Derived equivalence for quantum symplectic resolutions

Abstract

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson–Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon–Stafford (Gordon and Stafford in Adv Math 198(1):222–274, 2005; Duke Math J 132(1):73–135, 2006) and Kashiwara–Rouquier (Kashiwara and Rouquier in Duke Math J 144(3):525–573, 2008) as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities.

Appendix: Čech Complexes

In this section, we lay out the basics of the Čech complex for microlocalizations of ${\mathsf {A}}$. The story is essentially standard (cf. [55, 56] for the basics), but we do not know a reference that does everything we need in the form we need. As elsewhere in the paper, we assume that $G$ is a connected reductive group.

1.1 Modules and microlocalization

Let $S\subset \mathbf{C}[{\mathsf {W}}]$ be a subset of nonzero homogeneous elements (recall that $\mathbf{C}[{\mathsf {W}}]$ is graded via its identification with $\mathrm{gr }\,{\mathsf {A}}$). We let $\overline{S}$ denote the smallest multiplicatively closed subset $\overline{S}$ of $\mathbf{C}[{\mathsf {W}}]$ containing $S$, and let $\overline{S}_{\mathrm{sat }}$ denote any multiplicatively closed subset of ${\mathsf {A}}$ whose collection of principal symbols is exactly $\overline{S}$ (later we will make a specific such choice). We also let $\widetilde{S}$ denote the subset consisting of elements of $\overline{S}_{\mathrm{sat }}$ identified with homogeneous elements of the Rees algebra ${\mathcal {R}}= {\mathcal {R}}({\mathsf {A}})$.

Let $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$ denote the microlocalization of ${\mathcal {R}}({\mathsf {A}})$ at $\widetilde{S}$ as defined in [1], and let $Q^\mu _S({\mathsf {A}})$ denote the microlocalization of ${\mathsf {A}}$. More precisely, $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$ is a graded algebra over $\mathbf{C}[t]$ for which $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})/(t-1) = Q^\mu _S({\mathsf {A}})$. We next record a few basic properties of these algebras from [1]:

Lemma 7.1

• (1) For every $a\ne 0, Q^\mu _S({\mathsf {A}}) \cong \widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})/(t-a)$.

• (2) The algebra $Q^\mu _S({\mathsf {A}})$ comes equipped with a homomorphism ${\mathsf {A}}\rightarrow Q^\mu _S({\mathsf {A}})$ that makes $Q^\mu _S({\mathsf {A}})$ flat over ${\mathsf {A}}$ on both sides. Similarly, there is a natural graded homomorphism ${\mathcal {R}}\rightarrow \widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$ making $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$ flat over ${\mathcal {R}}$ on both sides.

• (3) All elements $s\in {\mathsf {A}}$ whose symbol lies in $\overline{S}$ become invertible in $Q^\mu _S({\mathsf {A}})$.

• (4) $Q^\mu _S({\mathsf {A}})$ comes equipped with a filtration $F_\bullet$ for which $F_k(Q^\mu _S({\mathsf {A}})) \cap {\mathsf {A}}= {\mathsf {A}}^k$. Moreover, $\mathrm{gr }_F(Q^\mu _S({\mathsf {A}})) \cong \overline{S}^{-1}\mathbf{C}[{\mathsf {W}}]\cong \widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})/(t)$.

• (5) If $M$ is an ${\mathsf {A}}$-module equipped with a good filtration, then $Q^\mu _S({\mathsf {A}})\otimes _{{\mathsf {A}}} M$ is naturally a filtered $Q^\mu _S({\mathsf {A}})$-module with associated graded isomorphic to $\overline{S}^{-1}\mathrm{gr }(M)$.

• (6) If $M$ is a finitely generated graded ${\mathcal {R}}({\mathsf {A}})$-module, then $\widetilde{Q}^\mu _{\widetilde{S}}(M): = \widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})\otimes _{{\mathcal {R}}({\mathsf {A}})} M$ is naturally a graded $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$-module with $\widetilde{Q}^\mu _{\widetilde{S}}(M)/t\widetilde{Q}^\mu _{\widetilde{S}}(M) \cong \overline{S}^{-1}(M/tM)$.

