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Equivariant Constructible Perverse Foundations

1 · Prerequisites

2 · Summary

Objective. This page builds the equivariant constructible derived category attached to an algebraic action of a product of complex general linear groups on a separated finite-type complex algebraic variety, together with the finite Stiefel frame resolutions that make its Hom and Ext groups computable. The category is defined through the simplicial action nerve and the complexes with Cartesian cohomology whose degree-zero restriction is algebraically constructible; its well-formedness, the conservativity of the forgetful restriction, the finite resolution-triple comparison and idempotent completeness are proved directly, with the coinduced simplicial injective model supplied by the locally split Cech descent input.

The second half develops the cohomological toolkit. A separate finite cohomology-gap lemma records the vanishing of the constant-sheaf cohomology of the general linear frame spaces in the Stiefel range; the acyclic-fiber lemma then converts that gap into bounded Hom comparisons, an interval full-faithfulness statement and the resolution-triple image criterion, with extension closure reserved for the strict acyclicity bound. The six equivariant operations, proper base change and the normalized dualizing object are assembled from the ordinary constructible calculus through the finite resolutions; the middle-perverse t-structure, its intersection complexes and the degreewise finiteness of equivariant Ext with its characteristic-zero scalar compatibility are transported through the same normalized bridge. The companion page computes the smallest nontrivial example, where the trivial action of the multiplicative group on a point already exhibits degreewise finiteness without total finiteness.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Equivariant constructible derived category of complex varieties

Definition

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 $G = \prod_{i} {GL}_{n_{i}} (C)$ be a finite product of complex general linear groups (Invertible matrices and the general linear group ${GL}_{n} (F)$ ) 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 $p \geq 0$ 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 $p \geq 1$ $(0 \leq i \leq p)$ and $s_{i} : Z_{p} \to Z_{p + 1}$ for $p \geq 0$ $(0 \leq i \leq p)$ be the face and degeneracy maps of the nerve of the translation groupoid $G \times X ⇉ X$ : writing $(g_{1}, \dots, g_{p}, x) \in Z_{p}$ , $d_{0} (g_{1}, \dots, g_{p}, x) = (g_{2}, \dots, g_{p}, x), d_{p} (g_{1}, \dots, g_{p}, x) = (g_{1}, \dots, g_{p - 1}, g_{p} x),$ $d_{i} (g_{1}, \dots, g_{p}, x) = (g_{1}, \dots, g_{i - 1}, g_{i} g_{i + 1}, g_{i + 2}, \dots, g_{p}, x) (0 < i < p),$ $s_{i} (g_{1}, \dots, g_{p}, x) = (g_{1}, \dots, g_{i}, e, g_{i + 1}, \dots, g_{p}, x) .$ These are continuous and satisfy the simplicial identities, so $Z_{∙}$ is a simplicial space in the sense of Simplicial objects, simplicial commutative rings and homotopy groups; equivalently, a simplicial operator $α : [p] \to [m]$ acts by a continuous map $Z_{α} : Z_{m} \to Z_{p}$ . In general this action nerve has no augmentation to $X$ : already the two faces $G \times X ⇉ X$ are projection and action and need not agree. For a free action with principal quotient $P \to P / G$ , its action nerve is instead augmented over $P / G$ , as in Locally split Cech nerves give actual derived Cartesian sheaf descent.

Simplicial sheaves. A simplicial sheaf of $k$ -vector spaces on $Z_{∙}$ is a family $F = (F_{p})_{p \geq 0}$ 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 $Z_{α}^{- 1} F_{p} \to F_{m}$ for every monotone $α : [p] \to [m]$ , 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 ${Sh}_{k} (Z_{∙})$ 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 $D^{b} ({Sh}_{k} (Z_{∙}))$ 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 $α : [p] \to [m]$ the structure morphism $Z_{α}^{- 1} H^{j} (A_{p}) \to H^{j} (A_{m})$ 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 $D^{b} ({Sh}_{k} (Z_{∙}))$ consisting of the complexes $A$ satisfying:

$A$ is bounded: the single complex $A$ has only finitely many nonzero cohomology simplicial sheaves $H^{j} (A)$ , i.e. there are integers $a \leq b$ 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 $A_{p}$ ;
$A$ has Cartesian cohomology; and
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 $D^{b} ({Sh}_{k} (Z_{∙}))$ ; they are not morphisms in the derived category of naive $G$ -equivariant sheaves. Set ${Ext}_{G}^{m} (A, B) = Hom (A, B [m])$ 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 $X$ ) $D_{G, c}^{b} (X; k) \to D^{b} ({Sh}_{k} (X))$ , $A \mapsto A_{0}$ ; 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.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Acyclic product fibers stabilize bounded constructible pullback Hom

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $f : Y \to S$ be an algebraic map that is a locally trivial bundle in the finite constructible local-product setting of Finite simplicial constructible sheaves have an actual derived incidence model and Complex algebraic constructible complexes have finite conical cochain models: $S$ and $Y$ are the analytic spaces of separated finite-type complex varieties, $Y$ trivializes over charts $V$ of $S$ carrying the finite conical cochain models of the second supplier as products $V \times F \to V$ , and the nonempty connected finite-type smooth fiber $F$ carries a finite deleted-boundary model. Suppose the constant-sheaf cohomology of the fiber is $k$ in degree zero and vanishes in degrees $1, \dots, r$ : $H^{0} (F; k) = k$ and $H^{j} (F; k) = 0$ for $1 \leq j \leq r$ , computed by the finite model. Finally, in the setting the pushforward of every bounded algebraically constructible complex considered below again has bounded algebraically constructible cohomology, with the finite models supplied by the two named lemmas. Then, for bounded algebraically constructible complexes of $k$ -sheaves:

