Y}$$\scriptstyle{\mathscr{L}f}$$\scriptstyle{\Delta_{f}}$$\scriptstyle{\Delta_{X}}$

which shows that $\mathscr{L}f$ is a pullback of $\Delta_{f}$. In particular,
if $f$ is representable and separated, then $\mathscr{L}f$ is a closed
immersion.

######  Proposition 4.4.

Let $f\colon X\to Y$ be an étale morphism of derived algebraic stacks such
that $f_{\mathrm{cl}}$ is stabilizer-preserving. Then $f$ is monodromy-free.

###### Proof.

Since $f$ is étale, the Cartesian diagram

implies that $X\times_{Y}X\to X$ is étale. Since the composition
$X\xrightarrow{\Delta_{f}}X\times_{Y}X\to X$ is the identity, we conclude that
$\Delta_{f}$ is étale. By diagram (4.1), since $\mathscr{L}f$ is a pullback of
$\Delta_{f}$, it must also be étale. But
$(\mathscr{L}f)_{\mathrm{cl}}=\mathscr{L}(f_{\mathrm{cl}})$ is an isomorphism,
since $f_{\mathrm{cl}}$ is stabilizer-preserving, so $\mathscr{L}f$ is an
equivalence. ∎

######  Lemma 4.5.

Suppose that $f\colon X\to Y$ is representable, separated and étale. Then
$\Delta_{f}$, hence $\mathscr{L}f$, is a closed and open immersion.

###### Proof.

By Remark 4.3, it suffices to show that an étale closed immersion $h\colon
S\to T$ is an open immersion. This reduces to the affine case since $h$ is
affine, and then to the classical case since $h$ is étale. This last case is
known, see [Stcks, Tag 025G]. ∎

We recall some facts about derived loop stacks from [BZN12]. Let $S^{1}$ be
the stackification of the presheaf that sends any $T\in\mathsf{Aff}$ to the
circle. Then it holds

$\displaystyle S^{1}\simeq B\mathbb{Z}\simeq*\bigsqcup_{*\sqcup*}*$

where $B\mathbb{Z}$ is the classifying space of the constant group scheme
$\mathbb{Z}$. Then for any derived stack $X$ it holds

$\displaystyle\mathscr{L}X\simeq\underline{\operatorname{Map}}(S^{1},X)$

For any $f\colon X\to Y$ it follows that $f^{*}\mathscr{L}Y$ is the stack over
$X$ which sends $T\to X$ to the space of commutative diagram

${T}$${X}$${S^{1}\times T}$${Y}$

which we think of as loops on $Y$ based at $f$. The morphism $\mathscr{L}X\to
f^{*}\mathscr{L}Y$ then sends $T\times S^{1}\to X$ to the loop $T\times
S^{1}\to X\to Y$, which is based at $f$. In particular, $\mathscr{L}f$ is a
morphism of derived group stacks. This also explains the term “monodromy-
free”.

######  Notation 4.6.

For a morphism $f\colon x\to y$ in an $\infty$-category $\mathsf{C}$, we write
$\mathsf{C}_{x/\cdot/y}$ for the double slice-category
$(\mathsf{C}_{/y})_{f/}$, so that an object of $\mathsf{C}_{x/\cdot/y}$ is a
factorization $x\to w\to y$ of $f$.

######  Remark 4.7.

Let $T$ be a derived stack. Throughout, we will write $B_{T}$ for the functor
$B_{T}\colon\operatorname{Grp}(\mathsf{Stk}_{/T})\to\mathsf{Stk}_{T/}$ which
sends a derived group stack $G$ over $T$ to the colimit of the bar-
construction $B(T,G)$ in $\mathsf{Stk}_{T}$. In analogy to the loop space-
suspension adjunction from topology, there is an adjunction

$\displaystyle
B_{T}:\operatorname{Grp}(\mathsf{Stk}_{T})\rightleftarrows\mathsf{Stk}_{T/}:\Omega_{T}$

such that $\Omega_{T}(T\to X)\simeq\mathscr{L}X\times_{X}T$, and with $B_{T}$
fully faithful, which we now explain.

We have a sequence of adjunctions

$\displaystyle\operatorname{Grp}(\mathsf{Stk}_{T})\rightleftarrows\operatorname{Epi}_{T/}(\mathsf{Stk}_{T})\rightleftarrows\mathsf{Stk}_{T/\cdot/T}\rightleftarrows\mathsf{Stk}_{T/}$

where: the first adjunction on the left is the adjoint equivalence from
Proposition 2.3; the second adjunction in the middle has left adjoint the
inclusion, and as right adjoint the functor which sends $g\colon T\to Y$ over
$T$ to the $\infty$-image $T\to\mathrm{im}(g)$; and the third adjunction has
left adjoint the forgetful functor, and as right adjoint the functor which
sends $T\to X$ to $T\to X\times T$.

The composition of all of these left adjoints is $B_{T}$. Write $\Omega_{T}$
for the right adjoint of $B_{T}$. The for $f\colon T\to X$ given, it holds
that

$\Omega_{T}(X)\simeq T\times_{\mathrm{im}(f,\mathrm{id}_{T})}T\simeq
T\times_{X\times T}T$

where the second equivalence follow from the fact that
$\mathrm{im}(f,\mathrm{id}_{T})\to X\times T$ is a monomorphism. Then the
first claim, $\Omega_{T}(T\to X)\simeq\mathscr{L}X\times_{X}T$, follows from
the Cartesian diagram

${\mathscr{L}X\times_{X}T}$${T}$${X}$${T}$${X\times T}$${X\times X}$

and the second claim follows from the familiar fact that $G\simeq
T\times_{BG}T$ for any $\infty$-group $G$ over $T$.

###  4.2. Twisted forms

We introduce $\infty$-bundles which are twisted versions of principal
$G$-bundles, for a given reductive group $G$. This will be used to describe
$[U/G]^{{\mathrm{max}}}$ in the cases relevant for the proof of Theorem 4.14.

######  Definition 4.8.

For $f\colon T\to BG$, say classified by the $G$-bundle $P\to T$, define the
inner form $\widetilde{G}_{f}$ of $G$ via the pullback

(4.2) ${\widetilde{G}_{f}}$${\mathscr{L}(BG)}$${T}$${BG.}$

Observe that $\widetilde{G}_{f}$ is a pullback of an $\infty$-group, hence is
an $\infty$-group itself. In the terminology from [NSS15, Prop. 4.10] it holds
that $\widetilde{G}_{f}$ is the $G_{\mathrm{ad}}$-fiber $\infty$-bundle
$P\times_{G}G_{\mathrm{ad}}$, since $\mathscr{L}(BG)=[G_{\mathrm{ad}}/G]$ by
definition. Clearly, we have $\widetilde{G}_{f}\simeq\Omega_{T}(BG)$.

Recall that the identity component $G^{0}$ is an open and closed, normal
subgroup of $G$, such that $G/G^{0}$ is finite, hence reductive. Since
$G^{0}\subset G$ is normal, the adjoint action on $G_{\mathrm{ad}}$ restricts
to the identity component, and we write $G^{0}_{\mathrm{ad}}$ for the
corresponding $G$-object. Then we define the _twisted form_
$\widetilde{G}_{f}^{0}$ of $G$ via the pullback

${\widetilde{G}_{f}^{0}}$${{[G^{0}_{\mathrm{ad}}/G]}}$${T}$${BG}$

The group structure on $G^{0}$ induces a group structure on
$\widetilde{G}_{f}^{0}$. The open and closed inclusion $G^{0}\to G$ induces a
morphism $\widetilde{G}_{f}^{0}\to\widetilde{G}_{f}$, which exhibits
$\widetilde{G}_{f}^{0}$ as open and closed subgroup of $\widetilde{G}_{f}$.

######  Lemma 4.9.

Let $G$ be an algebraic group of dimension $d$, and $\widetilde{G}_{f}$ an
inner form of $G$, classified by $f\colon T\to BG$ which has corresponding
$G$-bundle $P$.

  1. (i)

$\widetilde{G}_{f}$ and $\widetilde{G}_{f}^{0}$ are derived algebraic group
schemes that are smooth of dimension $d$ over $T$, and
$\widetilde{G}_{f}^{0}\to T$ has connected fibers.

  2. (ii)

The canonical map $B_{T}(\widetilde{G}_{f}^{0})\to T\times_{B(G/G^{0})}BG$ is
an equivalence.

###### Proof.

(i) Since $\mathscr{L}(BG)\simeq[G_{\mathrm{ad}}/G]$, it holds that
$\mathscr{L}(BG)$ is fiberwise an algebraic group scheme, smooth and of
dimension $d$ over $BG$, so by diagram 4.2 the same is true for
$\widetilde{G}_{f}\to T$. A similar argument goes for $\widetilde{G}_{f}^{0}$.

Pulling back $\widetilde{G}_{f}^{0}$ along the $G$-bundle $P\to T$, we obtain
$q\colon P\times G^{0}_{\mathrm{ad}}\to P$. Since $G^{0}$ is connected, $q$
has connected fibers.

(ii) Note that the bar-construction $B(*,G^{0})$ is $G$-equivariant, where $G$
acts on $B(*,G^{0})_{n}=(G^{0})^{\times n}$ diagonally and by conjugation.
Since $[-/G]$ commutes with colimits, we thus obtain a $G$-action on $BG^{0}$,
written $BG^{0}_{\mathrm{ad}}$, such that $[G^{0}_{\mathrm{ad}}/G]\to BG$
factors through $[BG_{\mathrm{ad}}^{0}/G]\to BG$.

Consider the diagram

${\widetilde{G}_{f}^{0}}$${{[G^{0}_{\mathrm{ad}}/G]}}$${B_{T}(\widetilde{G}_{f}^{0})}$${{[BG^{0}_{\mathrm{ad}}/G]}}$${BG}$${T}$${BG}$${B(G/G^{0})}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$${\scriptstyle(\alpha)}$${\scriptstyle(\gamma)}$${\scriptstyle(\delta)}$

By pulling back along $*\to BG$, we see that $(\delta)$ is Cartesian. Since
geometric realizations commute with pullbacks, also $(\alpha)$ is Cartesian.
Now because the rectangle $(\alpha)+(\gamma)$ is Cartesian and the morphisms
$\varphi,\psi$ are effective epimorphisms, it follows that the square
$(\gamma)$ is Cartesian as well, which implies the claim. ∎

###  4.3. A derived Luna étale slice theorem

We give a derived version of the étale local structure of algebraic stacks in
terms of quotient stacks.

######  Notation 4.10.

For $X\in\mathsf{Stk}$, write $\mathsf{Aff}^{\mathrm{\acute{e}t}}_{X}$ for the
$\infty$-category of affine étale morphisms $T\to X$.

Write $\lvert X\rvert$ for the equivalence class of morphisms
$x\colon\operatorname{Spec}k\to X$ with $k$ a field, where $x$ is equivalent
to $x^{\prime}\colon\operatorname{Spec}k^{\prime}\to X$ if there is a common
field extension $k,k^{\prime}\subset K$ such that
$x_{\mid\operatorname{Spec}K}\simeq x^{\prime}_{\mid\operatorname{Spec}K}$.
Consider the set of points, written $\lvert X\rvert$, as a topological space
by declaring $U_{0}\subset\lvert X\rvert$ to be open if there is an open
immersion $U\to X$ such such that $\lvert U\rvert\to\lvert X\rvert$ has image
$U_{0}$. Since open immersions are monomorphisms, passing from an open
immersion $U\to X$ to the corresponding open subset $\lvert
U\rvert\subset\lvert X\rvert$ constitutes an equivalence of partially ordered
sets. The induced map $\lvert X_{\mathrm{cl}}\rvert\to\lvert X\rvert$ is a
homeomorphism, as the following result shows.

######  Lemma 4.11 (Topological invariance).

Let $X$ be a quasi-compact derived algebraic stack. Then pulling back along
$X_{\mathrm{cl}}\to X$ induces an equivalence
$\mathsf{Aff}_{X}^{\mathrm{\acute{e}t}}\simeq\mathsf{Aff}_{X_{\mathrm{cl}}}^{\mathrm{\acute{e}t}}$.

###### Proof.

The affine case can be found in [GR17, Lem. 2.15].

For the general case, let $U\to X$ be a smooth atlas, where $U$ is a derived
scheme. We can assume that $U$ is affine since $X$ is quasi-compact. Then the
commutative diagram

${U_{\mathrm{cl}}}$${U}$${X_{\mathrm{cl}}}$${X}$

is Cartesian. Now the claim follows from the affine case by descent. ∎

Suppose that $X\in\mathsf{Stk}$ is such that $X_{\mathrm{cl}}$ admits a good
moduli space $X_{\mathrm{cl}}\to Y$. We say that a morphism $f\colon[U/G]\to
X$ is strongly étale if $U\in\mathsf{Aff}^{G}$, $f$ is étale and
$f_{\mathrm{cl}}$ fits in a Cartesian diagram

Observe that $f_{\mathrm{cl}}$ then has to be stabilizer-preserving and
therefore, by Proposition 4.4, also monodromy-free. Moreover, $f$ is
representable and separated by Example 4.2, since $U_{\mathrm{cl}}/\\!\\!/G$
is affine, hence separated. We record these observations in the following
proposition.

######  Proposition 4.12.

Let $X$ be a derived algebraic stack such that $X_{\mathrm{cl}}$ admits a good
moduli space. Any strongly étale morphism $f\colon[U/G]\to X$ is also
representable, separated and monodromy-free.

We now prove a derived Luna étale slice result.

######  Proposition 4.13.

Let $X$ be a derived algebraic stack such that $X_{\mathrm{cl}}$ is Noetherian
and admits a good moduli space. Then $X$ is covered by derived stacks of the
form $h\colon[U/G]\to X$, where $U\in\mathsf{Aff}^{G}$ is such that
$U^{G}\not=\emptyset$, $G$ is reductive, and $h$ is affine and strongly étale
(hence monodromy-free by Remark 4.4).

###### Proof.

Since $X_{\mathrm{cl}}$ is Noetherian, the set of closed points is dense in
$\lvert X_{\mathrm{cl}}\rvert$. By the Luna étale slice theorem for classical
algebraic stacks from [ER21, Thm. A.1], we therefore have a cover of
$X_{\mathrm{cl}}$ by stacks of the form
$h_{\mathrm{cl}}\colon[U_{\mathrm{cl}}/G]\to X_{\mathrm{cl}}$, where
$U_{\mathrm{cl}}$ is a classical $G$-scheme such that
$U_{\mathrm{cl}}^{G}\not=\emptyset$, the map $h_{\mathrm{cl}}$ is étale,
affine and stabilizer-preserving, and $G$ is a stabilizer of some closed point
in the image of $U_{\mathrm{cl}}\to X_{\mathrm{cl}}$.

Since $X_{\mathrm{cl}}$ is Noetherian, it is quasi-compact. By Lemma 4.11, we
thus have a Cartesian diagram

${{[U_{\mathrm{cl}}/G]}}$${V}$${X_{\mathrm{cl}}}$${X}$$\scriptstyle{h_{\mathrm{cl}}}$$\scriptstyle{f}$$\scriptstyle{h}$

where the vertical maps are affine and étale, and $f_{\mathrm{cl}}$ is an
isomorphism. By [Hal20, Lem. 4.2.5], the derived stack $V$ is of the form
$[U/G]$, for some derived $G$-scheme $U$, and $f$ is induced by a
$G$-equivariant map $f^{\prime}\colon U_{\mathrm{cl}}\to U$ such that
$f^{\prime}_{\mathrm{cl}}$ is an isomorphism.

The collection of morphisms $h\colon[U/G]\to X$, indexed by the elements
$h_{\mathrm{cl}}\colon[U_{\mathrm{cl}}/G]\to X_{\mathrm{cl}}$ appearing in the
cover coming from the Luna étale slice theorem, is an étale cover, because
$\lvert X_{\mathrm{cl}}\rvert\simeq\lvert X\rvert$. ∎

###  4.4. The derived locus $X^{{\mathrm{max}}}$ of maximal-dimensional
stabilizers

Throughout this subsection, let $X$ be a derived algebraic stack whose
classical truncation $X_{\mathrm{cl}}$ admits a good moduli space $Y$. We
moreover assume that $X_{\mathrm{cl}}$ is Noetherian. Recall that all stacks
have affine diagonal by assumption. In what follows, $G^{0}\subseteq G$
denotes the identity component of any (possibly disconnected) algebraic group
$G$.

