Equivariant Derived Categories

Introduction

The equivariant sheaves of Equivariant Sheaves and Descent form an abelian category $\mathrm{Sh}_G(X)$, and the derived category $D^b_G(X)$ of the equivariant sheaves is the equivariant derived category of the action. It carries a forgetful functor $\operatorname{For}:D^b_G(X)\to D^b(X)$ to the ordinary derived category of sheaves on $X$, induced by the exact forgetful functor of the equivariant structure, and the forgetful functor has both a left adjoint, the induction, and a right adjoint, the coinduction: these are the derived forms of the free and the cofree equivariant sheaves generated by an ordinary sheaf, and the pair of adjunctions $\operatorname{Ind}\dashv\operatorname{For}\dashv\operatorname{Coind}$ is the means by which the equivariant and the ordinary derived categories are compared. The equivariant derived category is the natural home of the equivariant cohomology, of the invariant part of a cohomology class and of the derived functors of the descent, and it is the operator-level counterpart of the element-level descent: the objects are the complexes with an equivariant structure, and the functors are the operations on them.

This article develops the equivariant derived category for a group acting on a space, the forgetful functor and its adjoints, the induced and coinduced objects, the exactness that makes the adjoints explicit, the six operations in the equivariant setting, and the boundary where the naive construction fails for a topological group and the Bernstein–Lunts refinement is required. It is the fifth article of the involutive layer of the category; the abelian category of equivariant sheaves and its descent are Equivariant Sheaves and Descent, the involution on cohomology is Cohomology with an Involution, the field-theoretic descent is Galois Descent for Sheaves, and the derived categories themselves are Part I, in Derived Categories and Derived Functors and Sheaf Cohomology.

The article uses no analysis and no geometry: the derived category is the derived category of an abelian category of sheaves, the groups are discrete unless the Bernstein–Lunts refinement is named, and no form, no norm and no derivative occurs. Throughout, a group $G$ acts on the space $X$, $\mathrm{Sh}_G(X)$ is the abelian category of equivariant sheaves of Equivariant Sheaves and Descent, $D^b_G(X)=D^b(\mathrm{Sh}_G(X))$ is its bounded derived category, $\operatorname{For}=\operatorname{For}_G$ is the forgetful functor, and $\operatorname{Ind}$ and $\operatorname{Coind}$ are its left and right adjoints. The group is discrete, so that $\mathrm{Sh}_G(X)$ is a Grothendieck category by Equivariant Sheaves and Descent, and it is written finite when the products are finite.

The Derived Category of Equivariant Sheaves

Definition. The equivariant derived category of a $G$-space $X$ is the derived category $D^b_G(X)=D^b(\mathrm{Sh}_G(X))$ of the abelian category of equivariant sheaves; its objects are the bounded complexes of equivariant sheaves and its morphisms are the morphisms in the derived category, in the sense of Derived Categories of Part I.

Proposition (identification with the groupoid). The category $\mathrm{Sh}_G(X)$ of equivariant sheaves is the category of sheaves on the action groupoid $X/\!/G$, and the equivariant derived category is $D^b_G(X)\cong D^b(X/\!/G)$; when the action is free and properly discontinuous the descent of Equivariant Sheaves and Descent is an equivalence of abelian categories and induces an equivalence of derived categories,

$$ D^b_G(X)\simeq D^b(X/G). $$

Proof. The identification of the abelian categories is that of Equivariant Sheaves and Descent; an equivalence of abelian categories induces an equivalence of their derived categories because it is exact and preserves injectives and acyclic resolutions. The descent equivalence is the theorem of Equivariant Sheaves and Descent for a free properly discontinuous action.

Proposition (Grothendieck structure). For a discrete group $G$ the category $\mathrm{Sh}_G(X)$ is a Grothendieck category with enough injectives and a generator, and $D^b_G(X)$ is well defined with the usual functoriality; the forgetful functor is exact and faithful.

Proof. A category of sheaves on a site is a Grothendieck category with enough injectives and a generator, and the action groupoid with discrete $G$ is a site of the requisite kind; the forgetful functor is exact and faithful by Equivariant Sheaves and Descent.

The Forgetful Functor and its Adjoints

Definition. The forgetful functor $\operatorname{For}:\mathrm{Sh}_G(X)\to\mathrm{Sh}(X)$ sends an equivariant sheaf to its underlying sheaf, forgetting the isomorphisms $\varphi_g$; it is exact by Equivariant Sheaves and Descent and induces an exact functor $\operatorname{For}:D^b_G(X)\to D^b(X)$ on the derived categories, the same symbol, which needs no derivation.

