B_{i}}\simeq
L_{X,A_{i}}\otimes_{A_{i}}B_{i}\simeq
L_{\operatorname{Spec}(A_{i}),A_{i}}\otimes_{A_{i}}B_{i}\simeq
L_{\operatorname{Spec}(A_{i}),B_{i}}$, which concludes the lemma.

To see this, note that $\operatorname{Der}_{B_{i}}(A_{i}/X,M_{i})\simeq 0$,
since $\operatorname{Spec}(A_{i})\hookrightarrow X$ is a monomorphism (see
Lemma 3.64). Therefore
$\operatorname{Der}_{B_{i}}(A_{i}/\mathbb{Z},M_{i})\simeq\operatorname{Der}_{B_{i}}(X/\mathbb{Z},M_{i})$
(since its fiber is $\operatorname{Der}_{B_{i}}(A_{i}/X,M_{i})$), the same
holds if we replace the point by $A_{i}$. Since the cotangent complexes at
$A_{i}$ exist, we get $L_{\operatorname{Spec}(A_{i}),A_{i}}\simeq L_{X,A_{i}}$
and after tensoring with $B_{i}$, we see that $L_{X,B_{i}}$ is a cotangent
complex for $X/\mathbb{Z}$ at $B_{i}$ if and only if
$\operatorname{Der}_{B_{i}}(A_{i}/\mathbb{Z},M_{i})\simeq\operatorname{Der}_{B_{i}}(X/\mathbb{Z},M_{i})$
but this we have seen above, i.e. we have equivalences

$\displaystyle\operatorname{Hom}_{B_{i}}(L_{X,B_{i}},M_{i})\simeq\operatorname{Hom}_{B_{i}}(L_{\operatorname{Spec}(A_{i}),B_{i}},M_{i})\simeq\operatorname{Der}_{B_{i}}(A_{i}/\mathbb{Z},M_{i})\simeq\operatorname{Der}_{B_{i}}(X/\mathbb{Z},M_{i}).$

We remark that by this construction and commutativity of $\tau_{\geq 0}$ with
limits, we have that $L_{X}$ is connective. In particular, we have shown that
$L_{X}$ is a cotangent complex for $X/\mathbb{Z}$. ∎

######  Remark 3.68.

Let us give another construction of a cotangent complex. Consider the functor
$L_{-}\colon\textup{AR}_{\mathbb{Z}}\rightarrow{\mathcal{D}}(R)$ given by the
usual cotangent complex seen as complex of abelian groups. We denote its right
Kan extension along the inclusion
$\textup{AR}_{R}\hookrightarrow\operatorname{dSch}^{\operatorname{op}}_{/\operatorname{Spec}(R)}$
with $\mathbb{R}L_{-}$. By the above proof, we see that for a derived scheme
$X$, we have $L_{X}\simeq\mathbb{R}L_{X}$ in ${\mathcal{D}}(R)$. In
particular, by stability of the derived $\infty$-category, the same holds for
the relative cotangent complex.

######  Definition 3.69.

We recall the notion of an obstruction theory for derived stacks respectively
morphism of derived stacks (see [TV08, 1.4.2.1, 1.4.2.2]).

  1. (i)

A derived stack $X$ is called infinitesimally cartesian or inf-cartesian if
and only if for every animated $R$-algebra $A$, connective $A$-module $M$ with
$\pi_{0}M=0$ and derivation $d\in\operatorname{Der}(A,M)$ the pullback square

${A\oplus_{d}M}$${A}$${A}$${A\oplus M,}$$\scriptstyle{d}$$\scriptstyle{s}$

where $s$ denotes the trivial derivation, induces a pullback square

${X(A\oplus_{d}M)}$${X(A)}$${X(A)}$${X(A\oplus
M).}$$\scriptstyle{d}$$\scriptstyle{s}$

A morphism $f\colon X\rightarrow Y$ of derived stacks is called
infinitesimally cartesian or inf-cartesian if and only if for every animated
$R$-algebra $A$, connective $A$-module $M$ with $\pi_{0}M=0$ and derivation
$d\in\operatorname{Der}(A,M)$ we have a pullback square

${X(A\oplus_{d}M)}$${Y(A\oplus_{d}M)}$${X(A)\times_{X(A\oplus
M)}X(A)}$${Y(A)\times_{Y(A\oplus M)}Y(A).}$

  2. (ii)

A derived stack $X$ has an obstruction theory if and only if it has a
cotangent complex and is infinitesimally cartesian.

A morphism of derived stacks $f\colon X\rightarrow Y$ has an obstruction
theory if and only if it has a cotangent complex and is infinitesimally
cartesian.

######  Definition 3.70 ([TV08, Def. 1.2.8.1]).

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks, we say $f$ is
formally smooth if for any $A\in\textup{AR}_{R}$, a connective $A$-module $M$
with $\pi_{0}M=0$, and derivation $d\in\operatorname{Der}_{R}(A,M)$ the
natural map

$\pi_{0}X(A\oplus_{d}M)\rightarrow\pi_{0}(X(A)\times_{Y(A)}Y(A\oplus_{d}M))$

is surjective.

######  Lemma 3.71.

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks.

  1. (1)

If $X$ and $Y$ have an obstruction theory, then $f$ has an obstruction theory.

  2. (2)

If $f$ has an obstruction theory, then for any morphism of derived stacks
$Z\rightarrow Y$ the base change $Z\times_{Y}X\rightarrow Z$ has an
obstruction theory.

  3. (3)

If for any $A\in\textup{AR}_{R}$ and any morphism
$\operatorname{Spec}(A)\rightarrow Y$ the base change
$X\times_{Y}\operatorname{Spec}(A)\rightarrow\operatorname{Spec}(A)$ has an
obstruction theory, then $f$ has an obstruction theory.

###### Proof.

This is [TV08, Lem. 1.4.2.3] but nevertheless we recall the proof.

The existence of the cotangent complex follows from Lemma 3.62.

Part 1 and 2 are clear by definition. For part 3 let $B$ be an animated ring,
$M$ a connective $B$-module with $\pi_{0}M=0$ and
$d\in\operatorname{Der}_{R}(B,M)$. We need to show that the diagram

${X(A\oplus_{d}M)}$${Y(A\oplus_{d}M)}$${X(A)\times_{X(A\oplus
M)}X(A)}$${Y(A)\times_{Y(A\oplus M)}Y(A)}$

is a pullback diagram. Let $x\in Y(A\oplus_{d}M)$, we claim that it suffices
to show that the induced morphism of the fibers of the two horizontal arrows
at $x$ is an equivalence.

Indeed, assume that we have a commutative diagram

${F}$${\ast}$${X(A\oplus_{d}M)}$${Y(A\oplus_{d}M)}$${X(A)\times_{X(A\oplus
M)}X(A)}$${Y(A)\times_{Y(A\oplus M)}Y(A)}$$\scriptstyle{x}$ (3.4.1)

where the upper square and the outer square is a pullback. Let us also
consider the pullback diagram

${Z}$${Y(A\oplus_{d}M)}$${X(A)\times_{X(A\oplus
M)}X(A)}$${Y(A)\times_{Y(A\oplus M)}Y(A)}$$\scriptstyle{\alpha}$

(note that naturally $\operatorname{fib}_{x}(\alpha)\simeq F$ as the outer
square of (3.4.1) is a pullback diagram). We have a naturally induced morphism
of fiber sequences (i.e. a commutative diagram of the form)

${F}$${X(A\oplus_{d}M)}$${Y(A\oplus_{d}M)}$${\operatorname{fib}_{x}(\alpha)}$${Z}$${Y(A\oplus_{d}M).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

The long exact homotopy sequence for fiber sequences in $\SS$ now implies the
claim.

That the upper square and the outer square of (3.4.1) are pullback diagrams
follows from the fact that the pullback of $f$ under the morphism
corresponding to $x$ has an obstruction theory. ∎

The following technical lemma shows, how liftings along square zero extensions
are linked to loops in the space of derivations. This is crucial, when dealing
with formal smoothness of morphisms. Lifts of morphisms along square zero
extensions are controlled by the cotangent complex which is, in some cases,
easier to handle.

######  Lemma 3.72.

Let $f\colon X\rightarrow Y$ be a morphism of derived stacks, and assume $f$
has an obstruction theory. Let $A\in\textup{AR}_{R}$, $M$ be a connective
$A$-module with $\pi_{0}M=0$ and $d\in\operatorname{Der}(A,M)$ a derivation.
Let $x\in X(A)\times_{Y(A)}Y(A\oplus_{d}M)$ be a point and $L(x)$ the fiber of
$X(A\oplus_{d}M)\rightarrow X(A)\times_{Y(A)}Y(A\oplus_{d}M)$ at $x$. There
exists an element
$\alpha(x)\in\pi_{0}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{f,x},M)$
such that
$L(x)\simeq\Omega_{\alpha(x),0}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{f,x},M)$,
where we consider the pullback diagram

${\Omega_{\alpha(x),0}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{f,x},M)}$${\ast}$${\ast}$${\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{f,x},M).}$$\scriptstyle{\alpha(x)}$$\scriptstyle{0}$

###### Proof.

This is [TV08, Prop. 1.4.2.6], but for the convenience of the reader we give a
proof.

First, note that $x$ corresponds to a diagram of the form

${\operatorname{Spec}(A)}$${X}$${\operatorname{Spec}(A\oplus_{d}M)}$${Y.}$

After composition with the natural maps, we get

${\operatorname{Spec}(A\oplus
M)}$${\operatorname{Spec}(A)}$${X}$${\operatorname{Spec}(A)}$${\operatorname{Spec}(A\oplus_{d}M)}$${Y,}$$\scriptstyle{d}$$\scriptstyle{s}$

which gives a point
$\alpha(x)\in\operatorname{Hom}_{\textup{dSt}_{A//Y}}(\operatorname{Spec}(A\oplus
M),X)\simeq\operatorname{Der}_{A}(X/Y,M)$. Using that $f$ is inf-cartesian, we
get a pullback diagram

${L(x)}$${\ast}$${\ast}$${\operatorname{Der}_{A}(X/Y,M).}$$\scriptstyle{\alpha(x)}$$\scriptstyle{0}$

To see this, note the following commutative diagram with pullback squares.