The goal is to show the following.

######  Theorem 4.14.

Let $d$ be the maximal dimension of stabilizers of points of $X$. Then there
exists a canonical closed immersion $X^{{\mathrm{max}}}\to X$ with the
following property: For any affine, strongly étale morphism $[U/G]\to X$,
where $U\in\mathsf{Aff}^{G}$ and $G$ is reductive with $\dim G=d$, there
exists a Cartesian square

Then $(-)_{\mathrm{cl}}$ commutes with $(-)^{\mathrm{max}}$, and
$X_{\mathrm{cl}}^{\mathrm{max}}$ is equivalent to the classical locus of
points of $X_{\mathrm{cl}}$ with maximal stabilizer dimension defined in
[ER21, Prop. C.5].

Since $\lvert X_{\mathrm{cl}}\rvert\simeq\lvert X\rvert$, the map
$x\mapsto\dim G_{x}$ is upper semi-continuous on $\lvert X\rvert$ by the same
argument as for the classical case, see [ER21]. We let $X^{\leq d}\to X$ be
the open immersion corresponding to the open subset of $\lvert X\rvert$
consisting of points $x$ with $\dim G_{x}\leq d$. Observe, if $X$ is classical
then so is $X^{\leq d}$, and in general it holds $(X^{\leq
d})_{\mathrm{cl}}\simeq(X_{\mathrm{cl}})^{\leq d}$.

######  Definition 4.15.

For $d\in\mathbb{N}$, let $X^{d}$ be the stack over $X$ such that $X^{d}(T\to
X)$ is the space of closed derived subgroup schemes $H^{0}\to
T\times_{X}\mathscr{L}X$ which are smooth over $T$ with connected fibers of
dimension $d$. Naturality in $T$ is defined by pullback. If $d$ is the maximal
dimension of stabilizers of $X$, we define $X^{{\mathrm{max}}}\coloneqq
X^{d}$.

######  Remark 4.16.

Let $T\in\mathsf{Stk}_{X}$ be of the form $[V/G]$ and $X$ of the form $[U/G]$,
where $U,V$ are affine derived $G$-schemes, and $G$ is reductive. Then it
holds that $X^{d}(T\to[U/G])$ is the space of derived closed subgroup stacks
$H^{0}\to T\times_{X}\mathscr{L}X$, such that after pulling back along
$q\colon V\to[V/G]$ it holds that $q^{*}H^{0}$ is a derived closed subgroup
scheme smooth over $V$ with connected fibers of dimension $d$.

In order to prove several properties satisfied by $X^{{\mathrm{max}}}$, we
will need a few auxiliary results.

######  Proposition 4.17.

Suppose that $X$ is of the form $[U/G]$ for an affine derived scheme $U$ and a
reductive group $G$. Then the derived stack $[U^{G^{0}}/G]$ has functor of
points

$\displaystyle[U^{G^{0}}/G](f\colon T\to
X)\simeq\mathsf{Stk}_{T/\cdot/BG}(T\times_{B(G/G^{0})}BG,X),$

###### Proof.

Write $\pi\colon BG\to B(G/G^{0})$ and $p\colon X=[U/G]\to BG$ for the
projections. Let $f\colon T\to X$ be given, and let $\pi_{!}$ be the left
adjoint of $\pi^{*}$, the functor which sends $S\to BG$ to $S\to B(G/G^{0})$.
Then it holds, using the universal property of the Weil restriction,

$\displaystyle\mathsf{Stk}_{X}(T,\pi^{*}\pi_{*}X)$
$\displaystyle\simeq\mathsf{Stk}_{BG}(T,\pi^{*}\pi_{*}X)\times_{\mathsf{Stk}_{BG}(T,X)}\\{f\\}$
$\displaystyle\simeq\mathsf{Stk}_{BG}(\pi^{*}\pi_{!}T,X)\times_{\mathsf{Stk}_{BG}(T,X)}\\{f\\}$
$\displaystyle\simeq\mathsf{Stk}_{T/\cdot/BG}(T\times_{B(G/G^{0})}BG,X).$

Let $p\colon X\to BG$ and $f\colon T\to X$ be given. Then we have a
commutative diagram

${f^{*}\mathscr{L}(X)}$${\mathscr{L}(BG)}$${T}$${BG}$$\scriptstyle{pf}$

and thus a morphism $h\colon f^{*}\mathscr{L}(X)\to\widetilde{G}_{pf}$ over
$T$ by definition of inner forms. In what follows, we will consider
$f^{*}\mathscr{L}(X)$ as derived stack over $\widetilde{G}_{pf}$ via $h$.

######  Proposition 4.18.

Suppose that $X$ is of the form $[U/G]$ for an affine derived scheme $U$ and a
reductive group $G$ of dimension $d$, write $p\colon[U/G]\to BG$ for the
projection. Then the derived stack $X^{d}$ has functor of points

$\displaystyle X^{d}(f\colon T\to
X)\simeq\operatorname{Grp}(\mathsf{Stk}_{\widetilde{G}_{pf}})(\widetilde{G}^{0}_{pf},f^{*}\mathscr{L}X).$

###### Proof.

Recall that $\widetilde{G}_{pf}^{0}\to\widetilde{G}_{pf}$ is a closed and open
subgroup which is smooth with connected fibers over $T$. Since
$f^{*}\mathscr{L}X\to\widetilde{G}_{pf}$ is a closed subgroup, it follows that
any homomorphism $\widetilde{G}_{pf}^{0}\to f^{*}\mathscr{L}X$ is a closed
immersion. We thus obtain a natural map

(4.3)
$\displaystyle\varphi\colon\operatorname{Grp}(\mathsf{Stk}_{\widetilde{G}_{pf}})(\widetilde{G}^{0}_{pf},f^{*}\mathscr{L}X)\to
X^{d}(f\colon T\to X)$

which we claim is an equivalence.

Let $H^{0}\to f^{\ast}\mathscr{L}X$ be any closed derived subgroup scheme,
smooth over $T$ and with connected fibers of dimension $d$. Consider the
diagram

Since $\beta$ is a closed and open immersion, the existence of the dotted
arrow can be checked on classical truncations, on which this holds by [ER21,
Appendix C], and on which moreover it is an isomorphism [ibid.]. Since $\beta$
is a monomorphism, the arrow $\alpha$ is canonically unique and satisfies that
$\alpha_{\mathrm{cl}}$ is an isomorphism. Since $H^{0},\widetilde{G}_{pf}^{0}$
are flat over $T$, also $\alpha$ is invertible. Clearly, the inverse of
$\alpha$ induces a morphism of derived group stacks $\widetilde{G}^{0}_{pf}\to
f^{*}\mathscr{L}X$ over $\widetilde{G}_{pf}$, which shows that $\varphi$ is
surjective on $\pi_{0}$.

By our argument above, the fiber of $\varphi$ over any point is the space of
automorphisms of the derived group stack $\widetilde{G}^{0}_{pf}$ over
$\widetilde{G}_{pf}$, which is contractible, since the morphism
$\widetilde{G}^{0}_{pf}\to\widetilde{G}_{pf}$ is a monomorphism. This
concludes the proof. ∎

Theorem 4.14 will now be an immediate consequence of the following.

######  Proposition 4.19.

Let $X,Y$ be a derived algebraic stacks such that
$X_{\mathrm{cl}},Y_{\mathrm{cl}}$ are Noetherian and admit good moduli spaces.

  1. (i)

$\lvert X^{d}\rvert\to\lvert X\rvert$ is injective, and identifies $\lvert
X^{d}\rvert$ with the subset $\lvert X\rvert^{d}\subset\lvert X\rvert$ of
points with stabilizer dimension $d$.

  2. (ii)

The construction $(-)^{d}$ is functorial in representable and separated
morphisms.

  3. (iii)

If $g\colon X\to Y$ is representable, separated and étale, then the natural
map $X^{d}\to X\times_{Y}Y^{d}$ is an equivalence.

  4. (iv)

Suppose that $X$ is of the form $[U/G]$ for an affine derived scheme $U$ and a
$d$-dimensional reductive group $G$. If $G^{0}$ acts trivially on $U$, then
$[U/G]^{d}\to[U/G]$ is an equivalence.

  5. (v)

In general, for $X=[U/G]$ as in the previous point but with possibly non-
trivial $G^{0}$-action on $U$, the morphism $[U^{G^{0}}/G]\to[U/G]$ induces an
equivalence $[U^{G^{0}}/G]\simeq[U/G]^{d}$.

  6. (vi)

The derived stack $X^{d}$ is algebraic.

  7. (vii)

The canonical map $(X^{d})_{\mathrm{cl}}\to X_{\mathrm{cl}}$ is the closed
substack described in [ER21, Appendix C].

  8. (viii)

$X^{d}\to X$ factors through $X^{\leq d}$ via a (necessarily unique) closed
immersion $X^{d}\to X^{\leq d}$.

###### Proof.

(i) Let $x\colon\operatorname{Spec}k\to X$ be given. Then

$\displaystyle(G_{x})^{0}_{\mathrm{cl}}\subset
G_{x}=\mathscr{L}(X)\times_{X}\operatorname{Spec}k$

is the unique closed subgroup of $G_{x}$ of dimension $d^{\prime}\coloneqq\dim
G_{x}$ which is smooth and connected. From this, (i) follows.

(ii) Let $g\colon X\to Y$ be representable and separated. For any $T\to X$, we
have a Cartesian square

(4.4)
${T\times_{X}\mathscr{L}X}$${\mathscr{L}X}$${T\times_{Y}\mathscr{L}Y}$${X\times_{Y}\mathscr{L}Y,}$$\scriptstyle{\mathscr{L}g}$

hence $T\times_{X}\mathscr{L}X\to T\times_{Y}\mathscr{L}Y$ is a closed
immersion by Remark 4.3. The claim follows.

(iii) Assume moreover that $g\colon X\to Y$ is étale. Then
$\mathscr{L}g\colon\mathscr{L}X\to X\times_{Y}\mathscr{L}Y$ is a closed and
open immersion by Lemma 4.5. From the diagram (4.4), it follows that
$\varphi\colon T\times_{X}\mathscr{L}X\to T\times_{Y}\mathscr{L}Y$ is an open
and closed immersion, hence a monomorphism, for any $f\colon T\to X$. Let a
$T$-point of $Y^{d}$ be classified by $\alpha\colon H^{0}_{Y}\to
T\times_{Y}\mathscr{L}Y$, and let $H^{0}_{X}\to T\times_{X}\mathscr{L}X$ be
the pullback of $\alpha$ along $\varphi$. Then the projection $\gamma\colon
H^{0}_{X}\to H^{0}_{Y}$ is an open and closed immersion over $T$, which is an
equivalence since $H^{0}_{Y}\to T$ has connected fibers. We have thus
described a $T$-point of $X^{d}$ such that composing with $\varphi$ recovers
$\alpha$.

(iv) Let $X=[U/G]$ with $U$ affine and $\dim G=d$, and suppose that $G^{0}$
acts trivially on $U$. Let $S_{U}$ be the derived stabilizer group scheme
$(G\times U)\times_{U\times U}U$ as in Example 4.1. Since $G^{0}$ acts
trivially on $U$, we have a natural map $G^{0}\times U\to S_{U}$ over $G\times
U$, and thus morphisms

$\displaystyle[G^{0}\times
U/G]\xrightarrow{\beta}\mathscr{L}X\xrightarrow{\alpha}[U/G]$

over $[U/G]$. Since $*\to BG$ is smooth, so is $U\to[U/G]$. Likewise,
$G^{0}\times U\to U$ is smooth, as it is the pullback of the smooth map
$G^{0}\to*$ along $U\to*$. It follows that $\alpha\beta$ is smooth. Clearly,
the fibers of $\alpha\beta$ are connected and of dimension $d$. Moreover,
since $U$ is affine, the morphism $S_{U}\to G\times U$ is separated, hence
$G^{0}\times U\to S_{U}$ is a closed immersion, since $G^{0}\times U\to
G\times U$ is. By Remark 4.16, we therefore have a morphism
$\varphi\colon[U/G]\to[U/G]^{d}$ over $[U/G]$. To see that it is an
equivalence, write $\chi\colon[U/G]^{d}\to[U/G]$ for the natural map. By
construction, it holds $\chi\varphi\simeq\mathrm{id}_{[U/G]}$.

To show that also $\varphi\chi\simeq\mathrm{id}_{[U/G]^{d}}$, let
$F^{0}\to\chi^{*}\mathscr{L}X$ be the derived closed subgroup stack
classifying the identity on $[U/G]^{d}$ as stacks over $[U/G]$. Then $F^{0}$
is universal in the following sense: for a $T$-point $f\in X(T)$, we have that
any $f^{\prime}\in X^{d}(T)$, say classified by $H^{0}\to f^{*}\mathscr{L}X$,
trivially satisfies $\chi f^{\prime}\simeq f$, and hence that
$H^{0}\simeq{f^{\prime}}^{*}F^{0}$ by definition of $(-)^{d}$. In particular,
for $\psi\colon[U/G]^{d}\to[U/G]^{d}$ any map over $[U/G]$, since
$\chi\psi\simeq\chi$, it follows that $\psi$ is classified by $F^{0}$ as well,
hence $\psi\simeq\mathrm{id}_{[U/G]^{d}}$. Since $\chi\varphi\chi\simeq\chi$,
the claim follows by applying this observation to $\psi\coloneqq\varphi\chi$.

(v) Let still $X=[U/G]$, put $Y\coloneqq[U^{G^{0}}/G]$, and write $p\colon
X\to BG$ for the projection. Since $Y\to X$ is representable and separated, we
have a natural map $Y^{d}\to X^{d}$ by (ii). Also, since $G^{0}$ acts
trivially on $U^{G^{0}}$, by (iv) the structure map $Y^{d}\to Y$ is an
equivalence. We therefore get a natural morphism $h\colon Y\to X^{d}$ over
$X$, which we claim to be an equivalence.

For any map $f\colon T\to X$, using Propositions 4.17 and 4.18, the morphism
$h(T):Y(T)\to X^{d}(T)$ is equivalent to

(4.5)
$\displaystyle\mathsf{Stk}_{T/\cdot/BG}(T\times_{B(G/G^{0})}BG,X)\xrightarrow{h(T)}\operatorname{Grp}(\mathsf{Stk}_{\widetilde{G}_{pf}})(\widetilde{G}_{pf}^{0},f^{\ast}\mathscr{L}X).$

We need to check that this morphism is an equivalence. To this end, observe
that

$X^{d}(T)\simeq\operatorname{Grp}(\mathsf{Stk}_{T})_{/\Omega_{T}(BG)}(\widetilde{G}^{0}_{pf},\Omega_{T}(X))\simeq\mathsf{Stk}_{T/\cdot/BG}(B_{T}(\widetilde{G}^{0}_{pf}),X)$

by the adjunction $B_{T}\dashv\Omega_{T}$ from Remark 4.7. Since
$B_{T}(\widetilde{G}_{f}^{0})\simeq T\times_{B(G/G^{0})}BG$ by Lemma 4.9, the
claim follows.

(vi) By the étale local structure of $X$ (Propositions 4.13 and 4.12), we can
reduce to the case where $X=[U/G]$ by (iii). The statement then follows from
(v).

(vii) Let $f\colon T\to X$ be given with $T$ a classical scheme, so that we
have a unique factorization $f\colon T\to X_{\mathrm{cl}}\to X$. The space
$(X^{d})_{\mathrm{cl}}(T)$ thus classifies derived closed subgroup schemes
$H^{0}\to T\times_{X}\mathscr{L}X$ over $T$. Since $T$ is classical and
$H^{0}\to T$ is smooth, we must have $H^{0}=(H^{0})_{\mathrm{cl}}$, and
$H^{0}\to T\times_{X}\mathscr{L}X$ is naturally equivalent to the closed
embedding of group schemes
$H^{0}\to(T\times_{X}\mathscr{L}X)_{\mathrm{cl}}=T\times_{X_{\mathrm{cl}}}(\mathscr{L}X)_{\mathrm{cl}}=T\times_{X_{\mathrm{cl}}}\mathscr{L}X_{\mathrm{cl}}$.
These are the $T$-points of $(X_{\mathrm{cl}})^{d}$ by [ER21, Prop. C.5].

