Equivariant constructible derived category of complex varieties

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field of characteristic zero, let $X$ be a separated finite-type complex algebraic variety (Classical algebraic prevarieties, regular maps, and varieties), regarded with its analytic topology as in Algebraically constructible complexes of analytic field sheaves, and let $G={\prod }_i{\mathrm{GL}}_{n_i}(\mathbb{C})$ be a finite product of complex general linear groups (Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$) acting algebraically on $X$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers). Equip the finite product $G$ with its usual topology and the product $G^p\times X$ with the product topology.

The action nerve. For $p\ge 0$ put $Z_p=G^p\times X$, where $G^0$ is a one-point space, so that $Z_0=X$, and let $d_i:Z_p\to Z_{p-1}$ for $p\ge 1$ $(0\le i\le p)$ and $s_i:Z_p\to Z_{p+1}$ for $p\ge 0$ $(0\le i\le p)$ be the face and degeneracy maps of the nerve of the translation groupoid $G\times X\rightrightarrows X$: writing $(g_1,\dots ,g_p,x)\in Z_p$, $d_0(g_1,\dots ,g_p,x)=(g_2,\dots ,g_p,x),d_p(g_1,\dots ,g_p,x)=(g_1,\dots ,g_{p-1},g_{p}x),$ $d_i(g_1,\dots ,g_p,x)=(g_1,\dots ,g_{i-1},g_i{g}_{i+1},g_{i+2},\dots ,g_p,x)(0<i<p),$ $s_i(g_1,\dots ,g_p,x)=(g_1,\dots ,g_i,e,g_{i+1},\dots ,g_p,x)\mathrm{.}$ These are continuous and satisfy the simplicial identities, so $Z_{\bullet }$ is a simplicial space in the sense of Simplicial objects, simplicial commutative rings and homotopy groups; equivalently, a simplicial operator $\alpha :[p]\to [m]$ acts by a continuous map $Z_{\alpha }:Z_m\to Z_p$. In general this action nerve has no augmentation to $X$: already the two faces $G\times X\rightrightarrows X$ are projection and action and need not agree. For a free action with principal quotient $P\to P\mathrm{/}G$, its action nerve is instead augmented over $P\mathrm{/}G$, as in Locally split Cech nerves give actual derived Cartesian sheaf descent.

Simplicial sheaves. A simplicial sheaf of $k$-vector spaces on $Z_{\bullet }$ is a family $F=(F_p{)}_{p\ge 0}$ in which $F_p$ is a sheaf of $k$-vector spaces on $Z_p$ (A sheaf on a topological space, Modules on a ringed space) together with structure morphisms $Z_{\alpha }^{-1}{F}_p\to F_m$ for every monotone $\alpha :[p]\to [m]$, subject to the identity and composition identities; a morphism $F\to G$ is a levelwise family of $k$-linear sheaf maps commuting with the structure morphisms. Kernels and cokernels are formed levelwise (cokernels after sheafification of the presheaf cokernel) and exactness is checked levelwise, so the category ${\mathrm{Sh}}_k(Z_{\bullet })$ of simplicial sheaves is abelian; under AC it carries explicit coinduced injective resolutions and a bounded derived category, with the conventions recorded in Locally split Cech nerves give actual derived Cartesian sheaf descent, whose bounded-below injective model uses Analytic field sheaves have injective derived models. Write $D^b({\mathrm{Sh}}_k(Z_{\bullet }))$ for that bounded derived category (Derived category of an abelian category), with its ordinary shift $[1]$ and cone triangles.

Cartesian cohomology. A complex $A$ of simplicial sheaves has Cartesian cohomology if for every monotone $\alpha :[p]\to [m]$ the structure morphism $Z_{\alpha }^{-1}{H}^j(A_p)\to H^j(A_m)$ is an isomorphism for every $j$; this is the condition that all structure morphisms of the simplicial sheaf $H^j(A)$ are isomorphisms, in the notation of Locally split Cech nerves give actual derived Cartesian sheaf descent.

Definition. The equivariant constructible derived category $D_{G,c}^b(X;k)$ is the full subcategory of $D^b({\mathrm{Sh}}_k(Z_{\bullet }))$ consisting of the complexes $A$ satisfying:

1. $A$ is bounded: the single complex $A$ has only finitely many nonzero cohomology simplicial sheaves $H^j(A)$, i.e. there are integers $a\le b$ with $H^j(A)=0$ for $j<a$ and $j>b$. This is uniform cohomological boundedness in the simplicial sheaf category; it is not a degreewise bound on the components $A_p$;
2. $A$ has Cartesian cohomology; and
3. the restriction $A_0$ of $A$ to the degree-zero space $Z_0=X$ has cohomology sheaves that are algebraically constructible with finite-dimensional stalks (Algebraically constructible complexes of analytic field sheaves).

Ext and restriction. Morphisms in $D_{G,c}^b(X;k)$ are morphisms in the ambient category $D^b({\mathrm{Sh}}_k(Z_{\bullet }))$; they are not morphisms in the derived category of naive $G$-equivariant sheaves. Set ${\mathrm{Ext}}_G^m(A,B)=\mathrm{Hom}(A,B[m])$ with the ordinary cochain shift. Evaluation at $p=0$ is exact because kernels, cokernels and cohomology of complexes of simplicial sheaves are formed levelwise, so it induces the forgetful functor (restriction to $X$) $D_{G,c}^b(X;k)\to D^b({\mathrm{Sh}}_k(X))$, $A\mapsto A_0$; its target is the bounded constructible category $D_c^b(X;k)$ of Algebraically constructible complexes of analytic field sheaves.

Well-formedness. This definition fixes only the ambient category, the object class and the conventions; the statements that make it usable — that $D_{G,c}^b(X;k)$ is a locally small triangulated category, that restriction to $X$ is conservative, that coinduced simplicial injectives compute its derived morphism groups, and that the finite general linear Stiefel resolutions realize it inside the bounded resolution triples on $X_N$ — are the content of Cartesian simplicial sheaves model actual equivariant derived Hom ↗, which is recorded in justified_by rather than in deps because that lemma depends on this definition. AC is inherited from the two construction suppliers named above and is used there for injective replacements; no further choice principle is introduced here, and the shift convention is the ordinary one of Derived category of an abelian category.