• (7) If $M$ is a filtered ${\mathsf {A}}$-module with Rees module $\widetilde{M}$, then for $a\ne 0, \widetilde{Q}^\mu _{\widetilde{S}}(\widetilde{M})/(t-a)\cong Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M$.

Define $\mathrm{Ind }_{\mathsf {A}}^{Q_S^\mu ({\mathsf {A}})}(M) = Q_S({\mathsf {A}})\otimes _{{\mathsf {A}}} M$. One has an adjoint pair of functors:

$$\begin{aligned} \mathrm{Ind }_{\mathsf {A}}^{Q_S({\mathsf {A}})} = Q_S({\mathsf {A}})\otimes _{{\mathsf {A}}} - : {\mathsf {A}}-\mathrm{mod } \leftrightarrows Q_S({\mathsf {A}})-\mathrm{mod }: \mathrm{Res }_{{\mathsf {A}}}^{Q_S({\mathsf {A}})}, \end{aligned}$$

where $\mathrm{Res }_{{\mathsf {A}}}^{Q_S({\mathsf {A}})}$ is the usual restriction functor. One gets a similar adjoint pair for graded ${\mathcal {R}}$-modules.

Lemma 7.2

We have

$$\begin{aligned} \mathrm{Ker }\big (\mathrm{Ind }_{\mathsf {A}}^{Q_S({\mathsf {A}})}\big )&= \Big \{ M \; \Big |\; SS(M) \subset \bigcup _{s\in S} V(s)\Big \}, \\ \mathrm{Ker }\big (\mathrm{Ind }_{\mathcal {R}}^{\widetilde{S}^{-1}({\mathcal {R}})}\big )&= \Big \{ M \; \Big |\; SS(M) \subset \bigcup _{s\in S} V(s)\Big \}. \end{aligned}$$

Proof

The equalities of the lemma are immediate from Lemma 7.1(5) for finitely generated modules, and follow for all modules by writing arbitrary modules as colimits of finitely generated ones. $\square$

1.2 Equivariant microlocal modules

Suppose that the connected reductive complex group $G$ acts on ${\mathsf {W}}$, that $\chi : G\rightarrow {\mathbb {G}}_m$ is a choice of character and that $S$ is a set of homogeneous (for the grading on $\mathbf{C}[{\mathsf {W}}]$ induced from $\mathrm{gr }\,{\mathsf {A}}$) $\chi ^{\mathbb {N}}$-semi-invariants (i.e., consists of elements $s$ each of which is $\chi ^{\ell (s)}$-semi-invariant for some $\ell (s)>0$). Then, $\overline{S}$ consists of $\chi ^{\mathbb {N}}$-semi-invariants, together with $1$, and we choose $\overline{S}_{\mathrm{sat }}$ to consist of $1$ together with all $\chi ^{\mathbb {N}}$-semi-invariants in ${\mathsf {A}}$ whose principal symbols lie in $\overline{S}$; this is a multiplicatively closed subset of ${\mathsf {A}}$. Note that then the $G$-action preserves the set of elements of $S, \overline{S}$ and $\overline{S}_{\mathrm{sat }}$ up to scalars; as a result, $G$ acts naturally on $Q_S^\mu ({\mathsf {A}})$ and $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$.

Recall that a vector $v$ in a $G$-representation $V$ is $G$ -rational (or simply rational if the group is understood) if there is a finite-dimensional rational $G$-subrepresentation $W\subseteq V$ with $v\in W$. Given a $G$-representation $V$, we let $V^{\mathrm{rat }}$ denote the subspace of $G$-rational vectors: it is a $G$-subrepresentation of $V$. For the following Lemma, it is convenient to note that a vector $v \in V$ is $G$-rational precisely if it lies in the image of a $G$-homomorphism $\rho :U \rightarrow V$ from a finite-dimensional rational $G$-representation $U$.

Lemma 7.3

If $V$ is a rational $G$-representation and $M$ is any $G$-representation, then $(M\otimes _{\mathbf{C}} V\big )^{\mathrm{rat }} = M^{\mathrm{rat }}\otimes _{\mathbf{C}} V$.