${L(x)}$${\ast}$${X(A\oplus_{d}M)}$${X(A)\times_{Y(A)}Y(A\oplus_{d}M)}$${\ast}$${\operatorname{Der}_{A}(X/Y,M)}$${\ast}$${X(A)\times_{Y(A)}Y(A\oplus
M)}$${X(A\oplus M)}$${X(A)\times_{Y(A)}Y(A\oplus
M)}$${X(A)\times_{Y(A)\times_{Y(A\oplus
M)}Y(A)}Y(A).}$$\scriptstyle{x}$$\scriptstyle{\alpha}$$\scriptstyle{0}$$\scriptstyle{\simeq}$

∎

######  Lemma 3.73.

Any affine derived scheme $X\simeq\operatorname{Spec}(B)$ has an obstruction
theory.

###### Proof.

Certainly, $X$ has a cotangent complex by $L_{X,x}\simeq L_{B}\otimes_{B}A$,
for $x\colon\operatorname{Spec}(A)\rightarrow X.$ So we are left to show that
$X$ is infinitesimally cartesian. But this follows from compatability of the
$\operatorname{Hom}$ functor with limits. ∎

######  Lemma 3.74.

Let $f\colon X\rightarrow Y$ be a morphism of affine derived schemes. If $f$
is smooth, then it is formally smooth.

###### Proof.

This follows from [TV08, Prop. 2.2.5.1], but for convenience of the reader, we
give a proof in our setting.

Let $A$ be an animated $R$-algebra, $M$ a connected $C$-module and
$d\in\operatorname{Der}(A,M)$. Let
$x\in\pi_{0}X(A)\times_{Y(A)}Y(A\oplus_{d}M)$ be a point. We have to show that
the fiber of

$X(A\oplus_{d}M)\rightarrow X(A)\times_{Y(A)}Y(A\oplus_{d}M)$

along $x$ is nonempty. By Lemma 3.73 and 3.72, it suffices to show that
$\pi_{0}\operatorname{Hom}(L_{B/C}\otimes_{B}A,M)$ is contractible, where
$X\simeq\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(C)\simeq Y$. By
Proposition 2.56 the $A$-module $L_{B/C}\otimes_{B}A$ is finite projective, so
especially a retract of a free module (see [Lur17, Cor. 7.2.2.9]) and
therefore $\pi_{0}\operatorname{Hom}(L_{B/C}\otimes_{B}A,M)$ is a retract of a
product of $\pi_{0}M$, which is zero by hypothesis on $M$. ∎

######  Remark 3.75.

We want to remark that Lemma 3.74 holds more generally. An $n$-geometric
morphism of derived stacks is smooth if any only if after restriction to
(Ring) via $t_{0}$ is locally of finite presentation and it is formally
smooth. This is a bit technical but a proof of this is given for example in
[TV08, Prop. 2.2.5.1].

The next proposition and corollary show, how smoothness of a geometric
morphism is linked to its cotangent complex. This can be seen as a
globalization of Proposition 2.56.

######  Proposition 3.76.

Let $f\colon X\rightarrow Y$ be an $n$-geometric morphism of derived stacks.
Then $f$ has an obstruction theory. Further if $f$ is smooth, then $f$ is
formally smooth and $L_{f}$ is perfect with Tor-amplitude in $[-n-1,0]$.

###### Proof.

The proof of this lemma in the spectral setting is given in [AG14, Prop.
4.45]. The proof in the derived setting is analogous. But for the convenience
of the reader, we give a proof.

We prove this lemma by induction over $n$. For the formal smoothness part we
will first reduce to the case where $Y$ is affine.

Indeed, let $B$ be an animated $R$-algebra. Note that we have to show that for
any point $x\in\pi_{0}(X(B)\times_{Y(B)}Y(B\oplus M))$ its fiber under
$X(B\oplus_{d}M)\rightarrow X(B)\times_{Y(B)}Y(B\oplus_{d}M)$ is nonempty. The
point $x$ corresponds to a commutative diagram of the form

${\operatorname{Spec}(B)}$${X}$${\operatorname{Spec}(B\oplus_{d}M)}$${Y.}$

After base change, we get a diagram of the form

${\operatorname{Spec}(B)}$${X\times_{Y}\operatorname{Spec}(B\oplus_{d}M)}$${\operatorname{Spec}(B\oplus_{d}M)}$${\operatorname{Spec}(B\oplus_{d}M)}$$\scriptstyle{\operatorname{id}}$

showing that, we can replace $f$ by the projection
$X\times_{Y}\operatorname{Spec}(B\oplus_{d}M)\rightarrow\operatorname{Spec}(B\oplus_{d}M)$,
in particular, we can assume $Y$ to be affine (this reduction is part of
[TV08, Prop. 2.2.5.1]).

Further for the existence of an obstruction theory, we may assume without loss
of generality that $Y\simeq\operatorname{Spec}(R)$ (see Lemma 3.71 and use
that affine schemes have an obstruction theory by Lemma 3.74).

Let $n=-1$, then $X\simeq\operatorname{Spec}(A)$ and each $A$ is a smooth
$R$-algebra. In particular, we see with Lemma 3.73 and Proposition 2.56 that
$L_{A/R}$ exists and is finite projective. The formal smoothness follows from
Lemma 3.74.

Now assume $n\geq 0$ and let $p\colon U\simeq\coprod_{i\in
I}\operatorname{Spec}(A_{i})\rightarrow X$ be an $n$-atlas, where $A_{i}$ are
smooth $R$-algebras. Let $B$ be a animated $R$-algebra, $M$ be a connective
$B$-module with $\pi_{0}M=0$ and $d\in\operatorname{Der}(B,M)$ a derivation.

Inf-cartesian. For this we will follow [TV08, Lem. 1.4.3.10].

By Lemma 3.66 any étale cover $B\rightarrow B^{\prime}$ gives a cartesian
square of the form

${B^{\prime}\oplus_{d}M}$${B^{\prime}}$${B^{\prime}}$${B^{\prime}\oplus
M,}$$\scriptstyle{d}$$\scriptstyle{s}$

which covers the square induced by the derivation. So to check if
$X(B\oplus_{d}M)\simeq X(B)\times_{X(B\oplus M)}X(B)$, we can pass to an étale
cover of $B$. Therefore, we may assume that any image $x_{1}\in X(B)$ of $x\in
X(B)\times_{X(B\oplus M)}X(B)$ under the projection, lifts to a point in
$u\in\operatorname{Spec}(A_{i})(B)$, for some $i$. Next, we claim that the
point $x$ lifts to a point
$y\in\operatorname{Spec}(A_{i})(B)\times_{\operatorname{Spec}(A_{i})(B\oplus
M)}\operatorname{Spec}(A_{i})(B)$.

To see this, consider the following commutative diagram

${\operatorname{Spec}(A_{i})(B)\times_{\operatorname{Spec}(A_{i})(B\oplus
M)}\operatorname{Spec}(A_{i})(B)}$${X(B)\times_{X(B\oplus
M)}X(B)}$${\operatorname{Spec}(A_{i})(B)}$${X(B).}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$

Let $F(p)$ (resp. $F(q)$) denote the fiber of $u$ (resp. $x_{1}$) under $p$
(resp. $q$). We get a natural morphism $g\colon F(p)\rightarrow F(q)$.
Moreover the fiber of $f$ along $x$ receives a natural morphism from
$\operatorname{fib}_{x}(g)$. Therefore, to see that
$\operatorname{fib}_{x}(f)$ is nonempty it is enough to show that
$\operatorname{fib}_{x}(g)$ is nonempty. But now $g$ is naturally identified,
per definition, with the morphism
$\Omega_{d^{\prime},0}\operatorname{Der}_{B}(A_{i},M)\rightarrow\Omega_{d^{\prime},0}\operatorname{Der}_{B}(X,M)$,
where $d^{\prime}$ is the derivation that is given by the image of $u$ (note
that $X(B)\rightarrow X(B\oplus M)\rightarrow X(B)$ is equivalent to the
identity). Thus the fiber of $g$ is given by
$\Omega_{d^{\prime},0}\operatorname{Der}_{B}(A_{i}/X,M)$, which is equivalent
to $\Omega_{d^{\prime},0}\operatorname{Hom}(L_{A_{i}/X,B},M)$ by induction
hypothesis. But now, again by induction hypothesis, we can find an étale cover
of $B$ such that $\pi_{0}\operatorname{Hom}(L_{A_{i}/X,B},M)=0$, since $M$ is
assumed to be connected, and therefore
$\Omega_{d^{\prime},0}\operatorname{Hom}(L_{A_{i}/X,B},M)$ is nonempty.

Now consider the commutative digram

${\operatorname{Spec}(A_{i})(B\oplus_{d}M)}$${\operatorname{Spec}(A_{i})(B)\times_{\operatorname{Spec}(A_{i})(B\oplus
M)}\operatorname{Spec}(A_{i})(B)}$${X(B\oplus_{d}M)}$${X(B)\times_{X(B\oplus
M)}X(B).}$$\scriptstyle{a}$

By induction hypothesis this square is a pullback sqaure, further $a$ is an
equivalence by affineness. Since $x$ lifts to a point in
$y\in\operatorname{Spec}(A_{i})(B)\times_{\operatorname{Spec}(A_{i})(B\oplus
M)}\operatorname{Spec}(A_{i})(B)$, we see that the fiber at $x$ is given by
the fiber of $a$ at $y$, which is nonempty and contractible (since affine
schemes are inf-cartesian by Lemma 3.73).

Existence. Let us look at the fiber $L_{f}$ of $L_{f\circ p}\rightarrow
L_{p}$. By induction hypothesis $L_{p}$ and $L_{f\circ p}$ exist and if $f$ is
smooth both are perfect with Tor-amplitude in $[-n,0]$ and $[0,0]$
respectively. In particular if $f$ is smooth then $L_{f}$ is perfect and has
Tor-amplitude in $[-n-1,0]$. We have to show that $L_{f}$ satisfies the
universal property of the cotangent complex for $f$ at any point. Let
$x\colon\operatorname{Spec}(B)\rightarrow X$ be a morphism. We may assume,
that $x$ factors through $p$, since $p$ is an effective epimorphism, so we can
pass to an étale cover $B$ which factors through $U$. Let
$y\colon\operatorname{Spec}(B)\rightarrow U$ be such a factorisation. We get a
map

$F\colon\operatorname{Der}_{y}(U,N)\rightarrow\operatorname{Der}_{x}(X,N)$

for any connective $B$-module $N$, which is surjective.