Theorem (the adjoints of the forgetful functor). The forgetful functor has a left adjoint $\operatorname{Ind}$ and a right adjoint $\operatorname{Coind}$, the induction and the coinduction, given on a sheaf $\mathcal{F}$ by

$$ \operatorname{Ind}(\mathcal{F})=\bigoplus_{g\in G}g_*\mathcal{F},\qquad \operatorname{Coind}(\mathcal{F})=\prod_{g\in G}g_*\mathcal{F}, $$

each with the equivariant structure under which $\varphi_h$ permutes the summands or the factors; on the derived categories the forgetful functor of the equivariant derived category has the derived adjoints

$$ \operatorname{Ind}\ \dashv\ \operatorname{For}\ \dashv\ \operatorname{Coind},\qquad \operatorname{Ind},\operatorname{Coind}:D^b(X)\to D^b_G(X), $$

and when $G$ is finite, $\operatorname{Ind}$ and $\operatorname{Coind}$ are exact and the derived adjoints are the adjoints themselves.

Proof. A morphism $\operatorname{Ind}(\mathcal{F})\to\mathcal{G}$ of equivariant sheaves is determined by its restriction to the $e$-summand, which is an arbitrary morphism $\mathcal{F}\to\operatorname{For}(\mathcal{G})$, and conversely an arbitrary morphism extends uniquely by equivariance: this is the universal property of the free equivariant sheaf, giving $\operatorname{Ind}\dashv\operatorname{For}$. A morphism $\mathcal{G}\to\operatorname{Coind}(\mathcal{F})$ is by the adjunction $g^*\dashv g_*$ a compatible family of morphisms $g^*\mathcal{G}\to\mathcal{F}$, which is to say a morphism $\operatorname{For}(\mathcal{G})\to\mathcal{F}$: this gives $\operatorname{For}\dashv\operatorname{Coind}$. Exactness for finite $G$: a finite direct sum or product of the exact functors $g_*$ is exact, so the functors pass to the derived categories without derivation; for infinite $G$ one takes the derived functors, which exist because $\mathrm{Sh}_G(X)$ has enough injectives.

Remark (the two adjoints are the free and the cofree objects). The induction is the free equivariant sheaf on an ordinary sheaf and the coinduction the cofree one; when $G$ is finite the two coincide, $\operatorname{Ind}\cong\operatorname{Coind}=\bigoplus_{g\in G}g_*\mathcal{F}$, which is the norm of a finite group action, and the two adjunctions collapse to the single self-adjointness of the norm. The distinction is genuine for an infinite group, where the free and the cofree objects differ.

Proposition (the adjoints on the invariants). The descent functor $q_*^{G}:\mathrm{Sh}_G(X)\to\mathrm{Sh}(X/G)$ has a left adjoint $q^*$ and, in the free properly discontinuous case, is an equivalence; on the derived categories, $Rq_*^{G}$ is right adjoint to $Lq^*=q^*$ and computes the derived invariants of an equivariant complex.

Proof. The adjunction $q^*\dashv q_*^{G}$ is that of Equivariant Sheaves and Descent, and its derived form is the standard one; the equivalence in the free case is the descent theorem, and the derived statements follow because an equivalence induces an equivalence of the derived categories.

The Six Operations in the Equivariant Setting

Proposition (the equivariant functors). An equivariant morphism $f:X\to Y$ of $G$-spaces induces functors between the equivariant derived categories, $f^*,f_*,f_!,f^!:D^b_G(-)\to D^b_G(-)$, the pullback and the pushforward with and without support, and the equivariant tensor product and $\mathcal{H}om$; they satisfy the base change and projection formulas of Derived Functors and Sheaf Cohomology, and they commute with the forgetful functor in the sense that $\operatorname{For}\circ f^*=f^*\circ\operatorname{For}$ and $\operatorname{For}\circ f_*=f_*\circ\operatorname{For}$ for the functors on the underlying spaces.

Proof. Each functor is constructed on the category of equivariant sheaves with its equivariant structure and then derived; the commutation with the forgetful functor is the fact that the forgetful functor is exact and compatible with the pullback and the pushforward of sheaves, and the formulas are those of Derived Functors and Sheaf Cohomology transported.

Remark (the equivariant cohomology). The equivariant cohomology of $X$ with coefficients in an equivariant complex $\mathcal{F}$ is $H^*(X,\operatorname{For}\mathcal{F})$ with the action of the group of Cohomology with an Involution; the derived category gives the natural setting for the operations on the equivariant classes, and the invariant part is the fixed part of the action. The derived refinement of the transfer of Cohomology with an Involution is the statement that the forgetful functor has a right adjoint, which produces the invariant part.