Proof

Since tensor products commute with colimits, we may assume that $V$ is finite-dimensional. If $u \in M\otimes V$ is a $G$-rational vector, pick a $G$-equivariant map $\rho :W \rightarrow M\otimes V$ from a finite-dimensional rational $G$-representation $W$ whose image contains $u$. We have a canonical morphism:

$$\begin{aligned} \Theta :\text {Hom}(W,M\otimes V) \rightarrow \text {Hom}(W\otimes V^{*},M), \end{aligned}$$

where if $f:W \rightarrow M\otimes V$, the linear map $\Theta (f)$ on $W\otimes V^{*}$ is given by the bilinear map $(w,\phi ) \mapsto (1\otimes \phi )(f(w))$, ($w \in W, \phi \in V^{*})$. The map $\Theta$ is evidently $G$-equivariant and hence the map $\Theta (\rho )$ is also. Thus to see that $u$ lies in $M^\mathrm{rat}\otimes V$, it suffices to note that $u$ lies in $\Theta (\rho )(W\otimes V^{*})\otimes V$, which is clear for example by writing the map $\rho$ in terms of a basis for $V$ and $W$. But it is clear that $M^{\mathrm{rat }}\otimes V \subseteq (M\otimes V)^{\mathrm{rat }}$ and hence the Lemma follows. $\square$

We write $\left( \widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}}), G\right) -\mathrm{Mod }$ for the category of weakly $G$-equivariant $\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}})$-modules.

Write $Z(S)\subset {\mathsf {W}}$ for the hypersurface defined as the union of hypersurfaces $V(s_i)$ for $s_i\in S$. Note that $Z(S)$ is the complementary closed subset to the open immersion $\mathrm{Spec }(\overline{S}^{-1}\mathbf{C}[{\mathsf {W}}])\hookrightarrow \mathrm{Spec }(\mathbf{C}[{\mathsf {W}}])$. By construction, $Z(S)$ is $G$-stable. Given a Lie algebra character $c:{\mathfrak {g}}\rightarrow \mathbf{C}$, let $({\mathsf {A}}, G, c)-\mathrm{mod }_{Z(S)}$ denote the localizing subcategory of $({\mathsf {A}}, G, c)-\mathrm{mod }$ whose objects are modules $M$ with $SS(M)\subseteq Z(S)$ (cf. Sect. 4.3).

Proposition 7.4

• (1) An object $M$ of $({\mathsf {A}}, G, c)-\mathrm{mod }$ lies in $({\mathsf {A}}, G, c)-\mathrm{mod }_{Z(S)}$ if and only if $\mathrm{Ind }_{\mathsf {A}}^{Q_S^\mu ({\mathsf {A}})}(M) = 0$.

• (2) The functor $\mathrm{Ind }_{\mathsf {A}}^{Q_S^\mu ({\mathsf {A}})}: ({\mathsf {A}}, G, c)-\mathrm{mod } \longrightarrow \big (\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}}), G\big )-\mathrm{Mod }$ factors uniquely through the quotient category $({\mathsf {A}}, G, c)-\mathrm{mod }\!/\!({\mathsf {A}}, G, c)-\mathrm{mod }_{Z(S)}$, and the induced functor

  $$\begin{aligned} ({\mathsf {A}}, G, c)-\mathrm{mod }/({\mathsf {A}}, G, c)-\mathrm{mod }_{Z(S)}\longrightarrow \big (\widetilde{Q}^\mu _{\widetilde{S}}({\mathsf {A}}), G\big )-\mathrm{Mod } \end{aligned}$$

  is faithful.

• (3) The quotient functor $\pi _S: ({\mathsf {A}}, G, c)-\mathrm{mod }\longrightarrow ({\mathsf {A}}, G, c)-\mathrm{mod }/({\mathsf {A}}, G, c)-\mathrm{mod }_{Z(S)}$ has right adjoint $\Gamma _S$ defined by $\Gamma _S\big (\pi _S(M)\big ) = (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M)^{\mathrm{rat }}.$ Moreover, the right adjoint $\Gamma _S$ is exact.

Moreover, analogous statements hold for $(G,c)$-equivariant graded ${\mathcal {R}}({\mathsf {A}})$-modules.

Proof

(1) follows from Lemma 7.2. (2) is immediate from the universal property of the quotient category.