To see the surjectivity of $F$, note that by induction hypothesis $p$ is
formally smooth. Further any element in $d\in\operatorname{Der}_{x}(X,N)$
corresponds to a diagram of the form

${\operatorname{Spec}(B\oplus
N)}$${X}$${\operatorname{Spec}(B).}$$\scriptstyle{d}$$\scriptstyle{x}$

Using the factorization by $U$, we get an element in
$U(B)\times_{X(B)}X(B\oplus N)$. By formal smoothness of $p$, we can lift this
element to an element in $U(B\oplus N)$ (note that $B\times_{0,B\oplus
N[1],0}B\simeq B\oplus N$). But by construction this element has to be a
derivation of $U$ at $y$.

The fiber of $F$ along the derivation, which is induced by
$\operatorname{Spec}(B\oplus
N)\rightarrow\operatorname{Spec}(B)\xrightarrow{x}X$ is given by
$\operatorname{Der}_{y}(U/X,N)$. Thus, we get a fiber sequence

$\operatorname{Hom}_{B}(L_{p},N)\rightarrow\operatorname{Hom}_{B}(L_{f\circ
p},N)\rightarrow\operatorname{Der}_{x}(X,N).$

After delooping202020For a pointed $\infty$-category ${\mathcal{C}}$ a deloop
of an object $c\in{\mathcal{C}}$ is an object $c^{\prime}\in{\mathcal{C}}$,
such that $c\simeq\Omega c^{\prime}$. For ${\mathcal{C}}=\SS$ the
$\infty$-category of spaces, there is a deloop for every object. This follows
from the effectivity of groupoid objects in $\SS$ (see [Lur09, Cor. 6.1.3.20])
(the map $x\rightarrow\ast$, where $x\in\SS$, defines a simplicial object,
which extends via the colimit to a Čech nerve). and surjectivity of $F$, we
see that $\operatorname{Der}_{x}(X,N)$ is the fiber of

$B\operatorname{Hom}_{B}(L_{p},N)\rightarrow B\operatorname{Hom}_{B}(L_{f\circ
p},N),$

where the prefix “$B$” denotes the deloop, and therefore
$\operatorname{Der}_{x}(X,N)\simeq\operatorname{Hom}_{B}(L_{f},N)$.

To see this, note that the map from $\operatorname{Der}_{x}(X,N)$ to the fiber
of $B\operatorname{Hom}_{B}(L_{p},N)\rightarrow
B\operatorname{Hom}_{B}(L_{f\circ p},N)$ is by the five-lemma an equivalence
on the homotopy groups.

Formal smoothness. This is part of [TV08, Prop. 2.2.5.1].

Assume $f$ is smooth and $x\colon\operatorname{Spec}(B)\rightarrow X$ is a
point. By the above $f$ has an obstruction theory. Therefore by Lemma 3.72 it
suffices to show that $\pi_{0}\operatorname{Hom}(L_{f,x},M)$ is contractible.
But this follows from

$\pi_{0}\operatorname{Hom}(L_{f,x},M)\simeq\pi_{0}(L_{f,x}^{\vee}\otimes_{B}M)\simeq\pi_{0}L_{f,x}^{\vee}\otimes_{\pi_{0}B}\pi_{0}M\simeq
0,$

by connectedness of $M$ (note that the above construction of the relative
cotangent complex implies that $L_{f}$ is perfect in the smooth case and
therefore dualizable and the dual is connective). ∎

######  Corollary 3.77.

Let $f\colon X\rightarrow Y$ be an $n$-geometric morphism of derived stacks.
Then $f$ is smooth if and only if $t_{0}f$ is locally of finite presentation
and $L_{f}$ is perfect and has Tor-amplitude in $[-n-1,0]$.

###### Proof.

The proof is the same as in the spectral setting presented in [AG14, Prop.
4.46] using Proposition 3.76. But for the convenience of the reader, we recall
the proof.

We may assume that $Y\simeq\operatorname{Spec}(A)$ is affine (use Lemma 3.62).
Let us fix an $n$-atlas $p\colon U\coloneqq\coprod_{i\in
I}\operatorname{Spec}(T_{i})\rightarrow X$. The "only if" part is Proposition
3.76 and the fact that the $T_{i}$ are smooth $A$-algebras by construction and
thus are locally of finite presentation (see Proposition 2.56).

For the "if" part assume the $\pi_{0}T_{i}$ are locally of finite presentation
over $\pi_{0}B$, $L_{f}$ exists, is perfect and has Tor-amplitude in
$[-n-1,0]$. We have a cofiber sequence

$p^{*}L_{f}\rightarrow L_{U/A}\rightarrow L_{p},$

where by construction $L_{f}$ and $L_{p}$ are perfect with Tor-amplitude in
$[-n-1,0]$ and $[-n,0]$ respectively (for the existence and Tor-amplitude of
$L_{p}$, we use Proposition 3.76). Therefore, $L_{U/A}$ is also perfect with
Tor-amplitude in $[-n-1,0]$. But since $U$ is the disjoint union of affines
the Tor-amplitude of $L_{U/A}$ is concentrated in $[0,0]$ and thus $L_{U/A}$
is perfect and finite projective, which implies that
$\operatorname{Spec}(T_{i})\rightarrow\operatorname{Spec}(A)$ is smooth (see
Lemma 2.56). ∎

######  Corollary 3.78.

Let $f\colon X\rightarrow Y$ be an $n$-geometric morphism of derived stacks
locally of finite presentation. Then $L_{f}$ is perfect.

###### Proof.

Per definition of perfect quasi-coherent modules over a derived stack, we have
to check that for any point $x\colon\operatorname{Spec}(A)\rightarrow X$ the
cotangent complex $L_{f,x}$ is a perfect $A$-module. By Lemma 3.62, we know
that the cotangent complex of the projection $\operatorname{pr}\colon
X\times_{Y,f\circ x}\operatorname{Spec}(A)\operatorname{Spec}(A)$ at the point
induced by $x$ is equivalent to $L_{f,x}$. So without loss of generality, we
may assume that $Y\simeq\operatorname{Spec}(B)$ is affine.

Since $f$ is $n$-geometric and locally of finite presentation, we know that
there exists an $n$-atlas $(p_{i}\colon\operatorname{Spec}(A_{i})\rightarrow
X)_{i\in I}$ such that $A_{i}$ are locally of finite presentation over $A$.
Since perfect quasi-coherent modules satisfy fpqc descent (see Remark 3.54),
we have that $L_{f}\in\operatorname{QCoh}_{\operatorname{perf}}(X)$ if and
only if each $p_{i}^{*}L_{f}$ is perfect. But by Lemma 3.63, we have the
following cofiber sequence

$p_{i}^{*}L_{f}\rightarrow
L_{\operatorname{Spec}(A_{i})/\operatorname{Spec}(A)}\rightarrow L_{p_{i}}.$

Since $A_{i}$ is locally of finite presentation over $A$, we know by
Proposition 2.57, that $L_{\operatorname{Spec}(A_{i})/\operatorname{Spec}(A)}$
is perfect and since by definition $p_{i}$ is smooth, we have with Proposition
3.76 that indeed $p_{i}^{*}L_{f}$ is perfect. ∎

The last part of this section is dedicated to show that a geometric derived
stack $X$ is automatically hypercomplete for the étale topology. The idea is
to show that for any animated $R$-algebra $A$, we have
$X(A)\simeq\lim_{n}X(A_{\leq n})$. Then we reduce to the case, where we look
at $n$-truncated sheaves, which are always hypercomplete.

######  Lemma 3.79.

Let $X$ be an $n$-geometric derived stack for some $n\geq-1$ and $A$ an
animated ring. Then the natural morphism $X\rightarrow\lim_{n}X\circ\tau_{\leq
n}$ is an equivalence.

###### Proof.

This is analogous to [Lur04, Prop. 5.3.7].

We will do this by induction over $n$. This is certainly true for if $X$ is
affine. So assume that $n\geq 0$ and let $p\colon U\coloneqq\coprod_{i\in
I}\operatorname{Spec}(A_{i})\rightarrow X$ be an $n$-atlas.

By definition $U\times_{X}U$ is $(n-1)$-geometric, this also holds for every
successive fiber product, i.e. every element of the Čech nerve
$\check{C}(U/X)_{\bullet}$ is $(n-1)$-geometric. Since $p$ is an effective
epimorphism, we have that the natural map
$\mathop{\textup{colim}}_{\Delta}\check{C}(U/X)_{\bullet}\rightarrow X$ is an
equivalence. By induction hypothesis, we have for every $[n]\in\Delta$ that
the natural map
$\check{C}(U/X)_{[n]}\rightarrow\lim\check{C}(U/X)_{[n]}\circ\tau_{\leq n}$ is
an equivalence. Thus, also its colimit under $\Delta$ is an equivalence, so we
get an induced commutative diagram