for $B \in D_{c}^{\geq b} (S; k)$ the cone of the unit $B \to R f_{*} f^{*} B$ belongs to $D^{\geq b + r + 1}$ ;
pullback induces an isomorphism $Hom (A, B [m]) ≅ Hom (f^{*} A, f^{*} B [m])$ whenever $A \in D_{c}^{\leq a} (S; k)$ , $B \in D_{c}^{\geq b} (S; k)$ and $m \leq b + r - a$ ;
if $a - b \leq r$ , then $f^{*}$ is fully faithful on the full subcategory $D_{c}^{[b, a]} (S; k)$ ;
if $a - b \leq r$ , then an object $C \in D_{c}^{[b, a]} (Y; k)$ lies in the image of $f^{*} : D_{c}^{[b, a]} (S; k) \to D_{c}^{[b, a]} (Y; k)$ if and only if the counit $f^{*} τ^{\leq a} R f_{*} C \to C$ is an isomorphism; the image is closed under direct summands, and it is closed under extensions when the stronger bound $r \geq a - b + 1$ holds;
no global tensor splitting $R f_{*} f^{*} B ≃ B \otimes R Γ (F; k)$ is asserted: the comparison is local and genuine characteristic-class transgressions are preserved.

Facts & Assumptions

Given: The field $k$ ; the algebraic bundle $f : Y \to S$ of the finite constructible local-product setting with nonempty connected finite-type smooth fiber $F$ ; integers $b \leq a$ and the acyclicity bound $r \geq 0$ of the Statement; the Axiom of Choice The Axiom of Choice; the derived category conventions of Derived category of an abelian category (in particular $H^{n} (X [k]) = H^{n + k} (X)$ and cones ending in $X [1]$ ) and the Hom conventions of Derived hom in the bounded setting.

[F1]

The local product clause of Finite simplicial constructible sheaves have an actual derived incidence model: for a local product projection $pr : V \times F \to V$ with supplied finite deleted-boundary models there is a natural comparison $R {pr}_{*} {pr}^{*} B ≃ B \otimes_{k} C^{*} (F; k)$ , natural under restriction to smaller product charts and taking the unit of the adjunction to $B \otimes (k \to C^{*} (F; k))$ ; the model $C^{*} (F; k)$ is a bounded complex of finite-dimensional $k$ -vector spaces, and the comparison commutes with every coefficient field extension. Its essential image description is by bounded finite-rank locally constant cohomology on the open simplices.

[F2]

The finite-model clause of Complex algebraic constructible complexes have finite conical cochain models: individually chosen compatible smooth Whitney strata, conical local products and finite deleted-boundary triangulation models for each operation, so that bounded algebraically constructible objects have bounded finite ordinary and compact-support cohomology, ordinary derived Hom and bounded constructible internal Hom with coefficient-extension compatibility, computed locally on the models and assembled by finite affine-open descent. In the setting of the Statement the same clause supplies the common finite models for the bundle chart and the pushforwards occurring below.

[F3]

The derived adjunction of Derived tensor–Hom and inverse/direct image for field sheaves: inverse image $f^{*} = f^{- 1}$ of $k$ -sheaves is exact, $f^{*} ⊣ R f_{*}$ on the bounded derived categories, with $R f_{*}$ computed on bounded-below $k$ -linear injective replacements.

[F4]

The injective model of Analytic field sheaves have injective derived models: under AC every sheaf category has functorial injective embeddings and bounded-below injective replacements, and morphisms of the bounded derived category are computed by these replacements, so the Hom groups of the Statement are set-sized and every distinguished triangle induces a long exact sequence of Hom groups.

[F5]

The canonical t-structure of Canonical t structure on a derived category, proved by The canonical pair is a t structure: $Hom (D^{\leq n}, D^{\geq n + 1}) = 0$ , $D^{\leq n} = D^{\leq 0} [- n]$ and $D^{\geq n} = D^{\geq 0} [- n]$ ; with the shift convention of Derived category of an abelian category, $D^{\geq c} [m] \subseteq D^{\geq c - m}$ and $D^{\leq c} [m] \subseteq D^{\leq c - m}$ . Canonical truncations exist and fit distinguished triangles (Canonical truncation of a complex, Canonical truncations fit a distinguished triangle), and the bounded derived category is triangulated with TR1-TR4 (The derived category inherits a triangulated structure, Localized cone triangles satisfy tr one through tr three).

[F6]

Constructibility: $D_{c}^{b} (-; k)$ consists of bounded complexes with algebraically constructible cohomology. Common finite algebraic refinements and preservation by algebraic inverse image are supplied by the finite constructible setting [F2]; cone closure on such a refinement is Analytic field sheaves have injective derived models. Vanishing of cohomology sheaves is local and an isomorphism is tested on stalks (A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk). This does not assert that algebraic constructibility is local for an arbitrary infinite analytic cover.

[F7]

Künneth over a field (Kunneth over a field): for complexes of $k$ -vector spaces with finite diagonals the cross product is a natural isomorphism $⨁_{p + q = n} H_{p} C \otimes_{k} H_{q} D ≅ H_{n} (C \otimes_{k} D)$ ; in particular the cohomology of $B \otimes_{k} C^{*} (F; k)$ is computed degreewise from the cohomologies of $B$ and of the finite fiber model.

[F8]

The sheaf-level adjunction of Inverse image is left adjoint to direct image on sheaves and the exactness of inverse image (Inverse image presheaf and inverse image sheaf, Direct image of a sheaf along a continuous map): the unit of $f^{*} ⊣ f_{*}$ at a sheaf is the natural map to the direct image, and all constructions are compatible with restriction to open subspaces.

[A1]

AC is used exactly through the injective replacements and the bounded derived Hom model of [F3] and [F4] and through the AC-assumed suppliers [F1] and [F2]; the cone, truncation, gluing and Hom-comparison arguments below are otherwise choice-free (The Axiom of Choice).