To prove (3), we begin by recalling that the right adjoint $\Gamma _S$ applied to $\pi _S(M)$ is constructed as the colimit over maps $M\rightarrow M'$ whose kernel and cokernel have singular support in $Z(S)$. If $\tau (M)$ is the maximal submodule with singular support in $Z(S)$, then $\Gamma _S(\pi _S(M))= \Gamma _S(\pi _S(M/\tau (M)))$; in particular, we may assume that $\tau (M) = 0$. In this case, $\Gamma _S(\pi _S(M))$ is determined by being maximal among all extensions of $M$ whose cokernel has singular support in $Z(S)$ and that contain no nonzero submodule with singular support in $Z(S)$. Consider the natural injective map $\iota : M\rightarrow (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M)^{\mathrm{rat }}$. If $M$ is finitely generated over ${\mathsf {A}}$, then Lemma 7.1(5) implies that $SS(\mathrm{coker }(\iota ))\subseteq Z(S)$; since the target of $\iota$ commutes with colimits in $M$, the same inclusion of singular support holds for arbitrary $M$. Now suppose $f: M\rightarrow M'$ is any map in $({\mathsf {A}}, G,c)-\mathrm{mod }$ whose cokernel has singular support in $Z(S)$. Then, the natural map $Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M\rightarrow Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M'$ is an isomorphism, so the same is true after passing to rational vectors. We thus obtain a map $g: M'\rightarrow (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M)^{\mathrm{rat }}$ so that $g\circ f = \iota$. Passing to the colimit over such $M'$, we immediately obtain a map $\Gamma _S(\pi _S(M))\rightarrow (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} M)^{\mathrm{rat }}$. Since neither domain nor target has a submodule with singular support in $Z(S)$, the universal property of $\Gamma _S$ implies that this map is an isomorphism.

To prove the exactness claim of (3), observe that $\mathrm{Ind }_{\mathsf {A}}^{Q_S^\mu ({\mathsf {A}})}$ is exact. Hence, it suffices to prove that $(-)^{\mathrm{rat }}$ is exact on the image of $\mathrm{Ind }_{\mathsf {A}}^{Q_S^\mu ({\mathsf {A}})}$. Left exactness is clear. To check right exactness, we may assume that $M$ and $N$ are finitely generated ${\mathsf {A}}$-modules and that $M\rightarrow N$ is surjective; moreover, it suffices to check for a surjective map ${\mathcal {R}}(M)\rightarrow {\mathcal {R}}(N)$ corresponding to good filtrations of $M$ and $N$. Now, by the construction of [1], the map $\widetilde{Q}_{\widetilde{S}}^\mu ({\mathsf {A}})\otimes {\mathcal {R}}(M)\rightarrow \widetilde{Q}_{\widetilde{S}}^\mu ({\mathsf {A}})\otimes {\mathcal {R}}(N)$ is the limit of localizations of the maps ${\mathcal {R}}(M)/t^k{\mathcal {R}}(M)\longrightarrow {\mathcal {R}}(N)/t^k{\mathcal {R}}(N)$, which are surjective maps of rational $G$-modules by construction; for simplicity, write $M_k\rightarrow N_k$ for the maps of localized modules. A vector $v \in \widetilde{Q}_{\widetilde{S}}^\mu ({\mathsf {A}})\otimes {\mathcal {R}}(N)$ is a rational vector if and only if there are a finite-dimensional rational $G$-module $V$ and vector $\widetilde{v}\in V$ and a compatible sequence of $G$-maps $V\rightarrow N_k$ for all $k$ such that the image of $\widetilde{v}$ corresponds to $v$ in the inverse limit. Since $G$ is reductive and the maps $M_{k+1}\rightarrow M_k$ and $M_k \rightarrow N_k$ are all surjective maps of rational $G$-modules, given such a sequence we can find a compatible sequence of $G$-equivariant lifts $V\rightarrow M_k$ whose composites to $N_k$ agree with the given maps, and such that the images of $\widetilde{v}$ in the $M_k$ form a compatible sequence of vectors. Hence, in the limit, we get a lift of $v$ to a rational vector in $\displaystyle \lim \limits _\leftarrow {M}_{k} = \widetilde{Q}_{\widetilde{S}}^\mu ({\mathsf {A}})\otimes {\mathcal {R}}(M)$ (here, we use that ${\mathcal {R}}(M)$ is finitely generated; otherwise, the microlocalization may not equal the tensor product). $\square$

Corollary 7.5

Suppose that the set $S$ as above satisfies $\displaystyle {\mathsf {W}}^\mathrm{uns} \subseteq \bigcup \nolimits _{s\in S} V(s)$ (where instability is taken with respect to $\chi$). Then the functors $\pi _S$ factor canonically

through exact functors $j_S^{*}$.