${\mathop{\textup{colim}}_{\Delta}\check{C}(U/X)_{[n]}}$${\mathop{\textup{colim}}_{\Delta}\lim_{n}\check{C}(U/X)_{[n]}\circ\tau_{\leq
n}}$${X}$${\lim_{n}X\circ\tau_{\leq n},}$

where the top arrow and left vertical arrow are equivalences and the right
vertical arrow is a monomorphism. Thus the bottom vertical arrow is an
equivalence if $\lim_{n}U\circ\tau_{\leq n}\rightarrow\lim_{n}X\circ\tau_{\leq
n}$ is an effective epimorphism. Let $x\in\lim_{n}X(A_{\leq n})$ and consider
the projection onto $X(A_{\leq 0})$, denoted by $x_{0}$. Then we can find an
étale cover $\widetilde{\pi_{0}A}$ of $A_{\leq 0}\simeq\pi_{0}A$ such that
$x_{0}$ has a lift in $U(\widetilde{\pi_{0}A})$. By Proposition 2.59 there is
an étale cover $\widetilde{A}$ of $A$ such that
$\pi_{0}\widetilde{A}\simeq\widetilde{\pi_{0}A}$. In particular, we see that
we can lift the image of $x_{0}$ in $X(\widetilde{A}_{\leq 0})$ under
$U(\widetilde{A}_{\leq 0})\rightarrow X(\widetilde{A}_{\leq 0})$. Now let
$x_{n}$ be the image of $x$ in $X(A_{\leq n})$. We will show the result by
induction. Assume the argument holds for $n-1$. In particular, let $u_{n-1}$
be the lift of $x_{n-1}$ under $U(\widetilde{A}_{\leq n-1})\rightarrow
X(\widetilde{A}_{\leq n-1})$. It is enough to prove that we can find a lift of
$(u_{n-1},x_{n})$ under $U(\widetilde{A}_{\leq n})\rightarrow
U(\widetilde{A}_{\leq n-1})\times_{X(\widetilde{A}_{\leq
n-1})}X(\widetilde{A}_{\leq n})$, since then for all $n\in\mathbb{N}_{0}$
there is a lift $u_{n}$ of $x_{n}$ compatible with the maps in the limit, i.e.
we get an element $u\in\lim_{n}U(\widetilde{A}_{\leq n})$ that maps to the
image of $x$ in $\lim_{n}X(\widetilde{A}_{\leq n})$. But this follows from
formal smoothness of $U\rightarrow X$ (see Proposition 3.76) and the fact that
the map $A_{\leq n}\rightarrow A_{\leq n-1}$ is a square zero extension (see
Lemma 2.52). ∎

######  Lemma 3.80.

Let $X\rightarrow Y$ be an $n$-geometric morphism and $A$ be a $k$-truncated
animated $R$-algebra. Then $X(A)\rightarrow Y(A)$ is $(n+k+1)$-truncated.

###### Proof.

This is a consequence of Lemma 3.79 and analogous to [Lur04, Cor. 5.3.8].

We have to show that the fiber of $X(A)\rightarrow Y(A)$ is
$(n+k+1)$-truncated. By Lemma 3.79 it suffices to show that for all
$n\in\mathbb{N}_{0}$ the map $X(A_{\leq j})\rightarrow X(A_{\leq
j-1})\times_{Y(A_{\leq n-1})}Y(A_{\leq j})$ is $(n+j+1)$-truncated, whenever
$j\leq k$. But from Lemma 2.52 and Lemma 3.72 the fiber of the previous map is
given by the loop of
$\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{X/Y,A},\pi_{j}A[j])$ which by
adjunction is $(n+j+1)$-truncated, since by definition $L_{X/Y}[n]$ is
connective212121Note that
$\pi_{n+j+1+k}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{X/Y,A},\pi_{j}A[j-1])\cong\pi_{0}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{X/Y,A}[n+j+1+k],\pi_{j}A[j+1])$
and since $L_{X/Y,A}[n+j+1+k]\in(\operatorname{Mod}_{A})_{\geq j+1+k}$ and
$\pi_{j}A[j+1]\in(\operatorname{Mod}_{A})_{\leq-j+1}$, we see by definition of
the $t$-structures that
$\pi_{n+j+1+k}\operatorname{Hom}_{\operatorname{Mod}_{A}}(L_{X/Y,A},\pi_{j}A[j+1])\cong
0$ for $k\geq 1$.. ∎

######  Lemma 3.81.

Let $X$ be an $n$-geometric stack, then $X$ is hypercomplete.

###### Proof.

This is a direct consequence of Lemma 3.79 and 3.80 and the fact that
truncated $\infty$-topoi are automatically hypercomplete. This is analogous to
[Lur04, Cor. 5.3.9] but anyway we will explain this.

By Lemma 3.79, we have $X\simeq X\circ\tau_{\leq n}$ and for any $k$-truncated
animated ring $A$, we have that $X(A)$ is $(n+k+1)$-truncated (see Lemma
3.80), in particular $X\circ\tau_{\leq n}$ is hypercomplete (see [Lur09, Lem.
6.5.2.9]) and since the $\infty$-topos of hypercomplete sheaves has limits, we
have that $X$ is hypercomplete. ∎

##  4\. The stack of perfect modules

In this section, we want to prove that the derived stack of perfect modules is
locally geometric. This was already proven in [TV07] in the model categorical
setting and in [AG14] in the spectral setting, but we recall the proof in its
entirety in our setting.

We recall some lemmas needed for the proof, as they will become important
later on when analyzing the substacks of derived $F$-zips.

######  Lemma 4.1.

Let $A$ be a commutative ring and $P$ be a perfect complex of $A$-modules and
let $n\in\mathbb{N}_{0}$. Further, for $k\in\mathbb{Z}$ let
$\beta_{k}\colon\operatorname{Spec}(A)_{\textup{cl}}\rightarrow\mathbb{N}_{0}$
be the function given by
$s\mapsto\dim_{\kappa(s)}\pi_{k}(P\otimes_{A}\kappa(s))$. Then
$\beta_{k}^{-1}([0,n])$ is quasi-compact open.

###### Proof.

By [Sta19, 0BDI] $\beta_{k}$ is upper semi-continuous and locally
constructible. As $\operatorname{Spec}(A)_{\textup{cl}}$ is affine it is
quasi-compact quasi-separated and so we see that $\beta_{k}^{-1}([0,n])$ is
quasi-compact open. ∎

######  Remark 4.2.

Let $A$ be a commutative ring and $P$ be a perfect complex of $A$-modules. Let
$I\subseteq\mathbb{Z}$ be a finite subset and for $k\in\mathbb{Z}$ let
$\beta_{k}$ be as in Lemma 4.1 . Assume that $\beta_{i}\not=0$ for $i\in I$
and zero everywhere else. Then using [Sta19, 0BCD,066N], we see that $P$ has
Tor-amplitude in $[\min(I),\max(I)]$.

######  Lemma 4.3.

Let $A$ be a commutative ring and $P$ be a perfect complex of $A$-modules.
Then there exists a quasi-compact open subscheme
$U\subseteq\operatorname{Spec}(A)_{\textup{cl}}$ with the following property,

  1. $\bullet$

an affine scheme morphism
$\operatorname{Spec}(B)_{\textup{cl}}\rightarrow\operatorname{Spec}(A)_{\textup{cl}}$
factors through $U$ if and only if $P\otimes_{A}B\simeq 0$.

###### Proof.

Let $\beta_{k}$ be as in Lemma 4.1. Then we set

$U\coloneqq\bigcap_{k\in\mathbb{Z}}\beta_{k}^{-1}(\\{0\\}).$

As $P$ is perfect, so in particular has finite Tor-amplitude, this
intersection has only finitely many pieces that are non equal to
$\operatorname{Spec}(A)_{\textup{cl}}$. Therefore, $U$ is a finite
intersection of quasi-compact opens in an affine scheme (see Lemma 4.1) and
thus quasi-compact open.

Now assume we have a morphism
$\operatorname{Spec}(B)_{\textup{cl}}\rightarrow\operatorname{Spec}(A)_{\textup{cl}}$
such that $P\otimes_{A}B=0$. Then certainly for any $b\in B$ and all
$i\in\mathbb{Z}$ we have
$\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}B\otimes_{B}\kappa(b))=0$. Let
$a\in\operatorname{Spec}(A)$ be the image of $b$. Then for all
$i\in\mathbb{Z}$ we have the following equalities

$\displaystyle\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}B\otimes_{B}\kappa(b))$
$\displaystyle=\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}\kappa(a)\otimes_{\kappa(a)}\kappa(b))$
$\displaystyle=\dim_{\kappa(b)}\pi_{i}(P\otimes_{A}\kappa(a))\otimes_{\kappa(a)}\kappa(b)$
$\displaystyle=\dim_{\kappa(a)}\pi_{i}(P\otimes_{A}\kappa(a)),$

where we use flatness of field extensions in the second equality. Therefore,
we see that $\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$ factors
through $U$.

For the other direction assume that
$\operatorname{Spec}(B)_{\textup{cl}}\rightarrow\operatorname{Spec}(A)_{\textup{cl}}$
factors through $U$. Then for any $b\in\operatorname{Spec}(B)$ and any
$i\in\mathbb{Z}$, we have that $\pi_{i}(P\otimes_{A}B\otimes_{B}\kappa(b))=0$.
By Remark 4.2, we see that $P\otimes_{A}B$ is given by a finite projective
module $M$ concentrated in one degree. The fiberwise dimension of $M$ is equal
to $0$ by assumption and thus by Nakayama $M=0$. ∎

The next lemma shows that the vanishing locus of perfect complexes is quasi-
compact open. This will be applied to the cofiber of morphisms of perfect
complexes. In particular, the locus classifying equivalences between fixed
perfect modules is therefore quasi-compact open.

######  Lemma 4.4.

Let $A\in\textup{AR}_{R}$ and $P$ be a perfect $A$-module. Define the derived
stack $V_{P}$ via $V_{P}(B)=$ full sub-$\infty$-category of
$\operatorname{Hom}_{\textup{AR}_{R}}(A,B)$ consisting of morphisms $u\colon
A\rightarrow B$ such that $P\otimes_{A,u}B\simeq 0$. This is a quasi-compact
open substack of $\operatorname{Spec}(A)$.

###### Proof.

This is [TV07, Prop. 2.23] translated to our setting. But for the convenience
of the reader, we give a proof.

Consider $Q\coloneqq P\otimes_{A}\pi_{0}A$. Then $Q$ is a perfect complex of
$\pi_{0}A$-modules. Lemma 4.3 shows that there is a quasi-compact open
subscheme $U\subseteq\operatorname{Spec}(\pi_{0}A)_{\textup{cl}}$, such that
for any point
$u\colon\operatorname{Spec}(R^{\prime})_{\textup{cl}}\rightarrow\operatorname{Spec}(\pi_{0}A)_{\textup{cl}}$,
where $R^{\prime}$ is a commutative ring, the module
$Q\otimes_{\pi_{0}A}R^{\prime}$ is isomorphic to $0$ if and only if $u$
factors through $U$.