(viii) Since $X^{\leq d}\to X$ is an open immersion, the factorization exists
uniquely by (i). To check that the induced map $X^{d}\to X^{\leq d}$ is a
closed immersion, we reduce to the classical case by (vii). By the argument
for (vii), the statement then follows from [ER21, Prop. C.5]. ∎

###### Proof of Theorem 4.14.

Everything follows from Proposition 4.13 and Proposition 4.19. ∎

The functoriality of $(-)^{d}$ shown in Proposition 4.19, together with the
functoriality of $(-)^{{\mathrm{max}}}$ in the classical case from [ER21,
Appendix C], has the following immediate consequence:

######  Corollary 4.20.

If $X^{\prime}\to X$ is étale, representable and separated, then
$X^{{\mathrm{max}}}\times_{X}X^{\prime}\simeq{X^{\prime}}^{{\mathrm{max}}}$,
provided that $X$ and $X^{\prime}$ have the same maximal stabilizer dimension.
If not, then the maximal stabilizer dimension of $X^{\prime}$ is strictly
smaller than the one for $X$, so that
$X^{{\mathrm{max}}}\times_{X}X^{\prime}\simeq\emptyset$.

##  5\. Intrinsic blow-ups as equivariant derived blow-ups

Let $U$ be a classical scheme with an action by a reductive group $G$. The
main aim of this section is to prove that the $G$-intrinsic blow-up of $U$,
introduced in [KLS17], has a natural derived enhancement which is obtained by
a corresponding derived blow-up construction, namely blowing up $U$ along its
derived fixed locus $U^{G}$.

To this end, we begin with a review of intrinsic blow-ups and then proceed to
establish the main result comparing intrinsic and derived blow-ups by using
standard equivariant local models for $U$. For simplicity, throughout, we
initially assume that $G$ is connected and then explain how to adapt the
constructions in the non-connected case. We conclude with some remarks and a
few instructive examples.

The operations of intrinsic blow-up and its derived enhancement will be
fundamental building blocks in our derived stabilizer reduction procedure in
Section 6.

###  5.1. Background on intrinsic blow-ups

Let $U$ be a classical affine scheme with an action of a reductive group $G$.
Assume for simplicity that $G$ is connected.

Take an equivariant closed embedding $U\to V$ into a smooth $G$-scheme $V$ and
let $I$ be the ideal defining $U$. Since $U\subseteq V$ is $G$-equivariant,
$G$ acts on $I$ and we have a decomposition $I=I^{\mathrm{fix}}\oplus
I^{\mathrm{mv}}$ into the fixed part of $I$ and its complement as
$G$-representations.

Let $V^{G}$ be the fixed point locus of $G$ inside $V$, defined by the ideal
generated by $\mathcal{O}_{V}^{\mathrm{mv}}$, and let
$\pi\colon\mathrm{bl}_{G}(V)\to V$ be the blow-up of $V$ along $V^{G}$. Let
$E\subseteq\mathrm{bl}_{G}(V)$ be its exceptional divisor and
$\xi\in\mathcal{O}_{\mathrm{bl}_{G}(V)}(E)$ the tautological defining equation
of $E$.

$G$-equivariance implies that
$\pi^{-1}(I^{\mathrm{mv}})\subseteq\xi\cdot\mathcal{O}_{\mathrm{bl}_{G}(V)}(-E)$
(cf. [KLS17, Section 2.2]), and consequently,

$\xi^{-1}\pi^{-1}(I^{\mathrm{mv}})\subseteq\mathcal{O}_{\mathrm{bl}_{G}(V)}(-E)\subseteq\mathcal{O}_{\mathrm{bl}_{G}(V)}.$

We define $I^{{\mathrm{intr}}}\subseteq\mathcal{O}_{\mathrm{bl}_{G}(V)}$ to be
the ideal sheaf

(5.1) $\displaystyle I^{{\mathrm{intr}}}=\text{ideal sheaf generated by
}\pi^{-1}(I^{\mathrm{fix}})\text{ and }\xi^{-1}\pi^{-1}(I^{\mathrm{mv}}).$

######  Definition 5.1 ([KLS17]).

The (classical) _$G$ -intrinsic blow-up_ of $U$ is the classical subscheme
$U^{{\mathrm{intr}}}_{\mathrm{cl}}\subseteq\mathrm{bl}_{G}(V)$ defined by the
ideal $I^{{\mathrm{intr}}}$. Observe that
$U^{{\mathrm{intr}}}_{\mathrm{cl}}\subset\mathrm{bl}_{G}(V)$ is $G$-invariant.
For $X=[U/G]$, we write
$X^{{\mathrm{intr}}}_{\mathrm{cl}}\coloneqq[U^{{\mathrm{intr}}}/G]$.

In [KLS17], it is proved by hands-on computation that
$U^{{\mathrm{intr}}}_{\mathrm{cl}}$ is intrinsic to the $G$-scheme $U$.
Moreover, by [KLS17, Sav20], the construction globalizes to any scheme $U$
with a $G$-action and a Zariski open cover by $G$-invariant subschemes.

######  Example 5.2.

Suppose that $G=\mathbb{G}_{m}$ is the one-dimensional torus acting on the
affine plane $V=\mathbb{C}^{2}_{x,y}$ with weights $1$ and $-1$ on the
coordinates $x$ and $y$ respectively, and $U\subseteq V$ is the closed
$G$-invariant subscheme cut out by the ideal $I=(x^{2}y,xy^{2})$.

$V^{{\mathrm{intr}}}_{\mathrm{cl}}$ is the blow-up of $V$ along the fixed
locus $V^{G}=\\{0\\}$ cut out by the ideal
$\mathcal{O}_{V}^{\mathrm{mv}}=(x,y)$. Writing $u,v$ for the homogeneous
coordinates on the exceptional divisor $\mathbb{P}^{1}$, we have that $G$ acts
linearly on $u,v$ with weights $1,-1$ respectively, and the blow-down map
$V^{{\mathrm{intr}}}_{\mathrm{cl}}\to V$ is locally given on coordinates by
$x\mapsto\xi u,y\mapsto\xi v$.

Since $I^{\mathrm{mv}}=(x^{2}y,xy^{2})$, the closed subscheme
$U^{{\mathrm{intr}}}_{\mathrm{cl}}\subseteq V^{{\mathrm{intr}}}_{\mathrm{cl}}$
is locally cut out by the ideal
$I^{{\mathrm{intr}}}=(\xi^{2}u^{2}v,\xi^{2}uv^{2})$.

We now briefly explain how one can proceed if $G$ is not connected.

Suppose that $U\to V$ is a $G$-equivariant embedding into a smooth $G$-scheme
$V$. Let, as before, $I$ be the ideal of $U$ in $V$. Let $G^{0}$ be the
connected component of the identity. This is a normal, connected subgroup of
$G$ of finite index. Let $I=I^{\mathrm{fix}}\oplus I^{\mathrm{mv}}$ be the
decomposition of $I$ into fixed and moving parts with respect to the action of
$G^{0}$. Using the normality of $G^{0}$, we see that the fixed locus
$V^{G^{0}}$ is a closed, smooth $G$-invariant subscheme of $V$, and also that
$I^{\mathrm{fix}},\ I^{\mathrm{mv}}$ are $G$-invariant.

Let $\pi\colon\mathrm{bl}_{V^{G^{0}}}V\to V$ be the blow-up of $V$ along
$V^{G^{0}}$ with exceptional divisor $E$ and local defining equation $\xi$.
Then, as before, take $I^{{\mathrm{intr}}}$ to be the ideal generated by
$\pi^{-1}(I^{\mathrm{fix}})$ and $\xi^{-1}\pi^{-1}(I^{\mathrm{mv}})$.
Everything is $G$-equivariant, and we define
$U^{{\mathrm{intr}}}_{\mathrm{cl}}$ as the subscheme of
$\mathrm{bl}_{V^{G^{0}}}V$ defined by the ideal $I^{{\mathrm{intr}}}$. At the
level of quotient stacks,
$X^{{\mathrm{intr}}}_{\mathrm{cl}}\coloneqq[U^{{\mathrm{intr}}}_{\mathrm{cl}}/G]$
is the intrinsic blow-up of $X\coloneqq[U/G]$.

###  5.2. Intrinsic blow-ups are classical truncations of derived blow-ups

Consider a derived affine scheme $U$, quasi-compact and locally of finite
presentation, with an action by a connected reductive group $G$ such that the
(derived) fixed locus $(U_{\mathrm{cl}})^{G}$ is non-empty. The following is
the main result of this section.

######  Theorem 5.3.

The $G$-intrinsic blow-up of the classical truncation $U_{\mathrm{cl}}$ is
naturally isomorphic, as a classical scheme with $G$-action, to the classical
truncation $(\operatorname{Bl}_{U^{G}}U)_{\mathrm{cl}}$ of the derived blow-up
of $U$ along the fixed locus $U^{G}$.

###### Proof.

The statement is local. Let $x\in(U_{\mathrm{cl}})^{G}$.

By Lemma 3.23, after possibly (Zariski) equivariantly shrinking $U$ around
$x$, we may reduce to the case where $U=\operatorname{Spec}R^{\prime}$ and
$R^{\prime}$ corresponds to a cdga in standard form (with homological
indexing)

(5.2) $\displaystyle R=[\dots\longrightarrow R_{2}\longrightarrow
R_{1}\longrightarrow R_{0}],$

where $R_{0}=\mathbb{C}[x_{1},...,x_{N}]$, $G$ acts linearly on the variables
$x_{i}$, and $R^{\prime}$ is generated in each positive degree by a
$G$-representation $V_{i}$ with variables $y_{j}^{i}$ corresponding to a basis
for each $V_{i}$.

Let $V=\operatorname{Spec}R_{0}$. By Proposition 3.24, we then know that
$U^{G}=\operatorname{Spec}S$, where $S$ is a cdga in standard form, with
$S_{0}=\mathcal{O}_{V^{G}}$, and $S$ generated in each positive degree $i$ by
the $G$-fixed variables of $V_{i}$.

Consider now the auxiliary scheme $F^{1}=\operatorname{Spec}S^{1}$, where
$S^{1}_{0}=\mathcal{O}_{V^{G}}$ and $S^{1}$ is generated in degree $1$ by the
$G$-fixed variables of $V_{1}$ and by all of the variables of $V_{i}$ for
$i\geq 2$.

There is a composition $U^{G}\to F^{1}\to U$, where the morphism $U^{G}\to
F^{1}$ is easily seen to be a sequence of cell attachments in degrees bigger
than $1$, using the fact that $V_{2}^{\mathrm{mv}}$ maps to $0$ in $S_{1}^{1}$
by $G$-equivariance.

In particular, Proposition 2.10 gives an isomorphism

(5.3)
$\displaystyle(\operatorname{Bl}_{U^{G}}U)_{\mathrm{cl}}\simeq(\operatorname{Bl}_{F^{1}}U)_{\mathrm{cl}},$

and we are thus reduced to calculating
$(\operatorname{Bl}_{F^{1}}U)_{\mathrm{cl}}$.

Let $A\subseteq R$ be the subalgebra of $R$ generated by $R_{0}$ and the
$G$-fixed variables in degree $1$.

Without loss of generality, let $x_{1},...,x_{l}$ be the moving variables in
$R_{0}$ and
$f_{1},...,f_{n}\in(x_{1},...,x_{l})=\left(R_{0}^{\mathrm{mv}}\right)$ the
images of the moving variables in $V_{1}$. Set

$B=A/(f_{1},...,f_{n}),\ D=A/(x_{1},...,x_{l})$

with the obvious cell attachment maps.

Let $Z=\operatorname{Spec}D,\ X=\operatorname{Spec}B,\
Y=\operatorname{Spec}A$. It is clear that $B$ is the subalgebra of $R$
generated by $R_{0}$ and $R_{1}$ and we have a morphism $U\to X$.

We then have $Z_{\mathrm{cl}}=(U_{\mathrm{cl}})^{G}$,
$X_{\mathrm{cl}}=U_{\mathrm{cl}}$ and a Cartesian square

(5.8)

Proposition 2.11 then implies that

(5.9)
$\displaystyle(\operatorname{Bl}_{F^{1}}U)_{\mathrm{cl}}\simeq(\operatorname{Bl}_{Z}X)_{\mathrm{cl}}.$

But, by (2.3), it immediately follows that
$(\operatorname{Bl}_{Z}X)_{\mathrm{cl}}$ is the $G$-intrinsic blow-up of
$X_{\mathrm{cl}}=U_{\mathrm{cl}}$, as we want, obtained using the equivariant
closed embedding $U_{\mathrm{cl}}\to V$.

Indeed, $V^{G}=\operatorname{Spec}R_{0}/(x_{1},\ldots,x_{l})$, since
$x_{1},\ldots,x_{l}$ are the moving variables among the $x_{i}$.

Letting $I$ be the ideal of the canonical inclusion $U_{\mathrm{cl}}\to V$,
$I$ is then the image of $R_{1}$, hence $I^{\mathrm{mv}}=(f_{1},\dots,f_{n})$.
Write $f_{i}=\sum_{ij}\lambda_{ij}x_{j}$. Thus in the Rees algebra

$\displaystyle
R_{V^{G}/V}^{G,\mathrm{ext}}=\frac{R_{0}[t^{-1},w_{1},\dots,w_{l}]}{(t^{-1}w_{1}-x_{1},\dots,t^{-1}w_{l}-x_{l})}$

it holds that $\xi^{-1}\pi^{-1}f_{j}=\sum_{ij}\lambda_{ij}w_{j}$, where
$\pi\colon\mathrm{bl}_{V^{G}}V\to V$ is the projection. Likewise,
$\pi^{-1}(I^{\mathrm{fix}})$ is generated by $\partial V_{1}^{\mathrm{fix}}$
and hence equals the image $\partial A_{1}$, where $\partial$ is the boundary
map.

We have that the intrinsic blow-up of $U_{\mathrm{cl}}$ is the projective
spectrum of the discrete quotient

$\displaystyle\frac{R_{0}[t^{-1},w_{1},\dots,w_{l}]}{(t^{-1}\underline{w}-\underline{x},\partial
A_{1},\sum_{1j}\lambda_{1j}w_{j},\dots,\sum_{nj}\lambda_{nj}w_{j})}\simeq\frac{\pi_{0}(A)[t^{-1},\underline{w}]}{(t^{-1}\underline{w}-\underline{x},\sum_{ij}\lambda_{ij}w_{j})}$

which we recognize as $\pi_{0}R^{G,{\mathrm{ext}}}_{Z/X}$, using (2.3).

∎

A special case of the theorem which is worth noting is when $U$ is classical
to begin with.

######  Corollary 5.4.

Let $U$ be a classical affine scheme, of finite type and locally of finite
presentation as a derived scheme, with a $G$-action. Then its classical
$G$-intrinsic blow-up $U^{{\mathrm{intr}}}_{\mathrm{cl}}$ is the classical
truncation of the derived blow-up $\operatorname{Bl}_{U^{G}}U$ of $U$ with
center the derived fixed locus $U^{G}$.

######  Remark 5.5.

Even though the above corollary is stated for a classical scheme of finite
type and of local finite presentation as a derived scheme, this is not a
restriction, and it still holds in the more general case of a scheme of finite
type. This is because the proof of Lemma 3.23 for a classical $G$-scheme of
finite type shows that one can obtain a local model $U=\operatorname{Spec}R$
as a possibly infinite sequence of equivariant cell attachments, which are
finitely many in each homological degree, i.e., a possibly infinite
equivariant standard form. This, however, does not affect the computations, as
is clear from our argument above. We elect to assume local finite presentation
for simplicity. The same applies to Section 6.

######  Remark 5.6.

Since $U^{G}$ is well-defined, it is now conceptually clear why the intrinsic
blow-up only depends on the $G$-scheme $U$ and not the auxiliary data used to
define it, which is a priori not obvious and had to be checked by hand in
[KLS17].

Another non-obvious consequence of the above results is the fact that the
classical truncation of the blow-up $\operatorname{Bl}_{U^{G}}U$ for a derived
$G$-scheme $U$ only depends on its classical underlying $G$-scheme
$U_{\mathrm{cl}}$.