Proof

1.1givenF1F2F3F8

The local comparison and the fiber model. Let $V \subseteq S$ be a product chart with trivialization $Y ∣_{V} ≅ V \times F$ and projection $pr : V \times F \to V$ . By [F1] there is a natural comparison $R {pr}_{*} {pr}^{*} B_{V} ≃ B_{V} \otimes_{k} C^{*} (F; k)$ for bounded $B_{V} \in D_{c}^{b} (V; k)$ , compatible with restriction to smaller charts and carrying the adjunction unit to $B_{V} \otimes (k \to C^{*} (F; k))$ . By [F2] the chart and the fiber carry finite deleted-boundary models, so $C^{*} (F; k)$ is a bounded complex of finite-dimensional vector spaces whose cohomology is $k$ in degree zero, in the sense that the structure map $k \to C^{*} (F; k)$ identifies $k$ with $H^{0}$ , and is zero in degrees $1 \leq j \leq r$ . The comparison is compatible with the global adjunction $f^{*} ⊣ R f_{*}$ of [F3] over the chart: restriction to $V$ is inverse image along the open inclusion, inverse image commutes with $f^{*}$ , $R f_{*}$ commutes with restriction to the open base for open inclusions by [F3] and [F8], and the unit is natural, so the restriction of the global unit $B \to R f_{*} f^{*} B$ to $V$ is the chart unit.

2.1F1F3F6F7step 1.1

The cone bound. Let $\tilde{C} : = Cone (k \to C^{*} (F; k))$ , a bounded complex of finite-dimensional vector spaces with $H^{0} (\tilde{C}) = 0$ and $H^{j} (\tilde{C}) = H^{j} (C^{*} (F; k)) = 0$ for $1 \leq j \leq r$ ; hence $H^{j} (\tilde{C}) = 0$ for all $j \leq r$ . Now let $B \in D_{c}^{\geq b} (S; k)$ , so $H^{p} (B) = 0$ for $p < b$ . By [F7] the cohomology of $B_{V} \otimes_{k} \tilde{C}$ is the direct sum of the terms $H^{p} (B_{V}) \otimes_{k} H^{q} (\tilde{C})$ with $p + q = n$ , and a nonzero such term needs $p \geq b$ and $q \geq r + 1$ , hence $n \geq b + r + 1$ ; therefore $H^{n} (B_{V} \otimes_{k} \tilde{C}) = 0$ for $n \leq b + r$ . Since the unit of the chart comparison is $B_{V} \otimes (k \to C^{*} (F; k))$ , its cone is $B_{V} \otimes_{k} \tilde{C}$ up to canonical isomorphism, so the cone of the chart unit lies in $D^{\geq b + r + 1}$ on every chart. The charts cover $S$ , the restriction of the global unit to each chart is the chart unit by step 1.1, and vanishing of the cohomology sheaves of a complex is a local condition [F6], so the cone of the global unit $B \to R f_{*} f^{*} B$ lies in $D^{\geq b + r + 1}$ , which proves claim 1.

3.1F3F4F5step 2.1

The Hom comparison. Let $B \to R f_{*} f^{*} B \to C \to B [1]$ be the distinguished triangle of the unit, with $C \in D^{\geq b + r + 1}$ by step 2.1, and let $A \in D_{c}^{\leq a} (S; k)$ and $m \leq b + r - a$ . By [F5], $C [m] \in D^{\geq b + r + 1 - m}$ and $C [m - 1] \in D^{\geq b + r + 2 - m}$ ; since $m \leq b + r - a$ gives $b + r + 1 - m > a$ and $b + r + 2 - m > a$ , orthogonality $Hom (D^{\leq a}, D^{\geq a + 1}) = 0$ gives $Hom (A, C [m]) = 0 = Hom (A, C [m - 1])$ . The long exact Hom sequence of [F4] applied to the triangle, shifted by $[m]$ , therefore shows that the canonical map $Hom (A, B [m]) \to Hom (A, R f_{*} f^{*} B [m])$ is a bijection. Under the derived adjunction of [F3] this map corresponds to the composition with the unit, $u \mapsto η_{B [m]} \circ u$ , whose adjoint is exactly pullback $u \mapsto f^{*} u$ ; since the adjunction bijection $Hom (f^{*} A, f^{*} B [m]) ≅ Hom (A, R f_{*} f^{*} B [m])$ carries $f^{*} u$ to $η_{B [m]} \circ u$ by naturality of the unit, the induced pullback map $Hom (A, B [m]) \to Hom (f^{*} A, f^{*} B [m])$ is a bijection. This proves claim 2.

3.2F3F4F5step 2.1

The image criterion. Assume $a - b \leq r$ and put $Φ = f^{*} : D_{c}^{[b, a]} (S; k) \to D_{c}^{[b, a]} (Y; k)$ and $Ψ = τ^{\leq a} \circ R f_{*}$ . For $B \in D_{c}^{[b, a]} (S; k)$ the cone of the unit $B \to R f_{*} f^{*} B$ lies in $D^{\geq b + r + 1} \subseteq D^{\geq a + 1}$ by step 2.1 and $b + r + 1 > a$ , so the truncated unit $B \to τ^{\leq a} R f_{*} f^{*} B$ is an isomorphism; its inverse gives $Ψ Φ ≅ id$ . Now let $C \in D_{c}^{[b, a]} (Y; k)$ . Since $C \in D^{\geq b}$ and $R f_{*}$ preserves lower bounds (a bounded-below complex can be resolved in degrees at least $b$ by [F4], and $f_{*}$ of a complex vanishing below $b$ vanishes below $b$ ), we get $R f_{*} C \in D^{\geq b} (S; k)$ , and by the setting $R f_{*} C$ has bounded constructible cohomology; hence $Ψ C = τ^{\leq a} R f_{*} C \in D_{c}^{[b, a]} (S; k)$ . Let $γ_{C} : Φ Ψ C \to C$ be the composite of $Φ (τ^{\leq a} R f_{*} C \to R f_{*} C)$ with the counit $ε_{C} : Φ R f_{*} C \to C$ of the adjunction. If $γ_{C}$ is an isomorphism, then $C ≅ Φ (Ψ C)$ with $Ψ C \in D_{c}^{[b, a]} (S; k)$ , so $C$ lies in the image of $Φ$ . Conversely, let $C = Φ B = f^{*} B$ with $B \in D_{c}^{[b, a]} (S; k)$ . Then $Ψ C = τ^{\leq a} R f_{*} f^{*} B$ , and the truncation $τ^{\leq a} R f_{*} f^{*} B \to R f_{*} f^{*} B$ factors, up to the canonical isomorphism $τ^{\leq a} R f_{*} f^{*} B ≅ B$ supplied by the previous paragraph, as the unit $B \to R f_{*} f^{*} B$ ; the triangle identity $ε_{f^{*} B} \circ f^{*} (η_{B}) = {id}_{f^{*} B}$ , together with exactness of $f^{*}$ , therefore shows that $γ_{f^{*} B}$ is an isomorphism. Hence for $C \in D_{c}^{[b, a]} (Y; k)$ the counit $γ_{C}$ is an isomorphism exactly when $C$ is a pullback, which is the first assertion of claim 4.