Let $f_{1},\dots,f_{n}\in\pi_{0}A$, such that the $\pi_{0}A_{f_{i}}$ covers
$U$. Then $V\coloneqq\mathop{\rm
Im}(\coprod_{i=1}^{n}\operatorname{Spec}(A[f_{i}^{-1}]))$, the image of
$\coprod_{i=1}^{n}\operatorname{Spec}(A[f_{i}^{-1}])\rightarrow\operatorname{Spec}(A)$,
is equivalent to $V_{P}$.

Indeed, take a morphism $u\colon A\rightarrow B$ in $\textup{AR}_{R}$. Then
$u\in V(B)$ if and only if there exists an $i$ such that $\pi_{0}u(f_{i})$ is
étale locally invertible in $\pi_{0}B$ (see Remark 3.42). This is equivalent
to $P\otimes_{A}\pi_{0}B\simeq 0$ by the choice of the $f_{i}$. This again is
equivalent to $P\otimes_{A}B\simeq 0$.

To see this, assume the $P\otimes_{A}B\not\simeq 0$ and take the minimal
$i\in\mathbb{Z}$ with $\pi_{i}(P\otimes_{A}B)\not\simeq 0$. Consider the Tor
spectral sequence

$E_{2}^{p,q}=\operatorname{Tor}_{p}^{\pi_{\ast}B}(\pi_{\ast}(P\otimes_{A}B),\pi_{0}B)_{q}\Rightarrow\pi_{p+q}(P\otimes_{A}\pi_{0}B).$

By the definition of the graded tensor product, we see that
$E^{0,q}_{2}=\pi_{q}(P\otimes_{A}B)$ for $q=i$ and $0$ for $q<i$. As explained
in Remark 2.55, we can choose a graded free resolution of $P\otimes_{A}B$ such
that each term of the resolution concentrated in degrees $\geq i$. Therefore,
the spectral sequence is concentrated in the quadrant where the left lower
corner is at $p=0$ and $q=i$. Thus, $\pi_{i}(P\otimes_{A}B)\cong
E_{2}^{0,i}\cong\pi_{i}(P\otimes_{A}\pi_{0}B)\cong 0$ contradicting the
assumption.

The geometricity follows from Lemma 3.28. ∎

Next, we show that the stack classifying morphisms between perfect
modules222222Note that for two perfect $A$-modules $P,Q$ over some animated
ring $A$, we have
$\operatorname{Hom}_{\operatorname{Mod}_{B}}(P\otimes_{A}B,Q\otimes_{A}B)\simeq\operatorname{Hom}_{\operatorname{Mod}_{A}}(P\otimes
Q^{\vee},B)$ as shown in the proof of Lemma 4.7. is actually geometric and in
good cases smooth. Since derived $F$-zips will come with two bounded perfect
filtrations (i.e. finite chains of morphisms of perfect modules), this lemma
is crucial for the geometricity of derived $F$-zips.

######  Lemma 4.5.

Let $A$ be an animated $R$-algebra. Let $P$ be a perfect $A$-module with Tor-
amplitude concentrated in $[a,b]$ with $a\leq 0$. Then the derived stack

$\displaystyle F^{A}_{P}\colon\textup{AR}_{A}$ $\displaystyle\rightarrow\SS$
$\displaystyle B$
$\displaystyle\mapsto\operatorname{Hom}_{\operatorname{Mod}_{A}}(P,B)$

is $(-a-1)$-geometric and locally of finite presentation over
$\operatorname{Spec}(A)$232323Certainly, we can view $F_{P}^{A}$ as a derived
stack over $R$ with a morphism to $\operatorname{Spec}(A)$. So for any
animated $R$-algebra $C$ that does not come with a morphism $A\rightarrow C$
the value of $F_{P}^{A}$ is empty.. Further, the cotangent complex of $F_{P}$
at a point $x\colon\operatorname{Spec}(B)\rightarrow F^{A}_{P}$ is given by

$L_{F_{P},x}\simeq P\otimes_{A}B.$

In particular, if $b\leq 0$, then $F_{P}$ is smooth.

###### Proof.

Before showing the geometricity, let us calculate the space of derivations of
$F_{P}$ and hence the cotangent complex.

Let $x\colon\operatorname{Spec}(B)\rightarrow F^{A}_{P}$ be a morphism of
derived stacks corresponding to a morphism $f\colon P\rightarrow B$ in
$\operatorname{Mod}_{A}$ and $M$ be a connective $B$-module. We have that
$\operatorname{Der}_{x}(F^{A}_{P}/A,M)$ is given by the fiber of
$\operatorname{Hom}_{A}(P,B\oplus M)\rightarrow\operatorname{Hom}_{A}(P,B)$ at
$f$. The underlying $R$-module of $B\oplus M$ is per construction the direct
sum of the underlying $R$-module of $B$ and of $M$. Therefore, any morphism
$P\rightarrow B\oplus M$ is uniquely up to homotopy characterized by a
morphism $P\rightarrow B$ and $P\rightarrow M$ and thus, we see that
$\operatorname{Der}_{x}(F^{A}_{P}/A,M)\simeq\operatorname{Hom}_{A}(P,M)\simeq\operatorname{Hom}_{A}(P\otimes_{A}B,M)$.
Hence, we have $L_{F_{P},x}\simeq P\otimes_{A}B$.

Let us conclude the rest of the proof, which we will prove by induction on
$a$.

If $a=0$, then $P$ is connective and
$F^{A}_{P}\simeq\operatorname{Hom}_{\textup{AR}_{A}}(-,\operatorname{Sym}_{A}P)$
which has the desired properties.

Now assume $a<0$. By Lemma 2.33 we have
$P\simeq\operatorname{fib}(Q\rightarrow M[a+1])$, where $Q$ has Tor-amplitude
in $[a+1,b]$ and $M$ is a finite projective $A$-module. Thus we get a fiber
sequence

$F^{A}_{M[a+1]}\rightarrow F^{A}_{Q}\rightarrow F^{A}_{P}\rightarrow
F_{M[a]}.$

By induction hypothesis $F_{Q}$ is $(-a-2)$-geometric and locally of finite
presentation. We will see that the map $p\colon F^{A}_{Q}\rightarrow
F^{A}_{P}$ is an effective epimorphism.

Indeed, note that the above fiber sequence and projectivity of $M$ imply that
$\pi_{0}p$ is surjective and thus $p$ is an effective epimorphism (see Remark
3.10).

The diagonal $F^{A}_{Q}\times_{F^{A}_{P}}F^{A}_{Q}$ is given by
$F^{A}_{Q\oplus_{P}Q}$242424Here $Q\oplus_{P}Q$ is defines as the pushout of
the morphism $P\rightarrow Q$ with itself., which will be $(-a-2)$-geometric
with smooth projections to $F^{A}_{Q}$.

To see this, note that we have a fiber sequence $Q\rightarrow
Q\oplus_{P}Q\rightarrow M[a+1]$ which has a retract. Thus the natural map
$Q\oplus_{P}Q\rightarrow Q\oplus M[a+1]$ is an equivalence on the level of
homotopy groups by the splitting lemma (the induced exact sequences are short
exact, using the retract) and therefore $Q\oplus_{P}Q\simeq Q\oplus M[a+1]$.
Hence, $F^{A}_{Q\oplus_{P}Q}\simeq F^{A}_{Q}\times F^{A}_{M[a+1]}$, which is
the pullback of $(-a-2)$-geometric stack and thus itself geometric.

Also the projection to $F^{A}_{Q}$ is smooth, because $F^{A}_{M[a+1]}$ is
smooth (the smoothness of $F^{A}_{M[a+1]}$ follows since
$L_{F^{A}_{M[a+1]},x}\simeq M[a+1]\otimes_{A}B$ at a point
$x\colon\operatorname{Spec}(B)\rightarrow F^{A}_{M[a+1]}$ and thus has Tor-
amplitude in $[a+1,0]$, which concludes (see Corollary 3.77)).

By Proposition 3.41, we see that $F^{A}_{P}\rightarrow\operatorname{Spec}(A)$
is a quasi-compact $(-a-1)$-geometric stack locally of finite presentation.

If $b\leq 0$, then $L_{F^{A}_{P}}$ is perfect with Tor-amplitude concentrated
in degree $[a,0]$ and therefore $F^{A}_{P}$ is smooth by Corollary 3.77. ∎

######  Remark 4.6.

A variant of Lemma 4.5 in the spectral setting can be found in [AG14, Thm.
5.2]. Alternatively, one can look at the proofs given in [TV07, Lem. 3.9] and
[TV07, 3.12] to construct a proof in the model categorical setting.

######  Lemma 4.7.

The diagonal map
$\textup{Perf}^{[a,b]}\rightarrow\textup{Perf}^{[a,b]}\times_{R}\textup{Perf}^{[a,b]}$
is $(b-a)$-geometric and locally of finite presentation.

###### Proof.

This is part of the proof of [AG14, Thm. 5.6] translated to our setting. For
the convenience of the reader, we give a proof.

A morphism
$\operatorname{Spec}(A)\rightarrow\textup{Perf}^{[a,b]}\times_{R}\textup{Perf}^{[a,b]}$
corresponds to two perfect modules $P,Q$ with Tor-amplitude concentrated in
$[a,b]$. The pullback under the diagonal classifies equivalences between $P$
and $Q$. This is an open, $0$-geometric substack of

$\displaystyle\operatorname{Hom}_{\operatorname{Mod}_{A}}(P\otimes_{A}Q^{\vee},-)\simeq\operatorname{Hom}_{\operatorname{Mod}_{A}}(P,Q\otimes_{A}-)\simeq\operatorname{Hom}_{\operatorname{Mod}_{-}}(P\otimes_{A}-,Q\otimes_{A}-),$

(note that perfect modules are dualizable).

To see this, note that for any morphism
$\operatorname{Spec}(B)\rightarrow\operatorname{Hom}_{\operatorname{Mod}_{A}}(P\otimes_{A}Q^{\vee},-)$,
given by a morphism $\varphi\colon P\otimes_{A}B\rightarrow Q\otimes_{A}B$,
the stack
$\operatorname{Equiv}(P,Q)\times_{\operatorname{Hom}_{\operatorname{Mod}_{A}}(P\otimes_{A}Q^{\vee},-)}\operatorname{Spec}(B)$
classifies morphisms $u\colon B\rightarrow C$, where
$\operatorname{cofib}\varphi\otimes_{B,u}C\simeq 0$, which is an open,
$0$-geometric substack of $\operatorname{Spec}(B)$ by Lemma 4.4.