###  5.3. A couple of examples

We give two examples which hopefully clarify the arguments in this section and
will also be useful guides for intuition. Both are derived enhancements of
Example 5.2 and we can directly verify that the classical truncation of the
derived blow-up in each of them coincides with the intrinsic blow-up described
in Example 5.2.

Throughout, we put $A\coloneqq\mathbb{C}[x,y]$ and $B\coloneqq
A/(xy^{2},yx^{2})$, so that

$(U\to V)\coloneqq(\operatorname{Spec}B\to\operatorname{Spec}A)$

is the derived critical locus of $f=x^{2}y^{2}$, considered as map
$V\to\mathbb{A}^{1}$. Write $Z=\operatorname{Spec}\mathbb{C}$, and consider
the closed embedding $Z\to U$ given by the quotient map $B\to
A/(x,y)\simeq\mathbb{C}$.

######  Example 5.7.

Lemma 2.12 gives us that $D_{Z/U}\simeq D_{Z/V}\times_{D_{U/V}}D_{U/U}$, and
Proposition 2.13, gives

$R_{Z/U}^{\mathrm{ext}}\simeq\frac{A[t^{-1},w_{1},w_{2}]}{(t^{-1}w_{1}-x,t^{-1}w_{2}-y,y^{2}w_{1},x^{2}w_{2})}\simeq\frac{A[It,t^{-1}]}{(xy^{2}t,yx^{2}t)}$

######  Example 5.8.

The Koszul complex defining the algebra $B$ looks like $A\to A\oplus A\to A$,
which gives us an element $(x,-y)\in A\oplus A$ that gives rise to a nonzero
element

$\displaystyle\sigma\in\pi_{1}B\simeq\pi_{0}\operatorname{Map}(\mathrm{LSym}(B[1]),B)$

and thus a map $f_{\sigma}\colon\mathrm{LSym}(B[1])\to B$. Let $C$ be the
pushout of $f_{\sigma}$ along the map $\mathrm{LSym}(B[1])\to B$ induced by
the zero-map $B[1]\to B$, so that $C=B/(\sigma)$.

Put $W=\operatorname{Spec}C$. Write $\tau$ for the element of
$R^{\mathrm{ext}}_{Z/U}$ in homological degree $1$, corresponding to the
composition $f_{\sigma}\colon\mathrm{LSym}(B[1])\to B\to
R_{Z/U}^{\mathrm{ext}}$. Then, again by Proposition 2.13, it holds

$R^{\mathrm{ext}}_{Z/W}\simeq R^{\mathrm{ext}}_{Z/U}/(\tau)$

In terms of the Koszul complex, $\tau$ corresponds to the element $(xt,-yt)$
in homological degree $1$ of the complex $R\to R\oplus R\to R$, where
$R=A[It,t^{-1}]$ for $I=(x,y)$

##  6\. Derived stabilizer reduction of Artin stacks

Let $X$ be a derived algebraic stack. Throughout this section, we assume that
$X$ is quasi-compact and locally of finite presentation, and that
$X_{\mathrm{cl}}$ admits a good moduli space $q\colon X_{\mathrm{cl}}\to M$.
Observe that $X_{\mathrm{cl}}$ is then of finite type, hence Noetherian.
Recall that all derived stacks (hence all classical stacks, good moduli
spaces, etc.) are assumed to have affine diagonal.

Under these assumptions, an intrinsic stabilizer reduction procedure was
developed in [KLS17, Sav20], producing a canonical sequence of classical Artin
stacks

(6.1) $\displaystyle X_{\mathrm{cl}}^{0}=X_{\mathrm{cl}},\
X_{\mathrm{cl}}^{1}\coloneqq\widehat{X}_{\mathrm{cl}}^{0},\ \ldots,\
X_{\mathrm{cl}}^{m}\coloneqq\widehat{X}_{\mathrm{cl}}^{m-1}$

where each $X_{\mathrm{cl}}^{i}$ is obtained by applying an operation called
“Kirwan blow-up” on the preceding stack $X_{\mathrm{cl}}^{i-1}$ (denoted by
the top hat), and the maximal stabilizer dimension of the stacks in the
sequence is strictly decreasing, so that $X_{\mathrm{cl}}^{m}$ has only finite
stabilizers. The Deligne–Mumford stack $\widetilde{X}_{\mathrm{cl}}\coloneqq
X_{\mathrm{cl}}^{m}$ is called the _intrinsic stabilizer reduction_ of
$X_{\mathrm{cl}}$.

In this section, we use the results obtained so far to upgrade this
construction to the derived context. More precisely, we define a derived
Kirwan blow-up operation whose classical truncation is the classical Kirwan
blow-up. This allows us to construct a derived stabilizer reduction procedure
by a canonical sequence of derived Artin stacks

(6.2) $\displaystyle X^{0}=X,\ X^{1}\coloneqq\widehat{X}^{0},\ \ldots,\
X^{m}\coloneqq\widehat{X}^{m-1}$

whose classical truncation is (6.1). We will define the derived
Deligne–Mumford stack $\widetilde{X}\coloneqq X^{m}$ to be the derived
stabilizer reduction of $X$.

###  6.1. Review of the classical case

We briefly recall the notions of intrinsic and Kirwan blow-up for a classical
Artin stack $X_{\mathrm{cl}}$ with a good moduli space $q\colon
X_{\mathrm{cl}}\to M$, as considered in [Sav20]. Let $M^{{\mathrm{max}}}$ be
the good moduli space of $X_{\mathrm{cl}}^{{\mathrm{max}}}$.

The classical intrinsic blow-up

$\pi^{{\mathrm{intr}}}\colon{X}_{\mathrm{cl}}^{{\mathrm{intr}}}\to
X_{\mathrm{cl}}$

of $X_{\mathrm{cl}}$ is defined by étale descent using affine, strongly étale
morphisms $[U_{\mathrm{cl}}/G]\to X_{\mathrm{cl}}$, where $G$ runs along all
reductive stabilizer groups of $X_{\mathrm{cl}}$ of maximum dimension. Given
such, ${X}_{\mathrm{cl}}^{{\mathrm{intr}}}$ is obtained through a
corresponding affine, étale, stabilizer-preserving cover by morphisms

(6.3)
$\displaystyle[U^{{\mathrm{intr}}}_{\mathrm{cl}}/G]\to{X}_{\mathrm{cl}}^{{\mathrm{intr}}}$

glued together with the complement $X_{\mathrm{cl}}\setminus
X_{\mathrm{cl}}^{{\mathrm{max}}}\subseteq X_{\mathrm{cl}}$ of
$X^{{\mathrm{max}}}$, which is unaffected by the intrinsic blow-ups of the
cover.

The _classical Kirwan blow-up_ is the open substack of _semi-stable points_
$\left(X_{\mathrm{cl}}^{{\mathrm{intr}}}\right)^{\mathrm{ss}}$ of the
intrinsic blow-up $X_{\mathrm{cl}}^{{\mathrm{intr}}}$, whose complement by
definition is the _unstable locus_
$(X_{\mathrm{cl}}^{{\mathrm{intr}}})^{\mathrm{us}}\subseteq
X_{\mathrm{cl}}^{{\mathrm{intr}}}$, defined in [Sav20].

We briefly remind the reader that it is determined as follows: Let $x\in
X_{\mathrm{cl}}^{\max}$ be any closed point with reductive stabilizer
$G=G_{x}$ of maximal dimension. Let $[U_{\mathrm{cl}}/G]\to X_{\mathrm{cl}}$
be a strongly étale morphism, whose image contains $x$, where
$U_{\mathrm{cl}}$ is classical, affine and equipped with a $G$-equivariant
closed embedding $U_{\mathrm{cl}}\to V$ into a smooth, affine, classical
$G$-scheme $V$. Write $\pi^{{\mathrm{intr}}}\colon V^{{\mathrm{intr}}}\to V$,
as usual, for the blow-up of $V$ along $V^{G^{0}}$, and $q\colon V\to
V/\\!\\!/G$ for the good moduli space morphism. By definition, we have a
$G$-invariant closed embedding $U^{{\mathrm{intr}}}\to V^{{\mathrm{intr}}}$.
The unstable locus of $V^{{\mathrm{intr}}}$ is then the strict transform of
the saturation of the closed subscheme $V^{G^{0}}/\\!\\!/G=q(V^{G^{0}})$
inside $V/\\!\\!/G$, i.e.

$\displaystyle(V^{{\mathrm{intr}}})^{\mathrm{us}}:=\overline{(\pi^{{\mathrm{intr}}})^{-1}\left(q^{-1}(V^{G^{0}}/\\!\\!/G)\setminus
V^{G^{0}}\right)},$

so that
$(V^{{\mathrm{intr}}})^{\mathrm{ss}}=V^{{\mathrm{intr}}}\setminus(V^{{\mathrm{intr}}})^{\mathrm{us}}$.
In [ER21], the quotient stack $[(V^{{\mathrm{intr}}})^{\mathrm{ss}}/G]$ is
referred to as the Reichstein transform of $[V/G]$. We then define
$(U^{{\mathrm{intr}}})^{\mathrm{ss}}=U\cap(V^{{\mathrm{intr}}})^{\mathrm{ss}}$.

One may check that this is independent of choices and hence well-defined. We
then take $(X_{\mathrm{cl}}^{{\mathrm{intr}}})^{\mathrm{ss}}$ to be the open
substack of $X_{\mathrm{cl}}^{{\mathrm{intr}}}$ whose points are in the image
of the composition
$[(U_{\mathrm{cl}}^{{\mathrm{intr}}})^{\mathrm{ss}}/G]\to[U_{\mathrm{cl}}^{{\mathrm{intr}}}/G]\to
X_{\mathrm{cl}}^{\mathrm{intr}}$ for all possible choices of $x\in
X_{\mathrm{cl}}^{\max}$ and strongly étale morphisms $[U_{\mathrm{cl}}/G]\to
X$ in the above.

An important property established in [Sav20] is the existence of a good moduli
space $(X^{\mathrm{intr}}_{\mathrm{cl}})^{\mathrm{ss}}\to\widehat{M}$ for the
Kirwan blow-up such that the natural induced morphism $\widehat{M}\to M$ is
proper. By construction, the maximum stabilizer dimension of points in
$(X^{\mathrm{intr}}_{\mathrm{cl}})^{\mathrm{ss}}$ is strictly lower than that
of $X_{\mathrm{cl}}$.

###  6.2. Derived intrinsic and Kirwan blow-ups

If $X$ is a derived Artin stack satisfying the assumptions of this section, we
have constructed earlier a canonical, closed immersion $X^{{\mathrm{max}}}\to
X$, which parametrizes points in $X$ with stabilizer of maximal dimension.

As in the classical case, we may thus give the following definition.

######  Definition 6.1.

The derived intrinsic blow-up of $X$ is defined as the stack over $X$

$\pi^{{\mathrm{intr}}}\colon
X^{{\mathrm{intr}}}\coloneqq\operatorname{Bl}_{X^{{\mathrm{max}}}}X\longrightarrow
X.$

As expected, the derived intrinsic blow-up is a canonical derived enhancement
of the classical intrinsic blow-up (which also justifies the notation).

######  Proposition 6.2.

The classical truncation of $X^{{\mathrm{intr}}}$ is the classical intrinsic
blow-up of the classical truncation $X_{\mathrm{cl}}$ of $X$.

###### Proof.

By the base change property of $X^{{\mathrm{max}}}$ along a cover by affine,
strongly étale morphisms from affine quotient stacks, we may reduce to the
case of a derived affine quotient stack $[U/G]$ for which
$[U/G]^{{\mathrm{max}}}=[U^{G^{0}}/G]$. But then the statement follows
immediately by Theorem 5.3 by working with the connected component $G^{0}$ and
$G$-equivariance. ∎

######  Lemma 6.3.

The derived stack $X^{{\mathrm{max}}}$ is locally of finite presentation.
Consequently, the morphism $\pi^{{\mathrm{intr}}}\colon X^{{\mathrm{intr}}}\to
X$ is locally of finite presentation.

###### Proof.

In [HKR22] it is shown that $\operatorname{Bl}_{X^{{\mathrm{max}}}}X$ is an
open substack of the deformation space $\mathscr{D}_{X^{{\mathrm{max}}}/X}$,
and that Weil restrictions preserve the property of being locally finitely
presented. Since $X$ is assumed to be locally of finite presentation, it thus
suffices to show that $X^{{\mathrm{max}}}$ is locally of finite presentation.

As before, taking slices and using Theorem 4.14, we reduce to the case
$X=[U/G]$, where $U$ is a derived $G$-scheme, locally of finite presentation.
Since $U^{G^{0}}$ is obtained by Weil-restricting along $BG\to*$, the claim
follows. ∎

Let $\mathcal{A}$ be a discrete, finitely generated, quasi-coherent, graded
$\mathcal{O}_{X_{\mathrm{cl}}}$-algebra, and write $q\colon X_{\mathrm{cl}}\to
Y$ for the good moduli space. Recall that the _saturated projective spectrum_
$\operatorname{Proj}^{q}_{X_{\mathrm{cl}}}\mathcal{A}$ (relative to $q$) is
the largest open substack of
$\operatorname{Proj}_{X_{\mathrm{cl}}}\mathcal{A}$ such that the natural map

$\operatorname{Proj}^{q}_{X_{\mathrm{cl}}}\mathcal{A}\to\operatorname{Proj}_{Y}q_{*}\mathcal{A}$

is well-defined, in which case it is a good moduli space [ER21, §3.1]. In case
$\mathcal{A}$ is the Rees algebra of a closed immersion, this is called the
_saturated blow-up_.

######  Definition 6.4.

Let $\mathcal{B}$ be a quasi-coherent, graded $\mathcal{O}_{X}$-algebra such
that $\pi_{0}\mathcal{B}$ is finitely generated. Then the _derived saturated
projective spectrum_ is the open substack
$\operatorname{Proj}^{q}_{X}\mathcal{B}$ of the derived projective spectrum
$\operatorname{Proj}_{X}\mathcal{B}$ such that

$(\operatorname{Proj}^{q}_{X}\mathcal{B})_{\mathrm{cl}}\to(\operatorname{Proj}_{X}\mathcal{B})_{\mathrm{cl}}\cong\operatorname{Proj}_{X_{\mathrm{cl}}}\pi_{0}\mathcal{B}$

is the saturated projective spectrum of $\pi_{0}\mathcal{B}$.

By Lemma 6.3 it holds that $\pi_{0}\mathcal{R}_{X^{{\mathrm{max}}}/X}$ is
finitely generated. The following therefore makes sense.

######  Definition 6.5.

The derived Kirwan blow-up of $X$ is defined as

$\pi\colon\widehat{X}\coloneqq\operatorname{Proj}^{q}_{X}\mathcal{R}_{X^{{\mathrm{max}}}/X}\longrightarrow
X.$

The _semi-stable locus_ of $X^{\mathrm{intr}}$ is the open substack
$(X^{\mathrm{intr}})^{\mathrm{ss}}\subseteq X^{{\mathrm{intr}}}$ whose
classical truncation is the classical semi-stable locus
$(X^{{\mathrm{intr}}}_{\mathrm{cl}})^{\mathrm{ss}}\subseteq
X^{{\mathrm{intr}}}_{\mathrm{cl}}$ discussed in § 6.1.

The connection with the classical Kirwan blow-up is as follows.

######  Proposition 6.6.

The derived Kirwan blow-up $\widehat{X}$ coincides with the semi-stable locus
$(X^{\mathrm{intr}})^{\mathrm{ss}}$ as open substacks of
$X^{{\mathrm{intr}}}$.

###### Proof.

It suffices to show that the semi-stable locus of
$X^{{\mathrm{intr}}}_{\mathrm{cl}}$ discussed in § 6.1 coincides with the
saturated blow-up
$\widehat{X}_{\mathrm{cl}}=\operatorname{Proj}^{q}_{X_{\mathrm{cl}}}\pi_{0}\mathcal{R}_{X^{{\mathrm{max}}}_{\mathrm{cl}}/X_{\mathrm{cl}}}$.
By [Sav20, Thm. 4.14], [ER21, Prop. 3.6] and since
$\widehat{X}_{\mathrm{cl}},(X^{{\mathrm{intr}}}_{\mathrm{cl}})^{\mathrm{ss}}$
are open substacks of $X^{{\mathrm{intr}}}_{\mathrm{cl}}$, we reduce to
$X=[U/G]$ with $U$ a classical affine scheme. Then, by definition of the semi-
stable locus and by [ER21, Prop. 3.6], we further reduce to the case where $U$
is smooth. Now the claim follows from [ER21, Prop. 4.5] and [Sav20, Thm. 4.5].
∎

We summarize the fundamental properties of the derived intrinsic and Kirwan
blow-up in the following theorem, which combines the above (and Proposition
4.19) with the properties of the corresponding classical blow-ups given in
[Sav20, Thm. 4.7].