4.1F3F5F6step 3.1

Full faithfulness. If $a - b \leq r$ , then $0 \leq b + r - a$ , so claim 2 with $m = 0$ gives that $Hom (A, B) \to Hom (f^{*} A, f^{*} B)$ is a bijection for all objects $A, B$ of $D_{c}^{[b, a]} (S; k)$ . Inverse image is exact and preserves the canonical t-structure and algebraic constructibility by [F3] and [F6], so $f^{*}$ maps $D_{c}^{[b, a]} (S; k)$ into $D_{c}^{[b, a]} (Y; k)$ ; a functor between categories that is bijective on all Hom sets is fully faithful, so claim 3 follows.

4.2step 3.2

Retracts. Let $C \in D_{c}^{[b, a]} (Y; k)$ be a direct summand of $Φ B = f^{*} B$ for some $B \in D_{c}^{[b, a]} (S; k)$ , with maps $i : C \to Φ B$ and $p : Φ B \to C$ satisfying $p i = {id}_{C}$ . The counit $γ$ is a natural transformation $Φ Ψ \to id$ ; naturality in the form $u \circ γ_{X} = γ_{Y} \circ Φ Ψ (u)$ for $u : X \to Y$ , applied to $u = i$ and $u = p$ , gives $p \circ γ_{Φ B} \circ Φ Ψ (i) = p \circ i \circ γ_{C} = γ_{C}$ and $γ_{C} \circ Φ Ψ (p) = p \circ γ_{Φ B}$ . By step 3.2 the map $γ_{Φ B}$ is an isomorphism. Define $δ : = Φ Ψ (p) \circ γ_{Φ B}^{- 1} \circ i : C \to Φ Ψ C$ . Then $δ \circ γ_{C} = Φ Ψ (p) \circ γ_{Φ B}^{- 1} \circ γ_{Φ B} \circ Φ Ψ (i) = Φ Ψ (p i) = {id}_{Φ Ψ C}$ and $γ_{C} \circ δ = p \circ γ_{Φ B} \circ γ_{Φ B}^{- 1} \circ i = p i = {id}_{C}$ , using the two displayed identities. Hence $γ_{C}$ is an isomorphism and, by step 3.2, $C$ lies in the image of $Φ$ , which proves closure under direct summands.

4.3F3F4F5F6step 3.2

Extensions under the strict bound. Assume $r \geq a - b + 1$ and let $C^{'} \to C \to C^{''} \to C^{'} [1]$ be a distinguished triangle in $D_{c}^{[b, a]} (Y; k)$ whose outer objects lie in the image: $C^{'} = f^{*} B^{'}$ and $C^{''} = f^{*} B^{''}$ with $B^{'}, B^{''} \in D_{c}^{[b, a]} (S; k)$ . Let $φ : C^{''} \to C^{'} [1]$ be its connecting morphism. Because $1 \leq b + r - a$ by the strict bound, claim 2 applies with $A = B^{''} \in D_{c}^{\leq a} (S; k)$ , $B = B^{'} \in D_{c}^{\geq b} (S; k)$ and $m = 1$ ; the isomorphism $Hom (B^{''}, B^{'} [1]) ≅ Hom (f^{*} B^{''}, f^{*} B^{'} [1])$ being pullback along $f^{*}$ , there is a unique $\overset{ˉ}{φ} : B^{''} \to B^{'} [1]$ with $f^{*} \overset{ˉ}{φ} = φ$ . By TR1 the morphism $\overset{ˉ}{φ}$ is completed to a distinguished triangle $B^{'} \to \overset{ˉ}{B} \to B^{''} \to B^{'} [1]$ in $D^{b} (S; k)$ ([F5]). The long exact cohomology sequence of this triangle and $B^{'}, B^{''} \in D^{[b, a]}$ show $H^{j} (\overset{ˉ}{B}) = 0$ for $j < b$ and for $j > a$ : the exact segment $H^{j} (B^{'}) \to H^{j} (\overset{ˉ}{B}) \to H^{j} (B^{''})$ has zero outer terms in both ranges. By [F6] the cone $\overset{ˉ}{B}$ of a morphism of bounded algebraically constructible complexes is bounded and algebraically constructible, so $\overset{ˉ}{B} \in D_{c}^{[b, a]} (S; k)$ . Exactness of $f^{*}$ [F3] turns the triangle on $\overset{ˉ}{φ}$ into a distinguished triangle with connecting morphism $f^{*} \overset{ˉ}{φ} = φ$ ; two distinguished triangles with the same connecting morphism are isomorphic on their third objects, because TR2 and TR3 applied to the identity on the source and target of the connecting morphism give a morphism between the two triangles whose first two components are identities and hence, by the five-lemma applied to the long exact Hom sequences of [F4], an isomorphism on the third objects. Thus $C ≅ f^{*} \overset{ˉ}{B}$ lies in the image of $Φ$ . This proves the extension-closure clause of claim 4; at the boundary $r = a - b$ the degree-one comparison $1 \leq b + r - a$ fails and the argument is deliberately not used.