Now $P\otimes Q^{\vee}$ is a perfect module of Tor-dimension $[a-b,b-a]$ (see
Lemma 2.33) and thus Lemma 4.5 concludes the proof. ∎

######  Definition 4.8.

Let $n\in\mathbb{N}$ and $A\in\textup{AR}_{R}$. We denote the
$\infty$-category of finite projective $A$-modules of rank $n$ with
$\operatorname{BGL}_{n}(A)$.

######  Lemma 4.9.

Let $n\in\mathbb{N}$. The functor $A\mapsto\operatorname{BGL}_{n}(A)$ from
$\textup{AR}_{R}$ to $\textup{Cat}_{\infty}$ satisfies fpqc descent.

###### Proof.

We already know that modules satisfy decent so it is enough to check that an
$A$-module $M$ is finite projective of rank $n$ if it is after base change to
an fpqc-cover $(A\rightarrow A_{i})_{i\in I}$. Note that
$\pi_{0}A\rightarrow\pi_{0}A_{i}$ is faithfully flat for every $i\in I$. Now
assume that $M\otimes_{A}A_{i}$ is finite projective of rank $n$. Then it is
in particular flat and we will show first that $M$ is flat over $A$. By
flatness, the natural map

$\pi_{j}A_{i}\otimes_{\pi_{0}A}\pi_{0}M\cong\pi_{j}A_{i}\otimes_{\pi_{0}A_{i}}\pi_{0}M\otimes_{\pi_{0}A}\pi_{0}A_{i}\cong\pi_{j}A_{i}\otimes_{\pi_{0}A_{i}}\pi_{0}(M\otimes_{A}A_{i})\rightarrow\pi_{j}(M\otimes_{A}A_{i})$

is an equivalence. By flatness of $A\rightarrow A_{i}$, we have
$\pi_{j}A_{i}\cong\pi_{j}A\otimes_{\pi_{0}A}\pi_{0}A_{i}$. Hence, we have
$\pi_{j}A_{i}\otimes_{\pi_{0}A}\pi_{0}M\cong\pi_{j}A\otimes_{\pi_{0}A}\pi_{0}A_{i}\otimes_{\pi_{0}A}\pi_{0}M$.
By faithfully flatness the map
$\pi_{j}A\otimes_{\pi_{0}A}\pi_{0}M\rightarrow\pi_{j}M$ is an equivalence if
and only if it so after base change to $\pi_{0}A_{i}$ for all $i\in I$. But
the above shows that this base change gives the map

$\pi_{j}A\otimes_{\pi_{0}A}\pi_{0}A_{i}\otimes_{\pi_{0}A}\pi_{0}M\rightarrow\pi_{j}(M\otimes_{A}A_{i})$

and flatness of $A_{i}$ over $A$ shows that
$\pi_{j}(M\otimes_{A}A_{i})\cong\pi_{j}M\otimes_{\pi_{0}A}A_{i}$ (see [Lur17,
Prop. 7.2.2.13]) so indeed,
$\pi_{j}A\otimes_{\pi_{0}A}\pi_{0}M\rightarrow\pi_{j}M$ is an equivalence.
Therefore, $M$ is flat over $A$.

Now a flat module is finite projective of rank $n$ if it is so on $\pi_{0}$
(see Lemma 2.25 and use the definition of finite projectiveness) but this
follows from classical faithfully flat descent. ∎

######  Remark 4.10.

The inclusion of $\SS\hookrightarrow\textup{Cat}_{\infty}$ is left adjoint to
the functor $(-)^{\simeq}$ that passes to the largest Kan complex contained in
an $\infty$-category (see [Lur09, Prop. 1.2.5.3]). Therefore if
$F\colon{\mathcal{C}}\rightarrow\textup{Cat}_{\infty}$ is a (hypercomplete)
sheaf for some Grothendieck topology on ${\mathcal{C}}$ then also
$F^{\simeq}\coloneqq(-)^{\simeq}\circ F$ is one.

######  Definition 4.11.

We define the derived stack classifying vector bundles as the stack

$\displaystyle\operatorname{BGL}_{n,R}\colon\textup{AR}_{R}$
$\displaystyle\rightarrow\SS$ $\displaystyle A$
$\displaystyle\mapsto\operatorname{BGL}_{n}(A)^{\simeq}.$

Further, we denote by $\operatorname{GL}_{n,R}$ the loop under the map
$\operatorname{Spec}(R)\rightarrow\operatorname{BGL}_{n,R}$, which is given
for an animated $R$-algebra $A$ by $\ast\mapsto A^{n}$, i.e. we have the
following pullback diagram in $\textup{dSt}_{R}$

${\operatorname{GL}_{n,R}}$${\operatorname{Spec}(R)}$${\operatorname{Spec}(R)}$${\operatorname{BGL}_{n,R}}$

(note that for a commutative ring $A$, we have that
$\operatorname{BGL}_{n,R}(A)$ is the groupoid of rank $n$ vector bundles on
$A$ and thus $\operatorname{GL}_{n,R}(A)$ is indeed given by the points of the
general linear group scheme of rank $n$).

######  Lemma 4.12.

Let $\operatorname{Proj}_{R}$ denote the derived stack classifying finite
projective modules. Then
$\operatorname{Proj}_{R}\simeq\coprod_{n\in\mathbb{N}}\operatorname{BGL}_{n,R}$,
in particular $\operatorname{Proj}_{R}$ is $1$-geometric and smooth.

Further, $\operatorname{GL}_{n,R}$ is an affine derived scheme and
$\operatorname{Proj}_{R}$ has an affine diagonal.

###### Proof.

The proof is the same as [TV08, Cor. 1.3.7.12] but for the convenience of the
reader, we give a sketch.

That
$\operatorname{Proj}_{R}\simeq\coprod_{n\in\mathbb{N}}\operatorname{BGL}_{n,R}$,
where $\operatorname{BGL}_{n,R}$ denotes the stack of finite projective
modules of rank $n$, is clear. So it suffices to show that
$\operatorname{BGL}_{R,n}$ is a $1$-geometric smooth stack. It is enough to
show that $\operatorname{GL}_{n,R}\rightarrow\operatorname{Spec}(R)$ is
$0$-geometric smooth. Then
$\operatorname{GL}_{n,R}\rightarrow\operatorname{Spec}(R)$ defines a $0$-Segal
groupoid (see [TV08, Def. 1.3.4.1]) and $\operatorname{BGL}_{R,n}$ is
$1$-geometric (see [TV08, Prop. 1.3.4.2]). That $\operatorname{BGL}_{n,R}$ is
smooth follows from the fact that the natural morphism
$\operatorname{Spec}(R)\rightarrow\operatorname{BGL}_{n,R}$ gives a $1$-atlas.

The claim about $\operatorname{GL}_{n,R}$ follows in the following way. The
stack $\operatorname{GL}_{n,R}$ is equivalent to the stack classifying
automorphisms of $R^{n}$, i.e.
$\operatorname{GL}_{n,R}(A)\simeq\operatorname{Equiv}_{A}(A^{n})$, for
$A\in\textup{AR}_{R}$. But by Lemma 4.4 this is a $0$-geometric open substack
of
$F_{R^{n^{2}}}\simeq\operatorname{Spec}(\operatorname{Sym}_{R}(R^{n^{2}}))$,
which is a $(-1)$-geometric smooth stack.

Alternatively, one could follow [TV08, Prop. 1.3.7.10] and show directly that
the inclusion $\iota\colon\operatorname{GL}_{n,R}\hookrightarrow
F_{R^{n^{2}}}$ is representable and étale by showing that for any point $x\in
F_{R^{n^{2}}}(A)$, we have
$\iota^{-1}(x)\simeq\operatorname{Spec}(A[\det(x)^{-1}])$. In particular, we
see in this way that $\operatorname{GL}_{n,R}$ is representable by an affine
derived scheme. This also shows that $\operatorname{BGL}_{n,R}$ has an affine
diagonal, since
$\operatorname{Spec}(R)\times_{\operatorname{Spec}(R)}\operatorname{Spec}(R)\rightarrow\operatorname{BGL}_{n,R}\times_{\operatorname{Spec}(R)}\operatorname{BGL}_{n,R}$
is a $1$-atlas by the above and so, we have a pullback diagram of the form

${\operatorname{GL}_{n,R}}$${\operatorname{Spec}(R)\simeq\operatorname{Spec}(R)\times_{\operatorname{Spec}(R)}\operatorname{Spec}(R)}$${\operatorname{BGL}_{n,R}}$${\operatorname{BGL}_{n,R}\times_{\operatorname{Spec}(R)}\operatorname{BGL}_{n,R},}$$\scriptstyle{\Delta}$

which shows that the diagonal is affine, since this can be tested after
passing to a cover by affines (see [TV08, Lem. 1.3.2.8]). ∎

######  Remark 4.13.

For the proof of the next theorem, we want to remark some generalities about
pullbacks of Kan complexes.

Let $X,Y$ be Kan complexes, i.e. elements in $\SS$, and assume we have a
morphism $X\rightarrow\operatorname{Fun}(\partial\Delta^{1},Y)$ in $\SS$. We
want to compute the following pullback in $\SS$

${W}$${\operatorname{Fun}(\Delta^{1},Y)}$${X}$${\operatorname{Fun}(\partial\Delta^{1},Y),}$$\scriptstyle{i}$

where $i$ is given by the restriction. In general this is a pullback in the
$\infty$-categorical sense, which can also be computed on the level of model
categories via the homotopy pullback (recall that $\SS$ is the
$\infty$-category associated to the model category of simplicial sets with the
usual model structure (weak equivalences are given by weak equivalences of the
underlying Kan complexes and fibrations are Kan fibrations)). We claim that
the homotopy pullback of the underlying Kan complexes is equivalent in $\SS$
to the ordinary pullback of simplicial sets if $i$ is a Kan fibration (i.e.
$i$ is a fibration in the model category of simplicial sets and since each
simplicial set involved is already a Kan complex they are per definition
fibrant).