######  Theorem 6.7.

Let $X$ be a derived algebraic stack satisfying the assumptions of this
section: $X$ is quasi-compact and locally of finite presentation, and
$X_{\mathrm{cl}}$ admits a good moduli space $q\colon X_{\mathrm{cl}}\to M$
(with affine diagonal by assumption).

  1. (i)

The Kirwan blow-up $\pi\colon\widehat{X}\to X$ also satisfies the assumptions
of this section.

  2. (ii)

The maximum stabilizer dimension of closed points in $\widehat{X}$ is strictly
smaller than that of $X$.

  3. (iii)

For any affine, strongly étale morphism $[U/G]\to X$ with $U$ affine, the base
change $\widehat{X}\times_{X}[U/G]$ is naturally isomorphic to the derived
Kirwan blow-up of $[U/G]$.

  4. (iv)

$\pi|_{\pi^{-1}(X^{\mathrm{s}})}$ is an isomorphism over the open locus
$X^{\mathrm{s}}$ of properly stable points, i.e., closed points $x\in\lvert
X\rvert$ with finite stabilizer such that $q^{-1}(q(x))=\\{x\\}$.

  5. (v)

The classical truncations of $X^{{\mathrm{intr}}}$ and $\widehat{X}$ are the
classical intrinsic and Kirwan blow-ups of the classical truncation
$X_{\mathrm{cl}}$ respectively.

######  Remark 6.8.

By Proposition 6.6, we can also use [ER21] to study $\widehat{X}$. For
example, [ER21, Prop. 3.4] produces an explicit, global description of the
good moduli space $\widehat{q}\colon\widehat{X}_{\mathrm{cl}}\to\widehat{M}$,
thus circumventing the gluing argument from [Sav20].

######  Example 6.9.

Let $Y\coloneqq[\mathbb{A}^{2}/\mathbb{G}_{m}]$, where $\mathbb{G}_{m}$ acts
on $\mathbb{A}^{2}$ with weights $1,-1$, and let $Z\to Y$ be the closed
substack corresponding to $\\{0\\}\to\mathbb{A}^{2}$. Since $Y^{\max}=Z$, from
[ER21, Ex. 4.6] we know that

$\widehat{Y}\cong(\operatorname{Bl}_{Z}Y)\setminus[(D_{1}\sqcup
D_{2})/\mathbb{G}_{m}]$

where $D_{1},D_{2}$ are the strict transforms of the coordinate axes.

###  6.3. Derived stabilizer reduction of Artin stacks

We are now in position to carry out the derived stabilizer reduction
procedure, as described in the introductory part of the section.

By Theorem 6.7, the Kirwan blow-up $\widehat{X}$ preserves the assumed
properties of $X$. In particular, we may repeatedly apply it until the maximum
stabilizer dimension becomes zero and we obtain a derived Deligne–Mumford
stack $\widetilde{X}\coloneqq X^{m}$ by the canonical sequence

(6.4) $\displaystyle X^{0}=X,\ X^{1}\coloneqq\widehat{X}^{0},\ \ldots,\
X^{m}\coloneqq\widehat{X}^{m-1}.$

######  Definition 6.10.

The derived Deligne–Mumford stack $\widetilde{X}\to X$ is called the _derived
stabilizer reduction_ of $X$.

It is clear by Theorem 6.7 that the classical truncation of $\widetilde{X}$ is
the intrinsic stabilizer reduction of $X_{\mathrm{cl}}$.

######  Remark 6.11.

By Remark 5.5, if $X$ is a classical Artin stack of finite type, then its
derived stabilizer reduction $\widetilde{X}$ is well-defined, with classical
truncation the intrinsic stabilizer reduction of $X_{\mathrm{cl}}$.

###  6.4. Connections with other approaches

We conclude with a brief discussion regarding the connections between the
following stabilizer reduction approaches in the literature:

  1. (i)

Kirwan’s original desingularization procedure for classical, smooth quotient
stacks obtained by Geometric Invariant Theory (GIT) [Kir85].

  2. (ii)

Edidin–Rydh’s canonical reduction of stabilizers for classical (possibly
singular) Artin stacks with good moduli spaces [ER21].

  3. (iii)

The intrinsic stabilizer reduction procedure introduced in [KLS17, Sav20] for
(possibly singular) GIT quotient stacks and Artin stacks with good moduli
spaces.

  4. (iv)

The derived stabilizer reduction procedure of the present paper.

In general, (ii) is a generalization of (i), while we have shown that (iv) is
always a derived enhancement of (iii).

Both (ii) and (iii) are sequences of two-step operations consisting of a blow-
up and the deletion of certain unstable points (this two-step operation in
(ii) is called a saturated blow-up, while in (iii) a Kirwan blow-up). The main
difference between (ii) and (iii) is in the blow-up being used, which can now
be succinctly formulated as follows, using Theorem 6.7. For a classical stack
$X$, the classical blow-up
$\operatorname{Bl}^{\mathrm{cl}}_{X_{\mathrm{cl}}^{{\mathrm{max}}}}X$ along
the classical locus $X_{\mathrm{cl}}^{{\mathrm{max}}}$ is used in (ii), while
the underlying classical stack
$(\operatorname{Bl}_{X^{{\mathrm{max}}}}X)_{\mathrm{cl}}$ of the derived blow-
up of $X$ (viewed as a derived stack) along the derived locus
$X^{{\mathrm{max}}}$ is used in (iii). Recall here that
$X^{{\mathrm{max}}}_{\mathrm{cl}}$ is both the classically defined locus and
the underlying classical stack of the derived locus.

For classical GIT quotient stacks that are smooth, all four constructions
coincide: approaches (i)–(iii) are literally identical, while in (iv) it
suffices to notice that the derived intrinsic blow-up of a classical smooth
Artin stack is the same as the classical intrinsic blow-up (since the derived
$X^{{\mathrm{max}}}$ is smooth and coincides with the classical
$X^{{\mathrm{max}}}$) so that the derived Kirwan blow-up is classical and
equals the saturated blow-up in each step.

However, in the singular case, the derived enhancement $X^{{\mathrm{max}}}$ of
the classical locus $X_{\mathrm{cl}}^{{\mathrm{max}}}$ is in general not
itself classical, as can be seen in the following example. Hence, the outputs
of (ii) and (iv) will typically differ.

######  Example 6.12.

Suppose that $G=\mathbb{G}_{m}=\mathbb{C}^{\times}$ is the one-dimensional
torus acting on the affine plane $V=\mathbb{C}^{2}_{x,y}$ with weights $1$ and
$-1$ on the coordinates $x$ and $y$ respectively, and that $U\subseteq V$ is
the closed $G$-invariant subscheme cut out by the ideal $I=(xy)$. Let
$X\coloneqq[U/G]$. Then the classical locus $X^{{\mathrm{max}}}_{\mathrm{cl}}$
is the quotient stack $\operatorname{Spec}\mathbb{C}[x,y]/(x,y)\times BG$.

To describe the derived locus $X^{{\mathrm{max}}}$, we can find a model $B$
for $\mathcal{O}_{U}$ in standard form, and then apply Proposition 3.24. In
this case, it is clear that we can take $B(0)=\mathbb{C}[x,y]$ and
$M_{0}=B(0)\otimes_{\mathbb{C}}W_{0}$, where $W_{0}$ is the trivial
$1$-dimensional $G$-representation and $M_{0}\to B(0)$ is multiplication by
$xy$. But then, $B_{G}$ is freely generated by $\mathbb{C}[x,y]/(x,y)$ in
degree $0$ and $W_{0}$ in degree $1$, therefore the derived
$X^{{\mathrm{max}}}\simeq\operatorname{Spec}B_{G}\times BG$ is not classical
and differs from its classical counterpart.

##  7\. Quasi-smooth stacks

In this short section, we consider the derived stabilizer reduction of a
quasi-smooth derived Artin stack $X$, satisfying the usual conditions of being
quasi-compact, locally of finite presentation, and $X_{\mathrm{cl}}$ admitting
a good moduli space $q\colon X_{\mathrm{cl}}\to M$.

Recall that $X$ is quasi-smooth if its cotangent complex ${\mathbb{L}}_{X}$ is
perfect of (homological) Tor-amplitude $[-1,1]$. The main result of the
section is as follows.

######  Theorem 7.1.

The derived stabilizer reduction $\widetilde{X}$ is a quasi-smooth
Deligne–Mumford stack. In particular, the intrinsic stabilizer reduction
$\widetilde{X}_{\mathrm{cl}}$ of $X_{\mathrm{cl}}$ admits a natural perfect
obstruction theory
${\mathbb{L}}_{\widetilde{X}}|_{\widetilde{X}_{\mathrm{cl}}}\to{\mathbb{L}}_{\widetilde{X}_{\mathrm{cl}}}$
in the sense of Behrend–Fantechi [BF97] and induces virtual fundamental
classes $[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{\ast}(\widetilde{X}_{\mathrm{cl}})$ and
$[\mathcal{O}_{\widetilde{X}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{X}_{\mathrm{cl}})$ in Chow and $K$-theory.

To prove the theorem, we make use of the following proposition.

######  Proposition 7.2.

Let $f\colon Z\to X$ be a closed embedding of quasi-smooth derived schemes.
Then $\operatorname{Bl}_{Z}X$ is a quasi-smooth derived scheme.

###### Proof.

The statement is local, so we may assume that $X=\operatorname{Spec}B$ and
$Z=\operatorname{Spec}D$ are affine and $f$ is induced by a map of rings $B\to
D$. Moreover, since $X$ and $Z$ are quasi-smooth, we may also assume that
$B=A/(f_{1},\ldots,f_{n})$ and $D=A/(g_{1},\ldots,g_{m})$ where
$A=\mathcal{O}_{V}$ for a smooth, classical affine scheme $V$ and
$f_{i},g_{j}\in\pi_{0}A$. It is then an immediate application of Proposition
2.13 that $\operatorname{Bl}_{Z}X$ is quasi-smooth as well. ∎

######  Remark 7.3.

The preceding proposition has also been used in recent work of Zhao [Zha23a,
Zha23b].

We may now deduce that quasi-smoothness is preserved by derived intrinsic
blow-ups.

######  Proposition 7.4.

Suppose that $X$ is a quasi-smooth derived Artin stack. Then $X^{\max}$ and
$X^{{\mathrm{intr}}}=\operatorname{Bl}_{X^{\max}}X$ are quasi-smooth.

###### Proof.

Both statements are étale local, so by Lemma 1.7 and Proposition 4.13 we may
assume that $X=[U/G]$ where $U=\operatorname{Spec}R\in\mathsf{Aff}^{G}$, $R$
is a $G$-cdga in standard form and $U^{G^{0}}$ is non-empty. The fact that $X$
is quasi-smooth implies that $R$ is generated over $R_{0}$ on a finite number
of generators in degree $1$, given by an $R_{0}$-module
$M_{1}=R_{0}\otimes_{\mathbb{C}}W_{1}$ for a $G$-representation
$W_{1}\in\mathsf{Rep}^{G}$. Proposition 3.24 then implies that $U^{G^{0}}$ is
of the same form and hence quasi-smooth. In particular,
$X^{\max}=[U^{G^{0}}/G]$ is quasi-smooth. Finally,
$\operatorname{Bl}_{U^{G^{0}}}U$ is quasi-smooth by Proposition 7.2 and,
applying Proposition 3.25,
$\operatorname{Bl}_{X^{\max}}X=[\operatorname{Bl}_{U^{G^{0}}}U/G]$ is quasi-
smooth. ∎

###### Proof of Theorem 7.1.

The derived Kirwan blow-up $\widehat{X}$ is an open substack of the derived
intrinsic blow-up $X^{\mathrm{intr}}$, so Proposition 7.4 implies that
$\widehat{X}$ is quasi-smooth. Since $\widetilde{X}$ is obtained via a
sequence of Kirwan blow-ups, it must be quasi-smooth. ∎

In Section 9, we will use this result to define virtual classes for moduli
stacks of sheaves on surfaces.

######  Remark 7.5.

Khan [Kha19] has defined virtual fundamental classes $[X]^{\mathrm{vir}}$ for
any quasi-smooth derived Artin stack $X$. For a quasi-smooth Deligne–Mumford
stack $X$, his construction agrees with the virtual fundamental classes in
Chow and $K$-theory defined using the machinery of perfect obstruction
theories. Writing $\pi\colon\widetilde{X}\to X$ for the natural projection
map, it is an interesting question to see how the virtual classes of the
stabilizer reduction $\widetilde{X}$ relate to those of $X$ via the map $\pi$,
especially in light of the potential enumerative applications in Section 9.

##  8\. $(-1)$-shifted symplectic stacks

$(-1)$-shifted symplectic derived Artin stacks play a fundamental role in
enumerative geometry and Donaldson–Thomas (DT) theory, since they naturally
arise as derived moduli stacks of sheaves and perfect complexes on Calabi–Yau
threefolds. In [KLS17, Sav20], it is shown that if $X$ is such a stack, then
the intrinsic stabilizer reduction $\widetilde{X}_{\mathrm{cl}}$ admits a
semi-perfect obstruction theory of virtual dimension zero and hence by [CL11]
a virtual fundamental cycle
$[\widetilde{X}_{\mathrm{cl}}^{\mathrm{\mathrm{vir}}}]\in
A_{0}(\widetilde{X}_{\mathrm{cl}})$, whose degree was defined to be the
generalized DT invariant via Kirwan blow-ups associated to $X_{\mathrm{cl}}$.
This was possible by rather cumbersome local computations.

In this section, we pursue a more detailed study of the derived stabilizer
reduction $\widetilde{X}$ to put this result on a much more robust footing.
Namely, we will show that, by truncating the derived tangent complex of
$\widetilde{X}$, there is a natural way to recover the above semi-perfect
obstruction theory. As an application, we construct generalized Vafa–Witten
invariants in Section 9.

###  8.1. Brief review of shifted symplectic geometry

Shifted symplectic structures were introduced by Pantev–Toën–Vaquié–Vezzosi in
[PTVV13]. Their definition is given in the affine case first, and then
generalized by showing that the local notion satisfies smooth descent. We use
the presentation from [PV18], but with different grading. See also [Rak20].

The local definition is as follows. Let $\mathbb{C}[\epsilon]$ be the graded
algebra, where $\epsilon$ is free in homogeneous degree 1 and homological
degree 1, hence with $\epsilon^{2}\simeq 0$. An object in
$\mathsf{Mod}^{\mathbb{Z}}_{\mathbb{C}[\epsilon]}$ is then a graded
$\mathbb{C}[\epsilon]$-module $\bigoplus_{p}M^{p}$, and multiplication by
$\epsilon$ induces maps $M^{p}\to M^{p+1}[1]$. Then the functor
$\mathsf{Alg}^{\mathbb{Z}}_{\mathbb{C}[\epsilon]}\to\mathsf{Alg}$, which sends
an algebra $\bigoplus_{p}B^{p}$ to $B^{0}$, has a left adjoint, written

$\mathrm{dR}_{(-)}\colon\mathsf{Alg}\to\mathsf{Alg}^{\mathbb{Z}}_{\mathbb{C}[\epsilon]}$

The algebra $\mathrm{dR}_{A}$ for $A\in\mathsf{Alg}$ is called the _derived de
Rham_ algebra. The underlying graded module of $\mathrm{dR}_{A}$ is

$\mathrm{dR}_{A}\simeq\bigoplus\nolimits_{p\geq
0}\Lambda^{p}_{A}({\mathbb{L}}_{A})[p]\simeq\mathrm{LSym}_{A}({\mathbb{L}}_{A}[1](-1))$

where $(-)(q)$ means twisting in homogeneous degree, so that
$(M(q))^{p}=M^{p+q}$.