5.1givenA1step 2.1step 3.1step 4.1step 3.2step 4.2step 4.3∎

Conclusion and the local nature of the comparison. Step 2.1 proves the cone claim 1; step 3.1 proves the Hom comparison claim 2, step 4.1 full faithfulness claim 3, and steps 3.2, 4.2 and 4.3 the image criterion and the closure properties of claim 4. The comparison used in step 2.1 is the local product comparison on product charts, not a global splitting: no isomorphism $R f_{*} f^{*} B ≃ B \otimes R Γ (F; k)$ is asserted, and the argument uses only the cone of the unit and the degreewise vanishing of the fiber model, so nonzero characteristic-class transgressions of the bundle are preserved. The only choice-theoretic inputs are those recorded in [A1].

Remarks

The corrected boundary is binding: at $r = a - b$ only the degree-zero Hom comparison, the image criterion and retract closure are asserted; extension closure needs the one extra degree of acyclicity. For the projection $C^{*} \to pt$ , $r = 0$ and the interval $[0, 0]$ , a nontrivial unipotent rank-two local system is an extension of two constant sheaves but is not a pullback from the point, so extension closure cannot be asserted at $r = a - b$ . Retract closure still holds there, and the locally split Cech essential-image argument used later uses degreewise full faithfulness of the local-section comparison, not this finite bound.
The two named suppliers of the setting are consumed with their stated hypotheses; their actual use in this proof is at steps 1.1 and 2.1, and the statement keeps the local finite-cochain route and does not assert the all-sheaf universal-acyclicity form of the source theorem.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

General linear frame spaces have the Stiefel cohomology gap

Statement

Assume the Axiom of Choice (The Axiom of Choice). Fix integers $s \geq 1$ , $n_{i} \geq 1$ and $N_{i} \geq n_{i} + 1$ for $1 \leq i \leq s$ , let $k$ be a field, and put $U_{N} = \prod_{i = 1}^{s} Inj (C^{n_{i}}, C^{N_{i}}), r_{N} = 2 \min_{i} (N_{i} - n_{i}),$ where $Inj (C^{n}, C^{N})$ is the open set of injective, equivalently full-rank, complex $N \times n$ matrices. Then:

each $Inj (C^{n}, C^{N})$ is homotopy equivalent to the Stiefel space $V_{n} (C^{N}) = {A \in C^{N \times n} : A^{*} A = I_{n}}$ of orthonormal frames (Stiefel spaces, Grassmannians, and tautological bundles);
each $V_{n} (C^{N})$ is a compact path-connected smooth manifold and is homeomorphic to a finite CW complex, and the map $V_{n} (C^{N}) \to V_{n - 1} (C^{N})$ forgetting the last column is a numerable locally trivial fiber bundle with fiber $S^{2 (N - n) + 1}$ , hence a Hurewicz and a Serre fibration;
$H^{0} (U_{N}; k) ≅ k$ and $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ .

No claim is made about the cohomology of $U_{N}$ above degree $r_{N}$ , and no GL-equivariance of the retractions or trivializations below is asserted.

Facts & Assumptions

Given: Integers $s, n_{i}, N_{i}$ with $n_{i} \geq 1$ , $N_{i} \geq n_{i} + 1$ ; the field $k$ ; the spaces $U_{N}$ , $Inj (C^{n}, C^{N})$ and $V_{n} (C^{N})$ with the subspace and product topologies; the Axiom of Choice The Axiom of Choice.

[F1]

Stiefel spaces and Gram-Schmidt (Stiefel spaces, Grassmannians, and tautological bundles, Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans): $V_{n} (C^{N})$ is the space of ordered orthonormal $n$ -frames in $C^{N}$ with the subspace topology of $(C^{N})^{n}$ ; Gram-Schmidt sends a linearly independent list $(v_{0}, \dots, v_{n - 1})$ to the orthonormal list $(e_{0}, \dots, e_{n - 1})$ spanning the same flags, by the displayed recursion.

[F2]

Submersions and regular level sets (Immersions, submersions, and constant-rank maps, A regular level set is locally a $C^{k}$ graph of dimension $m - n$ , Smooth manifolds and their smooth charts): a regular level set of a $C^{\infty}$ map between open subsets of Euclidean spaces is locally a $C^{\infty}$ graph over the kernel of the differential, hence a smooth submanifold of the ambient Euclidean space of the corresponding dimension.

[F3]

Compactness and triangulation (Assuming ${A C}_{ω}$ and $D C$ , compactness, sequential compactness, countable compactness, limit point compactness, completeness and total boundedness, pseudocompactness, closedness and boundedness, and the extreme-value property are equivalent for nonempty subsets of $R^{n}$ with $n \geq 1$ , Finite smooth triangulation of compact corners, relative to admissible boundary data, The geometric realization of an abstract simplicial complex, CW complex with closure finiteness and weak topology): a closed and bounded subset of a Euclidean space is compact; a compact smooth manifold, viewed as a manifold with corners with no corners, has a finite smooth triangulation, i.e. is homeomorphic to the realization of a finite abstract simplicial complex, and a finite simplicial complex is a finite CW complex whose cells are its open simplices.

[F4]

Bundles, paracompactness and fibrations (Locally trivial fiber bundle, Every compact space is paracompact, Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity, Numerable fiber bundles are hurewicz fibrations): a compact Hausdorff space is paracompact, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity, so a locally trivial bundle over a compact Hausdorff base is numerable, and every numerable fiber bundle with its ordinary product charts is a Hurewicz fibration, hence a Serre fibration; the pullback of a locally trivial bundle along a continuous map is a locally trivial bundle with the same fiber.