Indeed, this is a classical result and remarked in [Lur09, Rem. A.2.4.5] but
we will shortly sketch the idea behind it. Let ${\bf A}$ be a combinatorial
model category (e.g. the category of simplicial sets with the model structure
explained above) and let $I$ be the diagram category given by three objects
$\\{0,1,2\\}$ together with morphisms $0\rightarrow 2$, $1\rightarrow 2$ and
identities. One can attach the injective model structure onto the functor
category $\operatorname{Fun}(I,{\bf A})$ by defining weak equivalences to be
pointwise weak equivalences and defining cofibrations also pointwise (the
fibration are then given by certain lifting properties, which we will not
discuss on detail). The homotopy limit is defined as the right Quillen adjoint
of the constant functor ${\bf A}\rightarrow\operatorname{Fun}(I,{\bf A})$,
which is weakly equivalent to the the limit of a fibrant diagram in
$\operatorname{Fun}(I,{\bf A})$, i.e. for a diagram $a\rightarrow c\leftarrow
b$, the homotopy limit is defined as an object that is weakly equivalent to
ordinary limit of $a^{\prime}\rightarrow c^{\prime}\leftarrow b^{\prime}$,
where $a^{\prime}\rightarrow c^{\prime}$ and $b^{\prime}\rightarrow
c^{\prime}$ are fibrations, $c^{\prime}$ is fibrant and we have a commutative
diagram of the form

${a}$${c}$${b}$${a^{\prime}}$${c^{\prime}}$${b^{\prime},}$

where the vertical arrows are weak equivalences (this is analogous to the
theory of homotopy pushouts, which can be found in [Lur09, §A.2.4]). Now let
$I_{R}$ be the full subcategory of $I$ generated by the elements $0$ and $2$
and define $I_{L}$ as the full subcategory of $I$ generated by the elements
$1$ and $2$. The tuple $(I_{L},I_{R})$ makes $I$ into a Reedy category on $I$
(see [Lur09, §A.2.9] for more on Reedy categories). As explained in [Lur09,
Proposition A.2.9.19] there is a model structure on $\operatorname{Fun}(I,{\bf
A})$ corresponding to the Reedy structure on $I$ called the Reedy model
structure. Important for us is that a diagram $a\rightarrow c\leftarrow b$ is
fibrant for the Reedy model structure if $a,b$ are fibrant and either
$a\rightarrow c$ or $b\rightarrow c$ is a fibration. Further, as remarked in
[Lur09, Rem. A.2.9.23] the Reedy model structure and the injective model
structure are Quillen equivalent via the identity functor. Therefore, if we
have a fibrant diagram with respect to the Reedy model structure, then the
homotopy limit is per definition weak equivalent to the ordinary limit in
${\bf A}$. Since the homotopy limit in this case is the homotopy pullback, we
are done.

But that $i$ is a Kan fibration follows from [Lur21, Cor. 3.1.3.3] and hence
we can compute the above pullback in the $\infty$-category $\SS$ via the limit
of the underlying simplicial sets.

Now assume that $Y$ is an arbitrary $\infty$-category. We want to compute the
following pullback in $\SS$

${W}$${\operatorname{Fun}(\Delta^{1},Y)^{\simeq}}$${X}$${\operatorname{Fun}(\partial\Delta^{1},Y)^{\simeq},}$$\scriptstyle{i}$

where $i$ is naturally given by applying the functor $(-)^{\simeq}$ to the
restriction. If $i$ is a Kan fibration, then we can apply the argument above.
In general it may not be clear if $i$ is a Kan fibration. But in this case, we
have that the natural morphism
$F\colon\operatorname{Fun}(\Delta^{1},Y)\rightarrow\operatorname{Fun}(\partial\Delta^{1},Y)$
is an isofibration of $\infty$-categories (see [Lur21, 01F3]), meaning that it
is an inner fibration on the level of homotopy categories, we have the
following property: if $x\in h\operatorname{Fun}(\Delta^{1},Y)$ and we have an
isomorphism $u^{\prime}\colon y\xrightarrow{\sim}F(x)$ then, there exists a
$x^{\prime}\in h\operatorname{Fun}(\Delta^{1},Y)$ with an isomorphism $u\colon
x^{\prime}\rightarrow x$ such that $F(u)=u^{\prime}$. In particular, [Lur21,
Prop. 4.4.3.7] implies that $i$ is a Kan fibration.

######  Theorem 4.14.

The derived stack

$\displaystyle\textup{Perf}_{R}\colon\textup{AR}_{R}$
$\displaystyle\rightarrow\SS$ $\displaystyle A$
$\displaystyle\mapsto(\operatorname{Mod}_{A}^{\textup{perf}})^{\simeq}$

is locally geometric and locally of finite presentation.

To be more specific, we can write
$\textup{Perf}_{R}=\mathop{\textup{colim}}_{a\leq
b}\textup{Perf}_{R}^{[a,b]}$, where $\textup{Perf}_{R}^{[a,b]}$ is the moduli
space consisting of perfect modules which have Tor-amplitude concentrated in
degree $[a,b]$ and each $\textup{Perf}_{R}^{[a,b]}$ is $(b-a+1)$-geometric and
locally of finite presentation and the inclusion
$\textup{Perf}_{R}^{[a,b]}\hookrightarrow\textup{Perf}_{R}$ is a quasi-compact
open immersion. If $b-a\leq 1$ then $\textup{Perf}_{R}^{[a,b]}$ is in fact
smooth.

###### Proof.

The proof in the model categorical setting can be found in [TV07, Prop. 3.7]
and in the spectral setting in [AG14, Thm. 5.6]. The latter follows the former
with few changes for readability. We will follow the proof presented in the
latter using our setting.

We show that $\textup{Perf}^{[a,b]}$ is $n+1$-geometric, where $n=b-a$, by
induction over $n$.

For $n=0$, we are done, since then we have
$\operatorname{Proj}\simeq\textup{Perf}^{[a,a]}$ (see Lemma 2.33), which is
$1$-geometric and locally of finite presentation by Lemma 4.12.

Now let $n>0$ and assume $\textup{Perf}^{[a+1,b]}$ is $n$-geometric and
locally of finite presentation. Let $U$ be defined via the pullback diagram of
derived stacks

${U}$${\operatorname{Fun}(\Delta^{1},\operatorname{Mod}^{\operatorname{perf}})^{\simeq}}$${\textup{Perf}^{[a+1,b]}\times_{R}\textup{Perf}^{[a+1,a+1]}}$${\operatorname{Fun}(\partial\Delta^{1},\operatorname{Mod}^{\operatorname{perf}})^{\simeq}.}$$\scriptstyle{p}$

Let
$\operatorname{Spec}(A)\rightarrow\textup{Perf}^{[a+1,b]}\times_{R}\textup{Perf}^{[a+1,a+1]}$
be given by $(P,Q)$, where $P$ is a perfect $A$-module of Tor-amplitude
$[a+1,b]$ and $Q$ is the $a+1$ shift of a finite projective $A$-module, then
$p^{*}(P,Q)$ classifies morphisms between those, i.e.
$p^{*}(P,Q)\simeq\operatorname{Spec}(\operatorname{Sym}(P\otimes_{A}Q^{\vee}))$
(note that $P\otimes_{A}Q^{\vee}$ is perfect and has Tor-amplitude in
$[0,b-(a+1)]$ and thus is connective (see Lemma 2.33). Therefore $p$ is
$(-1)$-geometric and locally of finite presentation and with Lemma 4.7 and
3.38, we see that $U$ is $n$-geometric and locally of finite presentation.
Note that if $b-a\leq 1$ then $p$ is even smooth and using that
$\textup{Perf}^{[a+1,a+1]}$ is smooth, we see that $U$ is smooth.

By sending a morphism to its fiber, we get a morphism of derived stacks
$q\colon U\rightarrow\textup{Perf}^{[a,b]}$. Using Proposition 3.39 with Lemma
4.7, we see that $q$ is $n$-geometric, so it suffices to show that $q$ is also
smooth and an effective epimorphism.

That it is an effective epimorphism follows from Lemma 2.33. To check
smoothness let $\operatorname{Spec}(A)\rightarrow\textup{Perf}^{[a,b]}$ be a
morphism classified by a perfect $A$-module $P$ with Tor-amplitude in $[a,b]$.
Then $q^{-1}(P)(B)$, for some animated $A$-algebra $B$, consists of morphisms
of perfect $B$-modules $f\colon Q\rightarrow M[a+1]$ such that
$\operatorname{fib}(f)\simeq P\otimes_{A}B$, where $Q$ has Tor-amplitude in
$[a+1,b]$ and $M$ is finite projective. Since locally every finite projective
module is free of finite rank, we can decompose

$q^{-1}(P)\simeq\coprod_{m}q^{-1}(P)_{m},$

where $q^{-1}(P)_{m}$ is the substack of $q^{-1}(P)$, where the classified
morphisms have codomain given by the $a+1$ shift of free modules of rank $m$.
The stack $q^{-1}(P)_{m}$ is equivalent to the stack classifying morphisms
$A^{m}[a]\rightarrow P$, where the cofiber has Tor-amplitude in $[a+1,b]$,
which is equivalent to $A^{m}[a]\rightarrow P$ beeing a surjection on
$\pi_{a}$.

To see the equivalence of stacks let us look at $q^{-1}(P)_{m}(B)$. These are
all morphisms $f\colon Q\rightarrow B^{m}[a+1]$ such that
$\operatorname{fib}(f)\simeq P\otimes_{A}B$, again $Q$ is a perfect $B$-module
with Tor-amplitude in $[a+1,b]$. Since $\operatorname{Mod}_{A}$ is stable, we
see that $P\otimes_{A}B\rightarrow Q\rightarrow B^{m}[a+1]$ is a fiber diagram
if and only if it is a cofiber diagram and thus after shift we see that
$q^{-1}(P)_{m}(B)$ consists of morphisms $g\colon B^{m}[a]\rightarrow
P\otimes_{A}B$ such that its cofiber $\operatorname{cofib}(g)$ has Tor-
amplitude in $[a+1,b]$. By Lemma 2.33 $\operatorname{cofib}(g)$ has Tor-
amplitude in $[a,b]$. Since after tensoring $M^{m}[a]\rightarrow
P\otimes_{A}M\rightarrow\operatorname{cofib}(g)\otimes_{B}M$, where $M$ is a
discrete $\pi_{0}B$-module, we still have a cofiber sequence it is enough to
check that $\pi_{a}(g\otimes\operatorname{id}_{M})$ is a surjection. But since
$\pi_{a}(P\otimes_{A}M)=\pi_{a}(P)\otimes_{\pi_{0}A}M$ (use the degeneracy of
the Tor-spectral sequence at $(0,a)$) and the ordinary tensor product of
$\pi_{0}A$-modules preserves surjections it is enough to check that $\pi_{a}g$
is surjective. And therefore $q^{-1}(P)_{m}(B)$ consists of morphisms
$B^{m}[a]\rightarrow P\otimes_{A}B$, which are surjective on $\pi_{a}$
(obviously any morphism $B^{m}[a]\rightarrow P\otimes_{A}B$ with cofiber
having Tor-amplitude in $[a+1,b]$ is surjective on $\pi_{a}$).