Note that here (and elsewhere in this section) ${\mathcal {R}}-\mathrm{mod }$ denotes the category of graded ${\mathcal {R}}$-modules (and similarly for the equivariant categories).

1.3 Adjoints

Proposition 7.6

The functors $j_S^{*}$ of Corollary have right adjoints. More precisely,

• (1) the adjoints $j_{S*}$ are given by $j_{S*}M = \pi _c\circ \Gamma _S(M).$

• (2) Each $j_{S*}$ is exact.

Proof

Note that if an adjoint exists, then $\Gamma _c\circ j_{S*} = \Gamma _S$ by uniqueness of adjoints, and so $j_{S*} = \pi _c\circ \Gamma _c\circ j_{S*} = \pi _c\circ \Gamma _S$; and similarly for ${\mathcal {R}}$.

We give the proof only for ${\mathsf {A}}$; the proof for ${\mathcal {R}}$ is similar. We will show that the given formula defines a right adjoint. $\square$

We begin by proving:

Lemma 7.7

If $\widetilde{N} = \Gamma _S\pi _S(N)$, where $N$ is a $(G,c)$-equivariant ${\mathsf {A}}$-module, then $\Gamma _c\circ \pi _c(\widetilde{N}) = \widetilde{N}$. Similar statements hold for weakly equivariant and non-equivariant modules.

Proof

Since $\pi _S$ is a localization functor, we have $\pi _S\circ \Gamma _S = \mathrm{Id }$. It follows that $\Gamma _S\circ \pi _S(\widetilde{N}) = \widetilde{N}$. Now suppose that $f: \widetilde{N} \rightarrow N'$ is any map whose kernel and cokernel have unstable singular support. Then $\pi _S(\widetilde{N})\rightarrow \pi _S(N')$ becomes an isomorphism, and so adjunction defines a map $g: N'\rightarrow \Gamma _S\circ \pi _S(\widetilde{N}) = \widetilde{N}$ so that $g\circ f$ is the identity. Now $\Gamma _c\circ \pi _c(\widetilde{N})$ is the colimit over such maps $f$, and since split maps do not contribute to the colimit, the conclusion follows. $\square$

Returning to the proof of the proposition, we thus have

$$\begin{aligned} {\mathrm{Hom }}_{\mathcal {E}_{{\mathfrak {X}}}(c)}\big (\pi _c M, \pi _c\Gamma _S(\pi _S(N))\big )&= {\mathrm{Hom }}_{({\mathsf {A}}, G, c)}\big (M, \Gamma _c\pi _c\Gamma _S\pi _S(N)\big )\\&= {\mathrm{Hom }}_{({\mathsf {A}},G,c)}\big (M, \Gamma _S\pi _S(N)\big ) \;\; \text {by Lemma 7.7} \\&= \mathrm{Hom }\big (\pi _S(M),\pi _S(N)\big )\\&= \mathrm{Hom }\big (j_S^{*} \pi _c(M),\pi _S(N)\big ). \end{aligned}$$

Since $\pi _c$ and $\pi _S$ are essentially surjective, this proves (1). The exactness statement (2) is immediate from (1) by Proposition 7.4(3). This completes the proof of Proposition 7.6.