[F5]

The cohomological Serre spectral sequence (Cohomological Serre spectral sequence): for a Serre fibration $p : E \to B$ over a path-connected CW complex and a commutative unital ring $R$ , there is a natural first-quadrant spectral sequence $E_{2}^{a, b} ≅ H^{a} (B; H^{b} (p; R))$ , with differentials $d_{r} : E_{r}^{a, b} \to E_{r}^{a + r, b - r + 1}$ , strongly converging to $H^{a + b} (E; R)$ ; $H^{0} (p; R)$ is the constant system $R$ when the fibers are nonempty and connected.

[F6]

Cohomology of spheres (Homology of spheres, The universal coefficient theorem for cohomology over a PID): reduced homology of $S^{d}$ is $R$ in degree $d$ and zero otherwise; over a field $k$ the universal coefficient sequence gives $H^{0} (S^{d}; k) ≅ k$ , $H^{d} (S^{d}; k) ≅ k$ and $H^{j} (S^{d}; k) = 0$ for $1 \leq j \leq d - 1$ . Spheres of dimension at least one are path-connected (For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected, Paths, path-connected spaces and path components).

[F7]

Constant-sheaf comparison (Locally contractible metric spaces have singular constant sheaf cohomology): on a metric space with a contractible-open neighbourhood basis, the cohomology of the constant $k$ -sheaf agrees naturally with singular cohomology. Euclidean open subsets and smooth submanifolds have this property by their ball charts.

[F8]

A locally trivial bundle with nonempty path-connected fibre over a path-connected base has path-connected total space. Indeed, subdivide a base path into finitely many intervals lying in bundle charts, lift each interval using its product chart, and join the final lifted point to the desired point within the fibre. This follows directly from the product charts and path concatenation (Paths, path-connected spaces and path components, Locally trivial fiber bundle).

[A1]

AC supplies DC and countable choice by AC implies DC implies countable choice, meeting the explicit choice hypotheses in [F3] and [F4]. AC is used through the partition-of-unity, triangulation and Serre-spectral-sequence and constant-sheaf comparison suppliers of [F3]-[F7]; the Gram-Schmidt, level-set and degree bookkeeping arguments are choice-free (The Axiom of Choice).

Proof

1.1givenF1

The frame retraction. Fix $n \leq N$ and let $A \in Inj (C^{n}, C^{N})$ have columns $v_{0}, \dots, v_{n - 1}$ , a linearly independent list. Applying the Gram-Schmidt recursion of [F1] to $(v_{0}, \dots, v_{n - 1})$ gives an orthonormal list $(e_{0}, \dots, e_{n - 1})$ with $span (e_{0}, \dots, e_{k - 1}) = span (v_{0}, \dots, v_{k - 1})$ for every $k$ , so the upper triangular matrix $R$ with $R_{j k} = ⟨ v_{k}, e_{j} ⟩$ for $j < k$ , $R_{k k} = ∥ u_{k} ∥ > 0$ and $R_{j k} = 0$ for $j > k$ satisfies $A = Q R$ with $Q = (e_{0}, \dots, e_{n - 1}) \in V_{n} (C^{N})$ . The Gram-Schmidt formulas are continuous rational expressions with positive denominators on the full-rank locus, so $A \mapsto (Q (A), R (A))$ is continuous. Writing $R = D U$ with $D$ the diagonal part and $U$ unit upper triangular, the family $t \mapsto Q (A) ((1 - t) D + t I) ((1 - t) U + t I)$ is continuous, has invertible upper triangular second factor with positive diagonal for every $t \in [0, 1]$ , and connects $A$ to $Q (A)$ inside $Inj (C^{n}, C^{N})$ ; at $t = 1$ it is $Q (A)$ , so $A \mapsto Q (A)$ is a deformation retraction onto $V_{n} (C^{N})$ . This proves claim 1 and shows that $U_{N}$ is homotopy equivalent to $\prod_{i} V_{n_{i}} (C^{N_{i}})$ .

1.2givenF2F3

Compact smooth frame manifolds. Let $f (A) = A^{*} A$ from the open full-rank locus of $C^{N \times n}$ to the real vector space $Herm (n)$ . Its differential is $H \mapsto A^{*} H + H^{*} A$ ; for Hermitian $B$ , taking $H = \frac{1}{2} A (A^{*} A)^{- 1} B$ gives $A^{*} H = H^{*} A = B / 2$ , hence differential $B$ . Thus $I_{n}$ is a regular value and $V_{n} = f^{- 1} (I_{n})$ is a smooth submanifold by [F2]. It is closed in the ambient Euclidean space and bounded, since each frame column has norm one, so it is compact by [F3]. The subspace is Hausdorff and second countable, as required by the triangulation supplier. For $n = 0$ it is a point.

2.1F1F4step 1.2

The forgetting bundle. Fix $w \in V_{n - 1} (C^{N})$ and an orthonormal basis $c_{1}, \dots, c_{N - n + 1}$ of its orthogonal complement. For a nearby frame $w^{'}$ , put $P_{w^{'}} = I - w^{'} w^{' *}$ . The vectors $P_{w^{'}} c_{j}$ remain linearly independent near $w$ , since their Gram determinant is continuous and equals one at $w$ . Gram-Schmidt gives a continuously varying orthonormal basis $c_{j} (w^{'})$ of the complement of $w^{'}$ . The map $(w^{'}, z) \mapsto (w^{'}, \sum_{j} z_{j} c_{j} (w^{'}))$ for $z \in S^{2 (N - n) + 1} \subset C^{N - n + 1}$ trivializes the forgetting map; its continuous inverse takes the inner products with $c_{j} (w^{'})$ . This also covers $n = 1$ over the point $V_{0}$ . By step 1.2 the base is compact Hausdorff, so [F4] supplies a finite subordinate partition of unity and makes the bundle numerable, hence Hurewicz and Serre.