By this characterization the stack $q^{-1}(P)_{m}$ is an open substack of
$F^{A}_{(P^{\vee})^{m}[a]}$ (see Lemma 4.5 for notation).

To see this, let $\operatorname{Spec}(B)\rightarrow F_{(P^{\vee})^{m}[a]}$ be
given by a morphism $\xi\colon B^{m}[a]\rightarrow P\otimes_{A}B$. Let $Z$ be
the pullback of $\operatorname{Spec}(B)$ along the inclusion
$q^{-1}(P)_{m}\hookrightarrow F_{(P^{\vee})^{m}[a]}$. In particular, for any
animated $A$-algebra $C$, we have that $Z(C)$ consists of those morphisms
$f\colon B\rightarrow C$, such that $\pi_{a}f^{\ast}\xi$ is surjective. Since
$P\otimes_{A}B$ is perfect and has Tor-amplitude in $[a,b]$ its homotopy group
$\pi_{a}(P\otimes_{A}B)$ is finitely presented (see [Lur17, Cor. 7.2.4.5]).
Therefore, being surjective is an open condition on $\pi_{0}B$ (see [GW10,
Prop. 8.4]). Further refining by principal affine opens
$D(f_{i})\subseteq\operatorname{Spec}(\pi_{0}B)$, we get an open substack
$\bigcup\operatorname{Spec}(B[f_{i}^{-1}])$ of $\operatorname{Spec}(B)$. Now a
morphism $u\colon B\rightarrow C$ is in $Z(C)$ if and only if étale locally
there is an $i$ such that $\pi_{0}u(f_{i})$ is invertible. Therefore,
$Z\simeq\bigcup\operatorname{Spec}(B[f_{i}^{-1}])$.

Since $q^{-1}(P)_{m}$ is open in $F^{A}_{(P^{\vee})^{m}[a]}$, which itself is
smooth by Lemma 4.5, we see that $q^{-1}(P)$ is smooth over $A$, which
concludes the proof.

Indeed, let $P$ be a perfect $A$-module with Tor-amplitude in $[a,b]$. Then,
by Lemma 2.33, we can find a cofiber sequence $M[a]\rightarrow P\rightarrow
Q$, where $Q$ is perfect of Tor-amplitude $[a+1,b]$ and $M$ is finite
projective. Analogous to the above, we see that $P$ has Tor-amplitude in
$[a+1,b]$.

For the open immersion part it suffices by induction to show that for all
$a<b\in\mathbb{Z}$ the inclusion
$\textup{Perf}_{R}^{[a+1,b]}\hookrightarrow\textup{Perf}_{R}^{[a,b]}$ is an
open immersion. Let $A\in\textup{AR}_{R}$ and
$\operatorname{Spec}(A)\rightarrow\textup{Perf}_{R}^{[a,b]}$ be a morphism
classified by a perfect $A$-module $P$ of Tor-amplitude $[a,b]$. By Lemma
2.33, we have a fiber sequence of $A$-modules $P\rightarrow M[a+1]\rightarrow
Q$, where $Q$ is perfect of Tor-amplitude in $[a+1,b]$ and $M$ is finite
projective. Now $P$ has Tor-amplitude in $[a+1,b]$ if and only if $M\simeq 0$.
But by Lemma 4.4, we see that the vanishing locus of $M$ is a quasi-compact
open in $\operatorname{Spec}(A)$, which concludes the proof. ∎

######  Corollary 4.15.

Let $A$ be an animated $R$-algebra and let
$\operatorname{Spec}(A)\rightarrow\textup{Perf}_{R}$ be a morphism given by a
perfect $A$-module $P$. The cotangent complex $L_{\textup{Perf}_{R},A}$ is
perfect and if $A/R$ is étale then the cotangent complex a that point is given
by

$L_{\textup{Perf}_{R},A}\simeq(P\otimes_{A}P^{\vee})^{\vee}[-1].$

###### Proof.

This is analogous to [AG14, Cor. 5.9], but for the convenience of the reader
we recall the proof.

The first assertion follows from Theorem 4.14 with Corollary 3.78.

For the second assertion let $\Omega_{P}\textup{Perf}_{R}$ denote the loop of
$\textup{Perf}_{R}$ along $\operatorname{Spec}(A)\rightarrow\textup{Perf}_{R}$
classified by a perfect $A$-module $P$, i.e. the
$\Omega_{P}\textup{Perf}_{R}\coloneqq\operatorname{Spec}(A)\times_{\textup{Perf}_{R}}\operatorname{Spec}(A)$.
We know per definition of the cotangent complex that we have a pullback
diagram

${L_{\textup{Perf},A}}$${L_{A/R}}$${L_{A/R}}$${L_{\Omega_{P}\textup{Perf}_{R},*},}$

where $*$ is the point corresponding to the canonical map
$\operatorname{Spec}(A)\rightarrow\Omega_{P}\textup{Perf}_{R}$.

Let $T\in\textup{AR}_{A}$, then
$\Omega_{P}\textup{Perf}_{R}(T)\simeq\operatorname{Equiv}_{T}(P\otimes_{A}T)$,
where $\operatorname{Equiv}_{T}(P\otimes_{A}T)$ denotes the $T$-automorphisms
of $P\otimes_{A}T$. In particular $\Omega_{P}\textup{Perf}$ is an open
substack of

$\operatorname{Hom}_{A}((P\otimes_{A}P^{\vee})^{\vee},-)$

(after using adjunctions). By Lemma 4.5, we now have

$L_{\Omega_{P}\textup{Perf}_{R},\ast}\simeq(P\otimes_{A}P^{\vee})^{\vee}.$

Therefore, if $A/R$ is étale, we have $\Sigma L_{\textup{Perf}_{R},A}\simeq
L_{\Omega_{P}\textup{Perf}_{R},*}$, whe finishing the proof. ∎

## References

  * [AG12] Dima Arinkin and Dennis Gaitsgory.  Singular support of coherent sheaves, and the geometric langlands conjecture, 2012.
  * [AG14] Benjamin Antieau and David Gepner.  Brauer groups and étale cohomology in derived algebraic geometry.  Geom. Topol., 18(2):1149–1244, 2014.
  * [Ann18] Toni Annala.  Bivariant derived algebraic cobordism, 2018.
  * [Avr99] Luchezar L. Avramov.  Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology.  Ann. of Math. (2), 150(2):455–487, 1999.
  * [CS21] Kestutis Cesnavicius and Peter Scholze.  Purity for flat cohomology, 2021.
  * [GR17] Dennis Gaitsgory and Nick Rozenblyum.  A study in derived algebraic geometry. Vol. I. Correspondences and duality, volume 221 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2017.
  * [GW10] Ulrich Görtz and Torsten Wedhorn.  Algebraic geometry I.  Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010.  Schemes with examples and exercises.
  * [HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith.  Symmetric spectra.  J. Amer. Math. Soc., 13(1):149–208, 2000.
  * [Kha18] Adeel Khan.  Lecture notes: Descent in algebraic k-theory.  https://www.preschema.com/lecture-notes/kdescent/, WS 2017/2018.
  * [Lur04] Jacob Lurie.  Derived algebraic geometry.  ProQuest LLC, Ann Arbor, MI, 2004.  Thesis (Ph.D.)–Massachusetts Institute of Technology.
  * [Lur09] Jacob Lurie.  Higher topos theory, volume 170 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2009.
  * [Lur17] Jacob Lurie.  Higher algebra.  https://www.math.ias.edu/~lurie/papers/HA.pdf, 2017.
  * [Lur18] Jacob Lurie.  Spectral algebraic geometry.  https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf, 2018.
  * [Lur21] Jacob Lurie.  Kerodon.  https://kerodon.net, 2021.
  * [MLM94] Saunders Mac Lane and Ieke Moerdijk.  Sheaves in geometry and logic.  Universitext. Springer-Verlag, New York, 1994.  A first introduction to topos theory, Corrected reprint of the 1992 edition.
  * [PV15] Mauro Porta and Gabriele Vezzosi.  Infinitesimal and square-zero extensions of simplicial algebras, 2015\.  arXiv:1310.3573.
  * [Qui67] Daniel G. Quillen.  Homotopical algebra.  Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967.
  * [Sim96] Carlos Simpson.  Algebraic (geometric) $n$-stacks, 1996.  arXiv:alg-geom/9609014.
  * [SS03] Stefan Schwede and Brooke Shipley.  Stable model categories are categories of modules.  Topology, 42(1):103–153, 2003.
  * [Sta19] The Stacks Project Authors.  Stacks Project.  https://stacks.math.columbia.edu, 2019.
  * [TV07] Bertrand Toën and Michel Vaquié.  Moduli of objects in dg-categories.  Annales scientifiques de l’École Normale Supérieure, Ser. 4, 40(3):387–444, 2007.
  * [TV08] Bertrand Toën and Gabriele Vezzosi.  Homotopical algebraic geometry. II. Geometric stacks and applications.  Mem. Amer. Math. Soc., 193(902):x+224, 2008.
  * [Wil69] N. H. Williams.  On grothendieck universes.  Compositio Mathematica, 21(1):1–3, 1969.
  * [Yay22a] Can Yaylali.  Derived $F$-zips, 2022.
  * [Yay22b] Can Yaylali.  Derived $F$-zips.  PhD thesis, Technische Universität Darmstadt, Darmstadt, 2022.