The space of _$n$ -shifted, closed $p$-forms_ on $A$ is

$\mathcal{A}^{p,\mathrm{cd}}(A,n)\coloneqq\mathsf{Mod}^{\mathbb{Z}}_{\mathbb{C}[\epsilon]}(\mathbb{C}(-p)[p-n],\mathrm{dR}_{A})$

An element $\omega\in\mathcal{A}^{p,\mathrm{cd}}(A,n)$ thus induces an element
in

$\mathrm{dR}_{A}^{p}[n-p]\simeq\Lambda^{p}{\mathbb{L}}_{A}[n]$

In particular, for $p=2$, we obtain a map
$\omega\colon\mathbb{C}\to{\mathbb{L}}_{A}\wedge{\mathbb{L}}_{A}[n]$, hence a
morphism $\omega^{\flat}\colon{\mathbb{L}}_{A}^{\vee}\to{\mathbb{L}}_{A}[n]$
by adjunction. We say that $\omega$ is _non-degenerate_ if $\omega^{\flat}$ is
an equivalence.

######  Definition 8.1.

Suppose that $A\in\mathsf{Alg}^{G}$. Then the standard $G$-action on
${\mathbb{L}}_{A}$ induces a $G$-action on $\mathrm{dR}_{A}$, and
$\omega\in\mathrm{dR}_{A}$ is called _$G$ -invariant_ if it is invariant with
respect to this $G$-action.

The functor $\mathcal{A}^{p,\mathrm{cd}}(-,n)$ is a derived stack. For
$X\in\mathsf{Stk}$, the space of $n$-shifted, closed $p$-forms on $X$ is then

$\mathcal{A}^{p,\mathrm{cd}}(X,n)\coloneqq\mathsf{Stk}(X,\mathcal{A}^{p,\mathrm{cd}}(-,n))\simeq\lim\nolimits_{\operatorname{Spec}A\to
X}\mathcal{A}^{p,\mathrm{cd}}(A,n)$

An $n$-shifted, closed $p$-form $\omega$ on $X$ thus induces a homotopy
coherent family of $\omega_{A}\in\mathcal{A}^{p,\mathrm{cd}}(A,n)$, indexed
over all $\operatorname{Spec}A\to X$. It can be shown that, for $p=2$, this
again induces a morpshim
$\omega^{\flat}\colon{\mathbb{L}}^{\vee}_{X}\to{\mathbb{L}}_{X}[n]$, and that
$\omega$ is _non-degenerate_ if $\omega^{\flat}$ is an equivalence.

######  Definition 8.2 ([PTVV13, Def. 1.18], [PV18, Def. 1.10]).

An $n$-shifted, closed $2$-form $\omega$ on $X$ is called an $n$-shifted
symplectic structure if it is non-degenerate. If such $\omega$ exists, we say
that $X$ is $n$-shifted symplectic.

###  8.2. Local structure of $(-1)$-shifted symplectic stacks

We first introduce notation that will help us denote cdga’s more efficiently.
Recall that we use homological indexing notation.

######  Definition 8.3.

Let $G$ be a reductive group, $V$ a smooth $G$-scheme, $\mathcal{W}_{1}$ and
$\mathcal{W}_{2}$ be $G$-equivariant vector bundles on $V$ and
$\delta_{2}\colon\mathcal{W}_{2}\to\mathcal{W}_{1}$,
$\delta_{1}\colon\mathcal{W}_{1}\to\mathcal{O}_{V}$ be two $G$-equivariant
morphisms such that $\delta_{1}\delta_{2}=0$.

Then we write $A=K(G,V,\mathcal{W}_{1},\mathcal{W}_{2},\delta_{1},\delta_{2})$
for the $G$-equivariant sheaf of cdga’s generated by
$\mathcal{O}_{V},\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ in degrees $0,1$ and
$2$ respectively and whose differential is determined by the maps
$\delta_{1},\delta_{2}$.

For brevity we write $\mathcal{W}^{i}$ for $\mathcal{W}_{i}^{\vee}$, where
$i=1,2$, and $\delta^{2}$ for $\delta_{2}^{\vee}$.

In the case where $\mathcal{W}_{i}=W_{i}\otimes_{\mathbb{C}}\mathcal{O}_{V}$
for $G$-representations $W_{i}$, we also use the shorthand notation
$K(G,V,W_{1},W_{2},\delta_{1},\delta_{2})$.

######  Proposition 8.4.

Let $A=K(G,V,\mathcal{W}_{1},\mathcal{W}_{2},\delta_{1},\delta_{2})$ where $V$
is affine and write $U=\operatorname{Spec}A$. Then, after possibly (Zariski)
equivariantly shrinking around fixed points $x\in U^{G}$, $A$ can be written
in the form

$A=K(G,V,W_{1},W_{2},\delta_{1},\delta_{2})$

for some $G$-representations $W_{1},W_{2}$.

###### Proof.

Lemma 3.23 applies verbatim to prove the statement. ∎

The following lemma is a strengthening of the preceding proposition in the
$(-1)$-shifted symplectic case. Recall that a smooth scheme $V$ Zariski-
locally admits an étale morphism $V\to\mathbb{A}^{d}$ for some $d$. The local
sections $x_{i}\in\mathcal{O}_{V}$ induced by the corresponding map
$\mathcal{O}_{\mathbb{A}^{d}}\to\mathcal{O}_{V}$ are called _étale
coordinates_. For a $G$-action on $V$, we say that $G$ acts _linearly_ on
given étale coordinates, if there is a homomorphism $\varphi\colon
G\to\operatorname{GL}_{d}$ such that $V\to\mathbb{A}^{d}$ is $G$-equivariant
for the $G$-action on $\mathbb{A}^{d}$ induced by $\varphi$.

######  Lemma 8.5.

Let $U$ be an affine derived $G$-scheme, locally of finite presentation, such
that $X=[U/G]$ is a $(-1)$-shifted symplectic derived quotient stack, and let
$x\in U^{G}$. Let $\mathfrak{g}$ be the Lie algebra of the reductive group
$G$. Then after possible (Zariski) equivariant shrinking around $x$, we may
assume that $\mathcal{O}_{U}$ has a model
$A=K(G,V,W_{1},W_{2},\delta_{1},\delta_{2})$, where:

  1. (i)

$W_{1}$ is the vector space spanned by $\frac{\partial}{\partial x_{i}}$ for a
set of étale coordinates $\\{x_{i}\\}$ of $V$ on which $G$ acts linearly, so
that $W_{1}\otimes_{\mathbb{C}}\mathcal{O}_{V}\cong T_{V}$ as $G$-equivariant
vector bundles on $V$.

  2. (ii)

Under this identification, $\delta_{1}$ maps each $\frac{\partial}{\partial
x_{i}}$ to $\frac{\partial f}{\partial x_{i}}\in\mathcal{O}_{V}$ for a
$G$-invariant regular function $f\colon V\to\mathbb{A}^{1}$.

  3. (iii)

$W_{2}=\mathfrak{g}$ and the map
$\delta_{2}\colon\mathfrak{g}\otimes_{\mathbb{C}}\mathcal{O}_{V}\to
W_{1}\otimes_{\mathbb{C}}\mathcal{O}_{V}\cong T_{V}$ is the infinitesimal
derivative of the $G$-action on $V$.

###### Proof.

By Lemma 3.23, up to equivariant shrinking around $x$, we may assume that
$\mathcal{O}_{U}$ has a $G$-equivariant model $A$ in standard form, i.e.
generated by $\mathcal{O}_{V}$ in degree $0$, where $V$ is a smooth affine
$G$-scheme, and $G$-equivariant vector bundles
$\mathcal{W}_{i}=W_{i}\otimes_{\mathbb{C}}\mathcal{O}_{V}$ on $V$ in positive
degrees, where as usual $W_{i}=\mathcal{W}_{i}|_{x}$ is the restriction to
$x\in V$. We then have

$\displaystyle{\mathbb{L}}_{X}|_{x}\simeq{\mathbb{L}}_{A}|_{x}\cong\left[\ldots\longrightarrow\mathcal{W}_{2}|_{x}\longrightarrow\mathcal{W}_{1}|_{x}\longrightarrow\Omega_{V}|_{x}\longrightarrow\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V}\right]$

by Example 3.19 and the general formula for the cotangent complex of a cdga in
standard form (we conflate ${\mathbb{L}}_{A}$ with the explicit model from
[BBJ19, §2.3]).

Moreover, we may assume that $\operatorname{Spec}A$ is minimal at $x$, meaning
that the differentials of ${\mathbb{L}}_{A}|_{x}$ vanish. Since $X$ is
$(-1)$-shifted symplectic, there exists a quasi-isomorphism
${\mathbb{L}}_{A}|_{x}\cong{\mathbb{L}}_{A}|_{x}^{\vee}[1]$ and then
minimality implies that we must have $\mathcal{W}_{i}=0$ for $i>2$. Moreover,
we get $G$-equivariant isomorphisms $W_{1}\cong
T_{V}|_{x},W_{2}\cong\mathfrak{g}$, and we can pick a set of étale coordinates
on $V$ with a linear $G$-action using a basis for $\Omega_{V}|_{x}$ (as
$G$-module) and lifting through
$\mathrm{d}\colon\mathcal{O}_{V}\to\Omega_{V}$.

Consider now the smooth morphism
$\operatorname{Spec}A\to[\operatorname{Spec}A/G]$. The pullback to
$\operatorname{Spec}A$ of the shifted symplectic structure $\omega$ on $X$ is
a $G$-invariant $(-1)$-shifted closed $2$-form. Since $G$ is reductive, the
arguments of [BBJ19, Subsection 5.2] and the proof of [BBJ19, Theorem 5.18]
extend verbatim in this $G$-equivariant setting to show that the differential
$\delta_{1}$ maps $\frac{\partial}{\partial x_{i}}$ to $\frac{\partial
f}{\partial x_{i}}\in\mathcal{O}_{V}$ for a $G$-invariant regular function
$f\in\mathcal{O}_{V}^{G}$.

Finally, the pullback of the quasi-isomorphism
${\mathbb{L}}_{[A/G]}\cong{\mathbb{L}}_{[A/G]}^{\vee}[1]$ to
$\operatorname{Spec}A$ and then to $V$ (where we consider the natural morphism
of graded algebras $A\to\mathcal{O}_{V}$) gives rise to a $G$-equivariant
morphism of complexes

where the two middle vertical arrows are the identity maps and, by symmetry,
$(d\delta_{1})^{\vee}=d\delta_{1}$.

The differentials of both complexes vanish at $x$ and hence it follows that
the two outer vertical arrows must be isomorphisms after possible shrinking
around a $G$-invariant Zariski neighbourhood of $x$. Thus we may identify
$d\delta_{2}$ with the infinitesimal derivative $\sigma^{\vee}$ of the
$G$-action. By comparing degrees, $d\delta_{2}$ determines the morphism
$\delta_{2}\colon\mathfrak{g}\otimes_{\mathbb{C}}\mathcal{O}_{V}\to T_{V}$,
which thus must equal $\sigma^{\vee}$, so we have established (i)–(iii). ∎

######  Remark 8.6.

In fact, we can prove a slightly stronger statement giving an equivariant
version of the derived Darboux theorem of [BBBJ15]. Namely, using the
description of shifted symplectic structures on quotient stacks
$[\operatorname{Spec}A/G]$ [Yeu], the $(-1)$-shifted symplectic form on $X$
can be brought to a canonical Darboux form.

######  Remark 8.7.

We will actually only need conditions (i) and (iii) in the ensuing
computations in this paper.

###  8.3. An intrinsic blow-up computation

The intrinsic blow-up of a $(-1)$-shifted symplectic derived stack will in
general not be $(-1)$-shifted symplectic. An explicit example can be
constructed using the affine scheme $W$ in Example 5.8. Thus, in order to work
with the tangent complex of the derived stabilizer reduction of a
$(-1)$-shifted symplectic stack, we will need to get a better handle on the
local structure of its iterated intrinsic blow-ups. To this end, we introduce
further terminology.

######  Definition 8.8.

Let $A=K(G,V,\mathcal{W}_{1},\mathcal{W}_{2},\delta_{1},\delta_{2})$.

We say that $A$ satisfies property ($\dagger$) if there exists a $G$-invariant
Cartier divisor $D$ on $V$ and a commutative diagram of $G$-equivariant
morphisms

(8.7)

where $\Omega_{V}(-D)\to\mathfrak{g}^{\vee}(-D)$ is induced by the derivative
of the $G$-action,
$\mathfrak{g}^{\vee}(-D)\to\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V}$
is the natural inclusion,
$\mathcal{W}^{2}\to\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V}$ is
injective, and for any point in $V$ with closed $G$-orbit and reductive
stabilizer $H\subseteq G$, the composition

$\mathcal{W}^{1}|_{V^{H}}\to\mathfrak{g}^{\vee}(-D)\to\mathfrak{h}^{\vee}(-D)$

vanishes.

Let $X=[U/G]$, where $U$ is an affine derived scheme. If $U$ has a model $A$
in $G$-equivariant standard form, we write $\mathbb{T}_{X}|_{X_{\mathrm{cl}}}$
for the complex on the (classical) derived category of
$[\operatorname{Spec}\pi_{0}A/G]$ induced by pulling back
${\mathbb{L}}_{[\operatorname{Spec}A/G]}^{\vee}$ to $X_{\mathrm{cl}}$ (where
${\mathbb{L}}_{[\operatorname{Spec}A/G]}^{\vee}$ again refers to the explicit
model). Furthermore, we write $\tau^{[0,1]}(-)$ for the truncation in
homological degrees $[-1,0]$, which corresponds to the $[0,1]$-truncation in
cohomological indexing notation. We do this to better match with existing
notation on perfect obstruction theories. In particular,
$(\tau^{[0,1]}E)^{\vee}$ lives in cohomological degree $[-1,0]$, when
$\tau^{[0,1]}E$ is perfect.

######  Proposition 8.9.

Suppose that $A$ satisfies property $(\dagger)$ and
$X=[\operatorname{Spec}A/G]$ is Deligne–Mumford. Then the truncation
$E^{\bullet}=\tau^{[0,1]}\mathbb{T}_{X}|_{X_{\mathrm{cl}}}$ is a perfect two-
term complex, whose dual $E_{\bullet}\coloneqq(E^{\bullet})^{\vee}$ defines a
perfect obstruction theory111For us, a perfect obstruction theory means either
a morphism $E_{\bullet}\to({\mathbb{L}}_{X_{\mathrm{cl}}})_{\leq 1}$ or, as is
more standard, a morphism $E_{\bullet}\to{\mathbb{L}}_{X_{\mathrm{cl}}}$,
which is an isomorphism on $h^{0}$ and surjective on $h^{1}$ (in homological
grading). From now on, we work with the former, weaker version. on
$X_{\mathrm{cl}}$.

###### Proof.

By assumption, the injection
$\mathfrak{g}^{\vee}(-D)\to\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V}$
factors through the injection
$\mathcal{W}^{2}\to\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V}$ and
hence $\mathfrak{g}^{\vee}(-D)\to\mathcal{W}^{2}$ is injective. In particular,
it follows that the kernel of $\delta^{2}$ is equal to the kernel of the
morphism $\mathcal{W}^{1}\to\mathfrak{g}^{\vee}(-D)$. Since $X$ is
Deligne–Mumford, the dual $\Omega_{V}(-D)\to\mathfrak{g}^{\vee}(-D)$ of the
derivative is surjective, and thus so is
$\mathcal{W}^{1}\to\mathfrak{g}^{\vee}(-D)$. Therefore, the kernel
$\mathcal{F}\coloneqq\ker\delta^{2}$ is a vector bundle.

We then have (suppressing the restriction to $X_{\mathrm{cl}}$ on the right-
hand side)

$\mathbb{T}_{X}|_{X_{\mathrm{cl}}}\cong[\mathfrak{g}\otimes_{\mathbb{C}}\mathcal{O}_{V}\longrightarrow
T_{V}\xrightarrow{(d\delta^{1})^{\vee}}\mathcal{W}^{1}\xrightarrow{\delta^{2}}\mathcal{W}^{2}]$

in homological degrees $[-2,1]$ and, writing $T_{V/G}$ for the cokernel of the
derivative $\mathfrak{g}\otimes_{\mathbb{C}}\mathcal{O}_{V}\to T_{V}$,

$E^{\bullet}=\tau^{[0,1]}\mathbb{T}_{X}|_{X_{\mathrm{cl}}}\cong[T_{V/G}\longrightarrow\mathcal{F}],$

a two-term complex of vector bundles.