Remark (the topological group). For a topological group the naive category $\mathrm{Sh}_G(X)$ has too few objects — an equivariant sheaf for a connected group is often forced to be locally constant along the orbits, and the descent fails with no covering — and the derived category $D^b(\mathrm{Sh}_G(X))$ is not the correct equivariant derived category. The Bernstein–Lunts equivariant derived category $D^b_G(X)$ is constructed instead by resolving objects by complexes of "equivariant injectives" built from the simplicial space $G^\bullet\times X$ of the action, and it carries a forgetful functor with the same two adjoints and the six operations. The construction is the topological refinement of the present article and belongs to the later parts of the series; for a discrete group it agrees with $D^b(X/\!/G)$.

Proof (sketch). The simplicial space $G^{\bullet}\times X$ is the nerve of the action; a complex of sheaves on it with the descent conditions is an equivariant object, and the "equivariant injectives" are the complexes obtained by pushing forward from the nerve. The agreement with the discrete case is the identification of the nerve construction with the action groupoid when the group is discrete. This article does not develop the construction; it names it as the boundary of the naive theory.

Worked Cases

The Trivial Group

For $G=\{e\}$ the category $\mathrm{Sh}_G(X)$ is $\mathrm{Sh}(X)$, the equivariant derived category is $D^b(X)$, and the forgetful functor is the identity; the induction and the coinduction are the identity, and the adjunctions are the trivial ones. The case is the degenerate check and the boundary that the functor $\operatorname{For}$ be an equivalence exactly when the action is trivial.

The Free Involution

For $G=\mathbb{Z}/2$ acting freely with quotient $q$, the descent is an equivalence and induces $D^b_G(X)\simeq D^b(X/\langle\sigma\rangle)$; the forgetful functor becomes, under the equivalence, the pullback $q^*$, its right adjoint is the pushforward $q_*$, and the transfer of Cohomology with an Involution is the derived form of the adjunction. The example is the case in which every construction reduces to the ordinary derived category of the quotient.

The Constant Sheaf with a Character

For the constant equivariant sheaf attached to a one-dimensional representation $\chi$ of a finite group, the induction of the trivial sheaf is the equivariant sheaf $\bigoplus_{g}g_*\underline{\Bbbk}$ with the regular representation, and the coinduction agrees; the fixed part of the regular representation is one-dimensional, and the invariant cohomology of the induction is the cohomology of the quotient. The example shows the norm of a finite group action in the simplest case.

Summary

The equivariant derived category $D^b_G(X)$ is the derived category of the abelian category of equivariant sheaves, equivalently the derived category of sheaves on the action groupoid $X/\!/G$; for a free and properly discontinuous action the descent is an equivalence of abelian categories and of their derived categories, $D^b_G(X)\simeq D^b(X/G)$. The forgetful functor to the ordinary derived category is exact and has a left adjoint, the induction $\operatorname{Ind}(\mathcal{F})=\bigoplus_g g_*\mathcal{F}$, and a right adjoint, the coinduction $\operatorname{Coind}(\mathcal{F})=\prod_g g_*\mathcal{F}$, each with the equivariant structure that permutes the summands or the factors; for a finite group the two coincide and are exact, giving the norm, and for an infinite discrete group they differ and are derived. The descent functor $q_*^{G}$ is right adjoint to the pullback and computes the derived invariants, the six operations of the sheaf theory pass to the equivariant setting and commute with the forgetful functor, and the equivariant cohomology is the cohomology of the underlying complex with the action of the group.

For a topological group the naive derived category is too small and the Bernstein–Lunts refinement, built from the nerve of the action, is the correct equivariant derived category; it agrees with the discrete construction and is the boundary of the present article.

Summary of Notation

Symbol	Meaning
$\mathrm{Sh}_G(X)$	abelian category of $G$-equivariant sheaves; sheaves on $X/\!/G$
$D^b_G(X)=D^b(\mathrm{Sh}_G(X))$	equivariant derived category
$\operatorname{For}:D^b_G(X)\to D^b(X)$	forgetful functor; exact, faithful
$\operatorname{Ind}(\mathcal{F})=\bigoplus_{g\in G}g_*\mathcal{F}$	induction, left adjoint of the forgetful functor
$\operatorname{Coind}(\mathcal{F})=\prod_{g\in G}g_*\mathcal{F}$	coinduction, right adjoint of the forgetful functor
$\operatorname{Ind}\dashv\operatorname{For}\dashv\operatorname{Coind}$	the two adjunctions
$\operatorname{Ind}\cong\operatorname{Coind}=\bigoplus_{g}g_*\mathcal{F}$	norm, for a finite group
$q_*^{G}$	descent functor; right adjoint of $q^*$, computes derived invariants
$f^,f_,f_!,f^!$	equivariant six operations; commute with $\operatorname{For}$
$G^\bullet\times X$, Bernstein–Lunts $D^b_G(X)$	nerve of the action; refinement for a topological group