3.1F3F6F8step 2.1

Finite CW model and path-connectedness. By the triangulation theorem of [F3] applied to the compact smooth manifold $V_{n} (C^{N})$ , there is a finite simplicial complex $K_{n}$ and a homeomorphism $∣ K_{n} ∣ \to V_{n} (C^{N})$ . A finite simplicial complex is a finite CW complex: its cells are the open simplices, the characteristic maps are the inclusions of the closed simplices, closure finiteness holds because there are finitely many simplices, and the weak topology agrees with the topology of the realization because the realization is the finite union of its closed simplices. Hence $V_{n} (C^{N})$ is homeomorphic to a finite CW complex, as claimed in claim 2. For path-connectedness, induct on $n$ . The space $V_{0} (C^{N})$ is a point. For $n \geq 1$ the map $V_{n} (C^{N}) \to V_{n - 1} (C^{N})$ is the locally trivial bundle of step 2.1 with fiber the nonempty path-connected sphere $S^{2 (N - n) + 1}$ by [F6], and the base is path-connected by the induction hypothesis; the local connectedness argument of [F8] then makes $V_{n} (C^{N})$ path-connected. This completes claim 2.

4.1F4F5F6step 2.1step 3.1

The gap for one factor. Induct on $n$ , starting with $V_{0}$ a point. For $n \geq 1$ put $d = 2 (N - n) + 1$ . The forgetting bundle of step 2.1 has fibre $S^{d}$ and base homeomorphic to a path-connected finite CW complex by step 3.1. The induction hypothesis gives vanishing of the base cohomology in degrees $1$ through $2 (N - n + 1) = d + 1$ . In the Serre spectral sequence [F5], [F6] makes all fibre rows in degrees $1, \dots, d - 1$ zero; the degree-zero coefficient system is constant $k$ , since the fibre is connected. Consequently all $E_{2}$ terms of total degree $1, \dots, d - 1$ are zero, while $E_{2}^{0, 0} = k$ . No differential can enter $(0, 0)$ or leave it in this first-quadrant sequence. Strong convergence therefore gives $H^{0} (V_{n}; k) = k$ and $H^{j} (V_{n}; k) = 0$ for $1 \leq j \leq d - 1 = 2 (N - n)$ , completing the induction. No constancy of the top-degree fibre coefficient system is needed.

5.1F4F5step 1.1step 3.1step 4.1

The product. By step 1.1 it suffices to compute the cohomology of $\prod_{i = 1}^{s} V_{n_{i}} (C^{N_{i}})$ , and we induct on $s$ . For $s = 1$ this is step 4.1. For $s \geq 2$ let $P = \prod_{i < s} V_{n_{i}} (C^{N_{i}})$ and consider the projection $P \times V_{n_{s}} (C^{N_{s}}) \to P$ , a trivial bundle with fiber $V_{n_{s}} (C^{N_{s}})$ , hence numerable and a Serre fibration by [F4]. By step 3.1 each factor and, inductively, $P$ is homeomorphic to a finite CW complex, and a finite product of finite CW complexes is again a finite CW complex: identify $D^{a} \times D^{b}$ with $D^{a + b}$ by radial rescaling between their unit balls; the product characteristic maps into the product space then give cells $e \times e^{'}$ and a finite cell decomposition, closure finiteness holds because there are finitely many cells, and the weak topology of the product cell structure agrees with the product topology for finite compact Hausdorff complexes, where the compactly generated product topology coincides with the ordinary one. So [F5] applies to the projection with constant coefficient system in fiber degree $0$ . If $H^{0} (P; k) ≅ k$ and $H^{j} (P; k) = 0$ for $1 \leq j \leq r_{s - 1} = \min_{i < s} 2 (N_{i} - n_{i})$ by induction, and if $H^{0} (V_{n_{s}}; k) ≅ k$ , $H^{j} (V_{n_{s}}; k) = 0$ for $1 \leq j \leq 2 (N_{s} - n_{s})$ by step 4.1, then in total degree $m \leq r_{s} = \min (r_{s - 1}, 2 (N_{s} - n_{s}))$ every nonzero $E_{2}^{a, b}$ with $a + b = m$ has $b = 0$ , for $b \geq 1$ would need fiber cohomology in degree $b \leq r_{s} \leq 2 (N_{s} - n_{s})$ , which vanishes; the differential bookkeeping of step 4.1 then gives $H^{m} (P \times V_{n_{s}}; k) ≅ H^{m} (P; k)$ , which is $k$ for $m = 0$ and $0$ for $1 \leq m \leq r_{s}$ . This proves claim 3, $H^{0} (U_{N}; k) ≅ k$ and $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ , by the homotopy equivalence of step 1.1.

6.1F7step 1.1step 1.2step 4.1step 5.1

The cohomology convention. The full-rank spaces are Euclidean open subsets and the Stiefel spaces and their products are smooth metric manifolds with contractible-open chart bases. Thus [F7] identifies their constant-sheaf cohomology with the singular cohomology computed in steps 4.1 and 5.1; naturality carries the frame deformation retraction of step 1.1 to these comparisons. In particular the same gap holds for the constant-sheaf fibre cohomology used by the later derived pullback lemmas.

7.1givenA1step 1.1step 1.2step 2.1step 3.1step 4.1step 5.1step 6.1∎

Conclusion. Steps 1.1 and 5.1 give claims 1 and 3; steps 1.2, 2.1 and 3.1 give claim 2. AC is consumed only through the named compactness, triangulation, partition-of-unity, spectral-sequence and constant-sheaf comparison suppliers [F3]-[F7], as recorded in [A1].