Proposition 7.8

Let $M = j_{S*} \pi _S(N)$ where $N$ is a twisted equivariant ${\mathsf {A}}$-module or ${\mathcal {R}}$-module. Then

$$\begin{aligned} {\mathbb {R}}\Gamma _c(M) \cong (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} N)^{\mathrm{rat }}, \text { respectively } {\mathbb {R}}\Gamma _c(M) \cong (\widetilde{Q}_{\widetilde{S}}^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} N)^{\mathrm{rat }}. \end{aligned}$$

Proof

Since $j_{S*}$ and $\pi _S$ are exact, we have

$$\begin{aligned} ({\mathbb {R}}\Gamma _c)\circ (j_{S*}\pi _S) = {\mathbb {R}}(\Gamma _c\circ j_{S*}\circ \pi _S) = {\mathbb {R}}(\Gamma _c\pi _c\Gamma _S\pi _S). \end{aligned}$$

But Lemma 7.7 implies that $\Gamma _c\pi _c\Gamma _S\pi _S = \Gamma _S\pi _S$, and the latter is exact by Proposition 7.4, so we get $({\mathbb {R}}\Gamma _c)\circ (j_{S*}\pi _S) = \Gamma _S\pi _S$, as required. $\square$

1.4 Čech complex

Choose a finite collection $\{f_i \; |\; 0\le i\le n\}$ of nonzero homogeneous $G$-semi-invariants, so that $\displaystyle \bigcup \nolimits _{i=0}^n D(f_i) = {\mathsf {W}}^{ss}$. For each nonempty subset $I\subseteq \{0,\ldots ,n\}$ we let $S_I = \{f_i\; | \; i\in I\}$. For any weakly $G$-equivariant ${\mathsf {A}}$-module $M$, we get

$$\begin{aligned} M_I \overset{\mathrm{def }}{=} \Gamma _{S_I}\pi _{S_I}(M) = \big (Q_{S_I}^{\mu }({\mathsf {A}})\otimes _{\mathsf {A}}M\big )^{\mathrm{rat }}. \end{aligned}$$

Given $I\subseteq J\subseteq \{0,\ldots ,n\}$, we get a map $M_I\rightarrow M_J$ induced by the natural map $Q_{S_I}^\mu ({\mathsf {A}})\rightarrow Q_{S_J}^\mu ({\mathsf {A}})$. We form the Čech complex $\check{C}^\bullet (M)$ with

$$\begin{aligned} \check{C}^k(M) = \prod _{|I| = k+1} M_I, \end{aligned}$$

and differentials defined as in [33, Section III.4]. If $M$ lies in $({\mathsf {A}},G,c)-\mathrm{mod }$, then $\check{C}^\bullet (M)$ is a complex in the same category. We make analogous definitions for graded ${\mathcal {R}}$-modules $M$ using $\widetilde{Q}_{\widetilde{S}_I}^\mu ({\mathsf {A}})$ in place of $Q^\mu _{S_I}({\mathsf {A}})$ and let $\check{C}^\bullet (M)$ denote the corresponding Čech complex.

Proposition 7.9

For a $G$-invariant left ideal $I\subseteq {\mathsf {A}}, \check{C}^\bullet ({\mathsf {A}}/I) \cong \check{C}^\bullet ({\mathsf {A}})\otimes _{{\mathsf {A}}} ({\mathsf {A}}/I)$.

Proof

Picking generators of the ideal $I$ and using the fact that ${\mathsf {A}}$ is a rational $G$-representation, we may find a finite-dimensional rational representation $V$ of $G$ and an equivariant exact sequence ${\mathsf {A}}\otimes _{\mathbf{C}} V \rightarrow {\mathsf {A}}\rightarrow {\mathsf {A}}/I\rightarrow 0$. Tensoring with $Q_S^\mu ({\mathsf {A}})$ over ${\mathsf {A}}$ and applying $(-)^{\mathrm{rat }}$ for any $S$ gives an exact sequence (Proposition 7.4) $\big (Q_S^\mu ({\mathsf {A}})\otimes _{\mathbf{C}} V\big )^{\mathrm{rat }} \rightarrow Q_S^\mu ({\mathsf {A}})^{\mathrm{rat }} \rightarrow \big (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} {\mathsf {A}}/I\big )^{\mathrm{rat }}\rightarrow 0$. Now by Lemma 7.3, the first term of the sequence is just $Q_S^\mu ({\mathsf {A}})^{\mathrm{rat }}\otimes _{\mathbf{C}} V$, and hence its image in $Q_S^\mu (A)^{\mathrm{rat }}$ is $Q_S^\mu (A)^{\mathrm{rat }}I$. It follows that

$$\begin{aligned} (Q_S^\mu ({\mathsf {A}})\otimes _{{\mathsf {A}}} {\mathsf {A}}/I)^{\mathrm{rat }} \cong Q_S^\mu ({\mathsf {A}})^{\mathrm{rat }}/Q_S^\mu (A)^{\mathrm{rat }}I \cong Q_S^\mu ({\mathsf {A}})^{\mathrm{rat }}\otimes _{{\mathsf {A}}} ({\mathsf {A}}/I). \end{aligned}$$