To see that $E_{\bullet}=(E^{\bullet})^{\vee}$ defines a perfect obstruction
theory, let $I=\mathrm{im}(\delta_{1})$ be the ideal sheaf of
$X_{\mathrm{cl}}$ inside $V$. Since $\delta^{1}$ factors through
$\mathcal{F}$, we have the following commutative diagram

where both left-most vertical arrows are surjective, and thus a morphism

${\mathbb{L}}_{X}|_{X_{\mathrm{cl}}}\to({\mathbb{L}}_{X_{\mathrm{cl}}})_{\leq
1}$

since ${\mathbb{L}}_{X}|_{X_{\mathrm{cl}}}$ is equivalent to the top row, and
$({\mathbb{L}}_{X_{\mathrm{cl}}})_{\leq 1}$ to the bottom row in the diagram.

It follows that $E_{\bullet}=[\mathcal{F}^{\vee}\to\Omega_{V/G}]$ gives a
perfect obstruction theory on $X_{\mathrm{cl}}$, where
$\Omega_{V/G}=T_{V/G}^{\vee}=\ker(\Omega_{V}\to\mathfrak{g}^{\vee}\otimes_{\mathbb{C}}\mathcal{O}_{V})$.
∎

Let now $X=\operatorname{Spec}A$, where
$A=K(G,V,W_{1},W_{2},\delta_{1},\delta_{2})$ satisfies property ($\dagger$),
and let $x\in X$ be a point with closed $G$-orbit and (reductive) stabilizer
$H$ of maximal dimension.

Using Luna’s étale slice theorem [Dré04], we take a slice $x\in S\subseteq V$
for the $G$-action on $V$ at $x$ giving rise to an affine, strongly étale
morphism $\phi\colon S\times_{H}G\to V$.

The isomorphism $[S/H]\cong[S\times_{H}G/G]$ then gives an equivalence between
$H$-equivariant sheaves on $S$ and $G$-equivariant sheaves on $S\times_{H}G$.
We will freely use this identification henceforth.

The first part of the following proposition follows by definition, and the
second part from Proposition 3.24.

######  Proposition 8.10.

Let
$B=K(G,S\times_{H}G,W_{1},W_{2},\phi^{\ast}\delta_{1},\phi^{\ast}\delta_{2})$
and set $U=\operatorname{Spec}B$. Then $B$ also satisfies property ($\dagger$)
and the natural morphism $U\to X$ is $G$-equivariant with $[U/G]\to[X/G]$
strongly étale.

Moreover, if we write $W_{i}^{\mathrm{fix},H}$ for the fixed part of $W_{i}$
considered as an $H$-representation, $\mathcal{W}_{i}^{\mathrm{fix},H}$ for
the $G$-equivariant summand of $\mathcal{W}_{i}$ corresponding to
$W_{i}^{\mathrm{fix},H}\otimes_{\mathbb{C}}\mathcal{O}_{S}$ under the
isomorphism $[S/H]\cong[S\times_{H}G/G]$, and let

$C=K\left(G,S^{H}\times_{H}G,\mathcal{W}_{1}^{\mathrm{fix},H},\mathcal{W}_{2}^{\mathrm{fix},H},\delta_{1}|_{S^{H}\times_{H}G},\delta_{2}|_{S^{H}\times_{H}G}\right),$

then $[\operatorname{Spec}C/G]$ is $[U/G]^{{\mathrm{max}}}$.

We are now able to explicitly compute the blow-up of $U$ along
$U^{G}=\operatorname{Spec}C$, after setting up some more notation.

Without loss of generality, we may assume that $x_{1},\ldots,x_{l}$ are the
$H$-moving coordinates of $S$, which lift to minimal generators of the ideal
of $S^{H}\times_{H}G$ in $S\times_{H}G$ and on which the $H$-action is linear.

Pick bases $\\{\underline{w}_{1}^{f}\\}$ for $W_{1}^{\mathrm{fix},H}$ and
$\\{\underline{w}_{1}^{m}\\}$ for $W_{1}^{\mathrm{mv},H}$. Denote their union
by $\\{\underline{w}_{1}\\}$.

By $H$-equivariance, the images of $\\{\underline{w}_{1}^{m}\\}$ in
$S\times_{H}G$ under $\delta_{1}$ must land in the ideal
$(x_{1},\ldots,x_{l})$. Write $s_{i}^{\mathrm{mv}}=\sum_{k}\alpha_{ik}x_{k}$
for the image of the $i$-th basis element $w_{1,i}^{m}$ and
$s_{i}^{\mathrm{fix}}\in\mathcal{O}_{V}$ for the image of the $i$-th basis
element $w_{1,i}^{f}$.

Since $\delta_{2}|_{V^{H}}$ vanishes (using property ($\dagger$)), picking a
basis for $(W_{2})^{\mathrm{mv},H}$, the images of its elements in
$\mathcal{W}_{1}$ must land inside $(x_{1},\ldots,x_{l})\mathcal{W}_{1}$ by
$G$-equivariance. Write
$p_{j}^{\mathrm{mv}}\coloneqq\sum_{k,\ell}\beta_{jk\ell}x_{\ell}\cdot w_{1,k}$
for the image of the $j$-th basis element. Picking a basis for
$(W_{2})^{\mathrm{fix},H}$, write $p_{j}^{\mathrm{fix}}$ for the image of the
$j$-th basis element $w_{2,j}^{\mathrm{fix}}$.

######  Lemma 8.11.

Let $U=\operatorname{Spec}B$ and $U^{G}=\operatorname{Spec}C$ be as in the
previous proposition. Then

(8.8) $\displaystyle
R_{U^{G}/U}\simeq\frac{B[t^{-1},\underline{v}]}{(\underline{v}t^{-1}-\underline{x},\\{\sum_{k}\alpha_{ik}v_{k}\\},\\{\sum_{k,\ell}\beta_{jk\ell}v_{\ell}\cdot
w_{1,k}\\})}.$

###### Proof.

For brevity, denote (by slight abuse of notation) $V=S\times_{H}G$. Define:

  1. (i)

$Q$ to be the derived affine $G$-scheme with

(8.9) $\displaystyle\mathcal{O}_{Q}=\mathcal{O}_{U}/(\underline{x}).$

  2. (ii)

$Y$ to be the derived affine $G$-scheme with

(8.10) $\displaystyle\mathcal{O}_{Y}=\mathcal{O}_{V}/(\underline{s}^{f})$

obtained by the $G$-equivariant cell attachment
$W_{1}^{\mathrm{fix},H}[0]\to\mathcal{O}_{V}$ in degree $0$. Observe that we
can also write

(8.11)
$\displaystyle\mathcal{O}_{Q}=\mathcal{O}_{Y}/(\underline{x},\underline{s}^{\mathrm{mv}},\underline{0}),$

with the appropriate $G$-equivariant cell attachments corresponding to
$\underline{x},\underline{s}^{\mathrm{mv}}$ understood and the cell attachment
$W_{2}[1]\to\mathcal{O}_{Y}$ given by the zero morphism. This is possible due
to the fact that the natural morphism $W_{2}[1]\to\mathcal{O}_{U}$ restricts
to zero after quotienting out by the ideal $(\underline{x})$ by assumption.

By definition, we similarly have that

(8.12)
$\displaystyle\mathcal{O}_{U^{G}}=\mathcal{O}_{Y}/(\underline{x},\underline{0}^{\mathrm{fix}}),$

where the cell attachment for $\underline{x}$ is as above and we use the zero
cell attachment $W_{2}^{\mathrm{fix},H}[1]\to\mathcal{O}_{Y}$ for
$\underline{0}^{\mathrm{fix}}$.

We have an induced commutative square of closed embeddings

and hence by Lemma 2.12 it follows that

$R_{U^{G}/U}=R_{Q/U}\otimes_{R_{Q/Y}}R_{Z/Y}.$

Using (8.9), (8.10), (8.11), (8.12) and the ($G$-equivariant) finite quotient
formula of Proposition 3.26, we obtain

$\displaystyle R_{Q/U}$
$\displaystyle=\frac{\mathcal{O}_{U}[t^{-1},\underline{v}]}{(\underline{v}t^{-1}-\underline{x})}$
$\displaystyle R_{Z/Y}$
$\displaystyle=\frac{\mathcal{O}_{Y}[t^{-1},\underline{v},\underline{u}^{\mathrm{fix}}]}{(\underline{v}t^{-1}-\underline{x},\
\underline{u}^{\mathrm{fix}}t^{-1})}$ $\displaystyle R_{Q/Y}$
$\displaystyle=\frac{\mathcal{O}_{Y}[t^{-1},\underline{v},\
\underline{e}^{\mathrm{mv}},\underline{u}]}{(\underline{v}t^{-1}-\underline{x},\
\underline{e}^{\mathrm{mv}}t^{-1}-\underline{s}^{\mathrm{mv}},\
\underline{u}t^{-1})}.$

The morphism $R_{Q/Y}\to R_{Z/Y}$ maps $\underline{e}^{\mathrm{mv}}$ and
$\underline{u}^{\mathrm{mv}}$ to zero, while the morphism $R_{Q/Y}\to R_{Q/U}$
maps $\underline{e}_{i}^{\mathrm{mv}}$ to $\sum_{k}\alpha_{ik}v_{k}\in
R_{Q/U}$ and $u_{j}^{\mathrm{mv}}$ to
$\sum_{k,\ell}\beta_{jk\ell}v_{\ell}\cdot w_{1,k}\in\mathcal{R}_{Q/U}$.

Thus, the homotopy pushout diagram

immediately gives (8.8). ∎

We obtain the following immediate corollary.

######  Corollary 8.12.

In the above notation, write $Y=S\times_{H}G$ and $GY^{H}=S^{H}\times_{H}G$.
Then $Y^{{\mathrm{intr}}}=\operatorname{Bl}_{GY^{H}}Y$ and
$\pi\colon\widehat{Y}=(\operatorname{Bl}_{GY^{H}}Y)^{\mathrm{ss}}\to Y$ with
exceptional divisor $E$.

Consider the sheaf of cdga’s $\widehat{B}$ generated by
$\mathcal{O}_{\widehat{Y}}$ in degree $0$,
$\pi^{\ast}\mathcal{W}_{1}^{\mathrm{fix},H}\oplus\pi^{\ast}\mathcal{W}_{1}^{\mathrm{mv},H}(E)$
in degree $1$ and
$\pi^{\ast}\mathcal{W}_{2}^{\mathrm{fix},H}\oplus\pi^{\ast}\mathcal{W}_{2}^{\mathrm{mv},H}(2E)$
in degree $2$ with differentials induced by (8.8).

Then $\widehat{B}$ satisfies property $(\dagger)$ and the derived Kirwan blow-
up $\widehat{U}$ is given by $\operatorname{Spec}\widehat{B}$.

###### Proof.

We only need to check that property ($\dagger$) is satisfied. But this follows
verbatim from the argument used in the proof of [KLS17, Lemma 5.3]. ∎

###  8.4. Perfectness of the truncated tangent complex, virtual fundamental
cycle and virtual structure sheaf

Let now $X$ be a derived Artin stack satisfying our usual assumptions, which
include the existence of a good moduli space for the classical truncation
$X_{\mathrm{cl}}$.

######  Lemma 8.13.

Suppose that $X$ admits a strongly étale cover by quotient stacks
$[\operatorname{Spec}A/G]$ where $A$ satisfies property $(\dagger)$. Then the
same holds for its Kirwan blow-up $\widehat{X}$.

###### Proof.

This is immediate from Corollary 8.12. ∎

######  Corollary 8.14.

The derived stabilizer reduction $\widetilde{X}$ of a $(-1)$-shifted
symplectic derived Artin stack $X$ admits a strongly étale cover by quotient
stacks $[\operatorname{Spec}A/G]$ where $A$ satisfies property $(\dagger)$.

###### Proof.

By Proposition 4.13, we know that $X$ admits a strongly étale cover by affine
quotient stacks $[\operatorname{Spec}A/G]$. Using Lemma 8.5, we deduce that
after possible equivariant shrinking, $A$ may be taken to satisfy property
$(\dagger)$. Using the Fundamental Lemma [Alp10, Theorem 6.10], further
shrinking ensures that the morphism $[\operatorname{Spec}A/G]\to X$ remains
strongly étale.

Since $\widetilde{X}$ is obtained by $X$ through a sequence of Kirwan blow-
ups, the claim follows by Lemma 8.13 and induction. ∎

This corollary together with Proposition 8.9 imply the main result of this
section. To match up with cohomological indexing, we write $(-)^{\geq-n}$ for
the truncation $\tau_{\leq n}(-)$. Likewise, we write
$h^{k}(-)\coloneqq\pi_{-k}(-)$.

######  Theorem 8.15.

Let $X$ be a derived $(-1)$-shifted symplectic Artin stack of finite type and
$\widetilde{X}$ its derived stabilizer reduction. Then the truncation
$E^{\bullet}=\tau^{[0,1]}\mathbb{T}_{\widetilde{X}}|_{\widetilde{X}_{\mathrm{cl}}}$
is a perfect complex of virtual dimension $0$.

Moreover, let $\\{U_{\alpha}\to\widetilde{X}\\}$ be a strongly étale cover by
quotient stacks $[\operatorname{Spec}A/G]$ where $A$ satisfies property
$(\dagger)$. Write $E_{\bullet}\coloneqq(E^{\bullet})^{\vee}$.

Then the complexes $E_{\bullet}|_{(U_{\alpha})_{\mathrm{cl}}}$ are part of the
data of a semi-perfect obstruction theory on $\widetilde{X}_{\mathrm{cl}}$,
which recovers the semi-perfect obstruction theory on
$\widetilde{X}_{\mathrm{cl}}$ constructed in [Sav20], and also of an almost
perfect obstruction theory which recovers the one constructed in [KS22].

In particular, $E_{\bullet}$ together with $\widetilde{X}$ recover the
$0$-dimensional virtual fundamental cycle
$[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{0}(\widetilde{X}_{\mathrm{cl}})$ and virtual structure sheaf
$[\mathcal{O}_{\widetilde{X}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{X}_{\mathrm{cl}})$ of the intrinsic stabilizer reduction
$\widetilde{X}_{\mathrm{cl}}$, constructed in [Sav20, KS22].

###### Proof.

The statement on the perfectness of $E_{\bullet}$ follows immediately by
Proposition 8.9 and the preceding corollary. The virtual dimension then
follows from Lemma 8.5.

Now, by Proposition 8.9, the complexes
$E_{\bullet}|_{(U_{\alpha})_{\mathrm{cl}}}$ define perfect obstruction
theories $\phi_{\alpha}\colon
E_{\bullet}|_{(U_{\alpha})_{\mathrm{cl}}}\to{\mathbb{L}}_{(U_{\alpha})_{\mathrm{cl}}}^{\geq-1}$
on $(U_{\alpha})_{\mathrm{cl}}$. We may assume that the $U_{\alpha}$ are
affine schemes after étale base change, since $U_{\alpha}$ are
Deligne–Mumford.

Write $\mathbb{T}_{\widetilde{X}_{\mathrm{cl}}}^{\leq
1}=({\mathbb{L}}_{\widetilde{X}_{\mathrm{cl}}}^{\geq-1})^{\vee}$. Then, by
construction of the $\phi_{\alpha}$, we have commutative diagrams

where $\psi\colon
E^{\bullet}\to\mathbb{T}_{\widetilde{X}}|_{\widetilde{X}_{\mathrm{cl}}}$ is
the natural morphism, and $\rho$ is the dual of the composition
${\mathbb{L}}_{\widetilde{X}}|_{\widetilde{X}_{\mathrm{cl}}}\to{\mathbb{L}}_{\widetilde{X}_{\mathrm{cl}}}\to{\mathbb{L}}_{\widetilde{X}_{\mathrm{cl}}}^{\geq-1}$.

In particular, restricting to any closed point
$p\in(U_{\alpha})_{\mathrm{cl}}$ and taking $h^{1}(-)$, we obtain commutative
diagrams

(8.17)

We have a global obstruction sheaf defined by
$\mathrm{Ob}_{X_{\mathrm{cl}}}\coloneqq h^{1}(E^{\bullet})$, which glues
together the local obstruction sheaves
$\mathrm{Ob}_{\alpha}=h^{1}(E^{\bullet}|_{U_{\alpha}})$ and the transition
functions are the same as the ones used in [Sav20], by construction.