The result is then immediate from the definition of the Čech complex. $\square$

1.5 Rees modules

Suppose now that $M$ is an ${\mathsf {A}}$-module equipped with a good filtration. We obtain the associated Rees module ${\mathcal {R}}(M) = \oplus _k F_k(M)t^k \subseteq M[t, t^{-1}]$. This is an ${\mathcal {R}}= {\mathcal {R}}({\mathsf {A}})$-module; it is always torsion-free, hence flat, over $\mathbf{C}[t]$. Each microlocalization $Q_S^\mu ({\mathsf {A}})\otimes _{\mathsf {A}}M$ comes equipped with a filtration as well. Moreover, we have the following:

Proposition 7.10

Suppose that $M$ is an ${\mathsf {A}}$-module with good filtration. Then:

• (1) $\check{C}^\bullet \big ({\mathcal {R}}(M)\big )$ is a complex of torsion-free, hence flat, $\mathbf{C}[t]$-modules.

• (2) $\check{C}^\bullet \big ({\mathcal {R}}(M)\big )/(t-a)\cong \check{C}^\bullet (M)$ as ${\mathcal {R}}/(t-a){\mathcal {R}}\cong {\mathsf {A}}$-modules for every $a\ne 0$.

• (3) $\check{C}^\bullet \big ({\mathcal {R}}(M)\big )/(t)\cong \check{C}^\bullet (\mathrm{gr }(M))$ as $\mathbf{C}[{\mathsf {W}}]$-modules.

Here part (3) follows from Lemma 7.1(5).

Corollary 7.11

For any $({\mathsf {A}}, G)$-module $P$ with good filtration and any $i$, there is a natural $G$-equivariant isomorphism

$$\begin{aligned} \mathrm{gr }\big (H^i(\check{C}^\bullet P)\big ) \cong H^i((\check{C}^\bullet (\mathrm{gr }\, P)\big ) \end{aligned}$$

that is functorial in $P$.

Proof

This follows from the standard fact that cohomologies of complexes of flat modules commute with base change. $\square$

We record the following two lemmas that will be used in Sect. 5.2.

Lemma 7.12

(Lemma III.12.3 of [33]) Let $R$ be a noetherian ring and let $C^\bullet$ be a bounded-above complex of left $R$-modules such that for each $i, H^i(C^\bullet )$ is a finitely generated $R$-module. Then, there is a bounded-above complex $L^\bullet$ of finitely generated free $R$-modules and a quasi-isomorphism $g: L^\bullet \rightarrow C^\bullet$. Furthermore, if $S\subseteq R$ is a central subring of $R$ and all $C^i$ are flat over $S$, then for any $S$-module $M$, the induced map $g\otimes 1_M: L^\bullet \otimes M\rightarrow C^\bullet \otimes M$ is a quasi-isomorphism.

Lemma 7.13

Let ${\mathcal {R}}$ be a nonnegatively graded, noetherian, flat $\mathbf{C}[t]$-algebra (in particular, $\mathbf{C}[t]$ is central in ${\mathcal {R}}$). Suppose that $M$ is a graded ${\mathcal {R}}$-module of finite type and $N^\bullet$ is a complex of $\mathbf{C}[t]$-flat graded ${\mathcal {R}}$-modules. Then for any $a\in \mathbf{C}$,

$$\begin{aligned} \mathbf{C}[t]/(t-a)\otimes _{\mathbf{C}[t]}\mathrm{Hom }_{{\mathcal {R}}}(M, N^\bullet ) \cong \mathrm{Hom }_{{\mathcal {R}}/(t-a){\mathcal {R}}}(M/(t-a)M, N^\bullet /(t-a)N^\bullet ). \end{aligned}$$

1.6 Calculating with the Čech complex

Theorem 7.14

For every $M$ in $({\mathsf {A}},G,c)-\mathrm{mod }$, respectively $({\mathcal {R}},G,c)-\mathrm{mod }$, we have

• (1) $\pi _c M \simeq \pi _c \check{C}^\bullet (M)$, and

• (2) ${\mathbb {R}}\Gamma _c\circ \pi _c(M)\simeq \check{C}^\bullet (M)$.