It is now a simple verification to check that the local obstruction theories
satisfy the necessary compatibility condition to define a semi-perfect
obstruction theory as in [CL11] and that we hence get the same virtual cycle:
Let
$(U_{\alpha\beta})_{\mathrm{cl}}=(U_{\alpha})_{\mathrm{cl}}\times_{\widetilde{X}_{\mathrm{cl}}}(U_{\beta})_{\mathrm{cl}}$.
In the terminology of [KLS17, Definition 4.2], consider

an infinitesimal lifting problem at a closed point
$p\in(U_{\alpha\beta})_{\mathrm{cl}}$, where $\bar{\Delta}$ is an extension of
$\Delta$ by an ideal $I$. By [CL11, Lemma 2.6], this induces a canonical
obstruction element

$\displaystyle\omega(g,\Delta,\bar{\Delta})\in\mathrm{Ext}^{1}(g^{\ast}{\mathbb{L}}_{\widetilde{X}_{\mathrm{cl}}},I)=h^{1}(\mathbb{T}_{\widetilde{X}_{\mathrm{cl}}}^{\leq
1}|_{p})\otimes_{\mathbb{C}}I.$

We need to show that

$h^{1}(\phi_{\alpha}^{\vee}|_{p})(\omega(g,\Delta,\bar{\Delta}))=h^{1}(\phi_{\beta}^{\vee}|_{p})(\omega(g,\Delta,\bar{\Delta}))\\\
\in\mathrm{Ext}^{1}(g^{\ast}E_{\bullet},I)=h^{1}(E^{\bullet}|_{p})\otimes_{\mathbb{C}}I.$

But this follows immediately from diagram (8.17), since $h^{1}(\psi|_{p})$ is
an injection by the definition of $E_{\bullet}$ and Proposition 8.9 and thus
$h^{1}(\phi_{\alpha}^{\vee}|_{p})=h^{1}(\phi_{\beta}^{\vee}|_{p})$.

The claim for the almost perfect obstruction theory is similar and follows the
reasoning of [KS22, Subsection 5.4] identically. We leave the details to the
reader. ∎

##  9\. Sheaf-theoretic invariants

In this section, we explore immediate applications of our results to the
enumerative geometry of sheaves and complexes and Donaldson–Thomas theory in
particular.

Namely, we define virtual classes for moduli stacks of semi-stable sheaves and
complexes on smooth, projective surfaces.

Moreover, we obtain a fully derived upgrade of the construction of generalized
Donaldson–Thomas invariants via Kirwan blow-ups developed in [KLS17, Sav20].
These act as virtual counts of semi-stable sheaves and perfect complexes on
smooth, projective Calabi–Yau threefolds.

The robustness of derived geometry also allows us to apply the same method to
define new generalized Vafa–Witten invariants, enumerating semi-stable Higgs
pairs on projective surfaces.

###  9.1. Donaldson-type invariants of surfaces

Let $S$ be a smooth, projective surface and $\mathcal{M}$ a derived moduli
stack that parametrizes semi-stable sheaves or perfect complexes on $S$.
Possible choices of stability include Gieseker or slope stability [HL10] for
sheaves and Bridgeland stability [Bri07] for complexes. It is a standard fact
that $\mathcal{M}$ is then quasi-smooth.

If the genus $p_{g}=H^{0}(K_{S})$ of $S$ is positive or the rank of the
objects in question is zero, it might be necessary to consider a modification
of $\mathcal{M}$, for example, a “reduced” projective linear version as in
[Joy21], or fix the determinant, in order to remove certain trivial components
from the cotangent complex of $\mathcal{M}$. Without going into more detail,
we assume that this is the case so that $\mathcal{M}$ is a derived moduli
stack of semi-stable sheaves or perfect complexes on a smooth, projective
surface, which is of finite type with the good moduli space $M_{\mathrm{cl}}$
of $\mathcal{M}_{\mathrm{cl}}$ being proper.

By Theorem 7.1, the derived stabilizer reduction $\widetilde{\mathcal{M}}$ is
a proper quasi-smooth derived Deligne–Mumford stack and we have virtual
fundamental classes

$[\widetilde{\mathcal{M}}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{\ast}(\widetilde{\mathcal{M}}_{\mathrm{cl}}),\
[\mathcal{O}_{\widetilde{\mathcal{M}}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{\mathcal{M}}_{\mathrm{cl}}).$

######  Remark 9.1.

Following up on Remark 7.5, it would be interesting to compare these classes
with the virtual classes of $\mathcal{M}$ itself, constructed by Khan [Kha19],
and also with the homological virtual classes constructed in [Joy21].

We may thus define intersection-theoretic and $K$-theoretic Donaldson-type
invariants as follows.

######  Definition 9.2.

For any $\gamma\in H^{*}(\widetilde{\mathcal{M}}_{\mathrm{cl}},\mathbb{Q})$,
the intersection-theoretic generalized Donaldson-type invariant associated to
$\mathcal{M}$ is defined by the formula

$\mathrm{DT}(\mathcal{M},\gamma)=\int_{[\widetilde{\mathcal{M}}_{\mathrm{cl}}]^{\mathrm{vir}}}\gamma.$

Similarly, for any $\beta\in K^{0}(\widetilde{\mathcal{M}}_{\mathrm{cl}})$,
the $\beta$-twisted $K$-theoretic generalized Donaldson-type invariant
associated to $\mathcal{M}$ is defined by the formula

$\mathrm{DT}^{\mathrm{K-th},\beta}(\mathcal{M})=\chi\left(\widetilde{\mathcal{M}}_{\mathrm{cl}},[\mathcal{O}_{\widetilde{\mathcal{M}}_{\mathrm{cl}}}^{\mathrm{vir}}]\otimes_{\mathcal{O}_{\widetilde{\mathcal{M}}_{\mathrm{cl}}}}\beta\right).$

###  9.2. Donaldson–Thomas invariants of Calabi–Yau threefolds via Kirwan
blow-ups

Let $\mathcal{M}$ be a derived moduli stack parametrizing semi-stable sheaves
or perfect complexes on a smooth, projective Calabi–Yau threefold $W$, as
considered in [Sav20]. Possible choices of stability here are Gieseker
stability [HL10] for sheaves and PT, polynomial or Bridgeland stability
[Bri07, Bay09, PT09] for complexes.

As in [Sav20], $\mathcal{M}$ is $(-1)$-shifted symplectic and its classical
truncation $\mathcal{M}_{\mathrm{cl}}$ admits a good moduli space
$\mathcal{M}_{\mathrm{cl}}\to M_{\mathrm{cl}}$ with $M_{\mathrm{cl}}$ proper.

Let
$\mathcal{N}_{\mathrm{cl}}\coloneqq\mathcal{M}_{\mathrm{cl}}\\!\\!\fatslash\mathbb{G}_{m}$
be the rigidification of $\mathcal{M}_{\mathrm{cl}}$ by the scaling
automorphisms of sheaves or complexes. In [Sav20], it is shown that the
intrinsic stabilizer reduction $\widetilde{\mathcal{N}}_{\mathrm{cl}}$ of
$\mathcal{N}$ admits a natural semi-perfect obstruction theory of virtual
dimension zero. The degree of the induced virtual fundamental cycle
$[\widetilde{\mathcal{N}}_{\mathrm{cl}}]^{\mathrm{vir}}$ is defined to be the
generalized Donaldson–Thomas invariant via Kirwan blow-ups associated to
$\mathcal{M}$. By [KS22], the obstruction theory can be upgraded to an almost
perfect obstruction theory with induced virtual structure sheaf
$[\mathcal{O}_{\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{\mathcal{N}}_{\mathrm{cl}})$.

Our results above allow us to obtain a canonical derived enhancement as
follows.

######  Theorem 9.3.

There exists a derived Artin stack $\mathcal{N}$ with classical truncation
$\mathcal{N}_{\mathrm{cl}}$. The truncation
$E^{\bullet}=\tau^{[0,1]}\mathbb{T}_{\widetilde{\mathcal{N}}}|_{\widetilde{\mathcal{N}}_{\mathrm{cl}}}$
of the tangent complex is perfect, with induced virtual fundamental cycle
(using Theorem 8.15) equal to
$[\widetilde{\mathcal{N}}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{0}(\widetilde{\mathcal{N}}_{\mathrm{cl}})$ and induced virtual structure
sheaf equal to
$[\mathcal{O}_{\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{\mathcal{N}}_{\mathrm{cl}})$.

###### Proof.

The existence of $\mathcal{N}$ follows from [Hal20, Lem. 4.4.9], which
generalizes the rigidification construction to derived Artin stacks.

Now, observe that $\mathcal{M}$ admits a strongly étale cover by quotient
stacks $[\operatorname{Spec}A/G]$ where $A$ satisfies property $(\dagger)$.
Since $\mathcal{N}=\mathcal{M}\\!\\!\fatslash\mathbb{G}_{m}$, it follows that
$\mathcal{N}$ admits a strongly étale cover by the quotient stacks
$[\operatorname{Spec}A/(G/\mathbb{G}_{m})]$.

It is then clear that the arguments of Section 6 extend verbatim to show that
Kirwan blow-ups preserve the form of these étale covers and, in addition,
Deligne–Mumford stacks with such covers have perfect two-term truncation of
their tangent complex in amplitude $[0,1]$ (with cohomological grading), as
desired.

The equality of virtual fundamental cycles and virtual structure sheaves
follows identical arguments as for Theorem 8.15. ∎

######  Remark 9.4.

With minor modifications, the above also apply if one fixes the determinant of
the sheaves or complexes under consideration.

###  9.3. Stacks with a torus action

Let $X\in\mathsf{Stk}^{T}$ be a derived Artin stack with a $T$-action, for a
torus $T=\mathbb{G}_{m}\times\dots\times\mathbb{G}_{m}$.

######  Proposition 9.5.

Let $X\in\mathsf{Stk}^{T}$ be a derived stack with a $T$-action. Then
$X^{{\mathrm{max}}}$ admits a natural $T$-action and is a $T$-invariant closed
substack of $X$.

###### Proof.

Observe that the action map $X\times T\to X$ is affine, since $T$ is affine.
The claim then follows from functoriality of $(-)^{d}$ for representable and
separated morphisms by Proposition 4.19(ii) and $(X\times
T)^{{\mathrm{max}}}=X^{{\mathrm{max}}}\times T$. ∎

###  9.4. Vafa–Witten invariants of surfaces

Let $(S,\mathcal{O}_{S}(1))$ be a smooth, polarized projective surface and
$\pi\colon W=\mathrm{Tot}(K_{S})\to S$ the total space of its canonical bundle
with polarization $\pi^{\ast}\mathcal{O}_{S}(1)$. Let $T=\mathbb{G}_{m}$ be
the one-dimensional torus.

By the results of [TT20, TT17], there exists a $(-1)$-shifted symplectic
derived Artin stack $\mathcal{N}$, which parametrizes Gieseker semi-stable
Higgs pairs $(E,\phi\colon E\to E\otimes K_{S})$ on $S$ with fixed positive
rank and Chern classes, fixed determinant $\det E$ and zero trace
$\mathrm{tr}\ \phi=0$. $\mathcal{N}$ is a moduli stack of compactly supported
Gieseker semi-stable sheaves on $W$ with respect to the polarization
$\pi^{\ast}\mathcal{O}_{S}(1)$. It admits a natural $T$-action induced by
scaling the fibers of the projection map $W\to S$.

By construction, the classical truncation $\mathcal{N}_{\mathrm{cl}}$ is a GIT
quotient stack $[Q^{\mathrm{ss}}/G]$, where $Q$ is an appropriate Quot scheme
on $W$ and the $T$-action is induced by a $T$-action on $Q^{\mathrm{ss}}$,
which commutes with the $G$-action on $Q^{\mathrm{ss}}$. Moreover, the
$T$-fixed locus $\mathcal{N}_{\mathrm{cl}}^{T}$ has a good moduli space
$N_{\mathrm{cl}}^{T}$ which is a projective scheme.

We now see that, since $\mathcal{N}^{\max}$ is $T$-invariant by Proposition
9.5, $\widetilde{\mathcal{N}}$ admits a $T$-action such that the projection
$\widetilde{\mathcal{N}}\to\mathcal{N}$ is $T$-equivariant. Since
$N_{\mathrm{cl}}^{T}$ is proper, it follows that
$\widetilde{\mathcal{N}}_{\mathrm{cl}}$ is a proper Deligne–Mumford stack.
Moreover, Lemma 8.13, Corollary 8.14 and Theorem 8.15 apply $T$-equivariantly
and imply that
$\tau^{[0,1]}\mathbb{T}_{\widetilde{\mathcal{N}}}|_{\widetilde{\mathcal{N}}_{\mathrm{cl}}}$
is a $T$-equivariant perfect complex on
$\widetilde{\mathcal{N}}_{\mathrm{cl}}$ and the same is true for the $T$-fixed
component of the restriction to the fixed locus
$\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T}$, i.e.,
$\tau^{[0,1]}\mathbb{T}_{\widetilde{\mathcal{N}}}|_{{\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{T}}^{\mathrm{fix}}$,
which we know is the same as
$\tau^{[0,1]}\mathbb{T}_{\widetilde{\mathcal{N}}^{T}}|_{{\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{T}}$.
Write
$N^{\mathrm{vir}}\coloneqq\tau^{[0,1]}\mathbb{T}_{\widetilde{\mathcal{N}}}|_{{\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{T}}^{\mathrm{mv}}$
for the moving component.

In particular, by [Kie18, KS20], the $T$-fixed locus
${\widetilde{\mathcal{N}}_{\mathrm{cl}}}^{T}$ admits a virtual fundamental
cycle $[\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T}]^{\mathrm{vir}}$ and also a
virtual structure sheaf
$[\mathcal{O}_{\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T}}^{\mathrm{vir}}]\in
K_{0}^{T}(\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T})$. Thus, mirroring the
virtual torus localization formula [GP99, Lee04], we can give the following
definition.

######  Definition 9.6.

The numerical generalized Vafa–Witten invariant via Kirwan blow-ups associated
to $\mathcal{N}$ is defined by the formula

$\displaystyle\mathrm{VWK}(\mathcal{N})=\int_{\left[\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T}\right]^{\mathrm{vir}}}\frac{1}{e(N^{\mathrm{vir}})}.$

For any $\beta\in
K_{T}^{0}(\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T})\otimes_{\mathbb{Z}[t,t^{-1}]}\mathbb{Q}(t^{\frac{1}{2}})$,
the $\beta$-twisted $K$-theoretic generalized Vafa–Witten invariant via Kirwan
blow-ups associated to $\mathcal{N}$ is defined by the formula

$\displaystyle\mathrm{VWK}^{\mathrm{K-th},\beta}(\mathcal{N})=\chi_{t}\left(\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T},\frac{[\mathcal{O}_{\widetilde{\mathcal{N}}^{T}_{\mathrm{cl}}}^{\mathrm{vir}}]}{e(N^{\mathrm{vir}})}\otimes_{\mathcal{O}_{\widetilde{\mathcal{N}}_{\mathrm{cl}}^{T}}}\beta\right)\in\mathbb{Q}(t^{\nicefrac{{1}}{{2}}}).$

Here $t$ denotes the torus parameter and $\chi_{t}$ has the same meaning as in
[Tho20, Subsection 2.4].

It is interesting to investigate whether this definition satisfies the wall-
crossing formulas of [TT17, Joy21]. Deformation invariance of
$\mathrm{VWK}(\mathcal{N})$ follows from a $T$-equivariant version of the
arguments in [Sav20].

######  Remark 9.7.

The reason for the localization to $\mathbb{Q}(t^{\nicefrac{{1}}{{2}}})$ above
is to preserve consistency with the notation used in [Tho20], as well as the
classical case, where it might be necessary to consider half-integer weights
for the torus action. This happens already with the square root of the virtual
canonical bundle in the classical case.

######  Remark 9.8.

Observe that $\mathcal{N}$ does not have connected stabilizers, so we have to
perform the derived stabilizer reduction procedure with non-connected
stabilizers in this case.

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