Proof

It suffices to prove (1) for finitely generated modules $M$ since both sides commute with colimits. If $M$ is an ${\mathsf {A}}$-module, it can be equipped with an equivariant good filtration to give a finitely generated equivariant graded ${\mathcal {R}}$-module $\widetilde{M}$ for which $\widetilde{M}/(t-1)\widetilde{M} = M$. Since the functor $\mathbf{C}[t](t-1)\otimes -$ descends to the quotient categories and $\mathbf{C}[t](t-1)\otimes \big (\check{C}^\bullet (\widetilde{M})\big ) \simeq \check{C}^\bullet (M)$ (by Proposition 7.10), it will suffice to show that for a finitely generated ${\mathcal {R}}$-module $M$, the statement of (1) holds.

Let ${\mathcal {R}}-\mathrm{mod }^\mathrm{uns}$ denote the full subcategory of objects with unstable singular support. We have a commutative diagram:

with faithful rows. To check that the canonical map $\pi _c M \rightarrow \pi _c \check{C}^\bullet (M)$ is a quasi-isomorphism, then, it is enough to check after applying the functor $\mathrm{incl }$, hence to check that $\overline{\pi } M\simeq \overline{\pi } \check{C}^\bullet (M)$. Choose a finite, free graded ${\mathcal {R}}$-module resolution $F^\bullet \rightarrow M$. We get a map $\overline{\pi } F^\bullet \rightarrow \overline{\pi } \check{C}^\bullet (F^\bullet )$, and if this is a quasi-isomorphism (replacing the second double complex by its totalization), then the conclusion follows for $M$. Since both $\overline{\pi }$ and $\check{C}^\bullet$ commute with colimits, the quasi-isomorphism for $F^\bullet$ reduces to the corresponding statement for ${\mathcal {R}}$ itself.

We have $\mathbf{C}[t]/(t)\otimes _{\mathbf{C}[t]}\check{C}^\bullet ({\mathcal {R}}) \cong \check{C}^\bullet (\mathrm{gr }({\mathsf {A}}))$ and thus the associated graded of ${\mathcal {R}}\rightarrow \check{C}^\bullet ({\mathcal {R}})$ is the natural map $\mathbf{C}[{\mathsf {W}}]\rightarrow \check{C}^\bullet (\mathbf{C}[{\mathsf {W}}])$. The target of this last map is a complex of $\mathbf{C}[{\mathsf {W}}]$-modules that computes $H^\bullet ({\mathsf {W}}^{ss}, \mathcal {O})$. Since ${\mathsf {W}}$ is affine we have

$$\begin{aligned} \mathrm{supp }(H^i({\mathsf {W}}^{ss}, \mathcal {O}))\subseteq {\mathsf {W}}^\mathrm{uns} \;\;\;\; \text {for}\, i>0, \text {implying} \\ SS\big (\mathrm{Cone }[{\mathcal {R}}\rightarrow \check{C}^\bullet ({\mathcal {R}})]\big ) \subseteq {\mathsf {W}}^\mathrm{uns}. \end{aligned}$$

It follows that $\overline{\pi }{\mathcal {R}}\xrightarrow {\simeq } \overline{\pi }\check{C}^\bullet ({\mathcal {R}})$ is a quasi-isomorphism, proving (1).

For part (2), we have: ${\mathbb {R}}\Gamma _c(\pi _c(M)) \simeq {\mathbb {R}}\Gamma _c(\pi _c\check{C}^\bullet (M))\simeq \check{C}^\bullet (M)$, where the first isomorphism comes from part (1) of the present theorem and the second isomorphism follows from Proposition 7.8 and Proposition 7.6(1). $\square$