Remarks

The statement is the cohomological input consumed by the equivariant Stiefel approximation lemma of this page; it asserts the gap $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ and nothing above $r_{N}$ , where the characteristic classes of the bundle live.
The argument is a local finite-cell computation: the sphere bundle is a numerable bundle over a finite CW model, so the Serre spectral sequence has finite pages and the low-degree bookkeeping uses only the base degree with fiber degree zero.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

General linear Stiefel approximations stabilize bounded constructible Hom

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $G = \prod_{i} {GL}_{n_{i}} (C)$ be a finite product of complex general linear groups with $n_{i} \geq 1$ (Invertible matrices and the general linear group ${GL}_{n} (F)$ ), acting algebraically on a separated finite-type complex algebraic variety $X$ (Classical algebraic prevarieties, regular maps, and varieties, Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers). For integers $N_{i} \geq n_{i} + 1$ put $U_{N} = \prod_{i} Inj (C^{n_{i}}, C^{N_{i}}), B_{N} = \prod_{i} Gr (n_{i}, N_{i}), X_{N} = (U_{N} \times X) / G,$ where $U_{N}$ has the right action $u \cdot g = u \circ g$ and the quotient uses $(u, x) \cdot g = (u g, g^{- 1} x)$ . For the action nerve use the equivalent left action $g \cdot (u, x) = (u g^{- 1}, g x)$ , and set $r_{N} = 2 \min_{i} (N_{i} - n_{i})$ . Let $p_{N} : G^{∙} \times P_{N} \to G^{∙} \times X$ , with $P_{N} = U_{N} \times X$ and $p_{N} = (i d, {pr}_{X})$ , be the simplicial map of the two action nerves (Equivariant constructible derived category of complex varieties), and let $Q_{N} : D_{G, c}^{b} (X; k) \to D_{c}^{b} (X_{N}; k)$ be the composite of $p_{N}^{*}$ with the locally split Čech descent equivalence for the principal bundle $P_{N} \to X_{N}$ (Locally split Cech nerves give actual derived Cartesian sheaf descent). Then, for bounded algebraically constructible $A, B$ in $D_{G, c}^{b} (X; k)$ :

Hom comparison. If $A \in D_{c}^{\leq a}$ and $B \in D_{c}^{\geq b}$ , then pullback induces an isomorphism ${Hom}_{G} (A, B [m]) ≅ {Hom}_{X_{N}} (Q_{N} (A), Q_{N} (B) [m])$ whenever $m \leq b + r_{N} - a$ ;
full faithfulness. On an ordinary cohomological interval $D_{c}^{[b, a]}$ of width $a - b \leq r_{N}$ , the functor $Q_{N}$ is fully faithful;
image. On such an interval the image of $Q_{N}$ consists exactly of the objects $H$ whose principal-bundle pullback $q_{N}^{*} H$ lies in the image of ${pr}_{X}^{*} : D_{c}^{[b, a]} (X; k) \to D_{c}^{[b, a]} (P_{N}; k)$ , i.e. of the resolution triples whose frame-bundle pullback comes from $X$ ; the image is closed under retracts, and under extensions when $r_{N} \geq a - b + 1$ ;
coherence. For two resolutions $N, M$ , use the common product resolution $U_{N} \times U_{M}$ and its associated quotient $X_{N, M} = (U_{N} \times U_{M} \times X) / G$ with projections $π_{N} : X_{N, M} \to X_{N}$ and $π_{M} : X_{N, M} \to X_{M}$ . The pullbacks $π_{N}^{*} Q_{N} (A)$ and $π_{M}^{*} Q_{M} (A)$ are canonically identified on intervals of width at most $\min (r_{N}, r_{M})$ , and their Hom comparisons commute with natural transformations and satisfy the cocycle identity after pullback to a triple product resolution. No identification of $U_{N} \times U_{M}$ with $U_{N + M}$ or direct isomorphism of functors with different target categories is asserted;
trivial group. If $G$ is trivial then $U_{N}$ and $B_{N}$ are points and $X_{N} = X$ , $r_{N} = \infty$ , and $Q_{N}$ is the identity of $D_{c}^{b} (X; k)$ .

The finite local comparisons do not assert a global tensor splitting for an arbitrary bundle with fibre $U_{N}$ , such as the comparison projections $X_{N, M} \to X_{N}$ and $X_{N, M} \to X_{M}$ .

Facts & Assumptions

Given: The group $G$ , the variety $X$ , the integers $N_{i} \geq n_{i} + 1$ , the spaces $U_{N}, B_{N}, P_{N}, X_{N}$ , the action nerve of the definition, the simplicial map $p_{N}$ , and the functor $Q_{N}$ of the Statement; the Axiom of Choice The Axiom of Choice; the bounded derived category conventions of Derived category of an abelian category and Derived hom in the bounded setting.

[F1]

Frame and quotient conventions are those of Stiefel spaces, Grassmannians, and tautological bundles, A free group action has no nonidentity element fixing a point and Principal g bundle and associated fiber bundle. Full-rank frames are an open matrix locus. The algebraic principal charts and the associated quotient charts are constructed directly in step 1.1; they are not inferred from the orthonormal Stiefel-space definition.

[F2]

Fiber gap (General linear frame spaces have the Stiefel cohomology gap): each factor and the product carry the finite models whose cohomology satisfies $H^{0} (U_{N}; k) ≅ k$ and $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ , and the frame space is homotopy equivalent to the product of Stiefel manifolds.

[F3]

The acyclic-fiber lemma (Acyclic product fibers stabilize bounded constructible pullback Hom): for a locally trivial bundle in the finite constructible local-product setting with $r$ -acyclic fiber, the cone of the unit of the adjunction lies in $D^{\geq b + r + 1}$ on $D_{c}^{\geq b}$ , the induced pullback map on Hom groups is an isomorphism in the stated range, the functor is fully faithful on intervals of width at most $r$ , and the image is characterized by the truncated counit and closed under retracts, with extension closure for width strictly below $r$ .

[F4]

The finite constructible local-product setting (Finite simplicial constructible sheaves have an actual derived incidence model, Complex algebraic constructible complexes have finite conical cochain models): bounded algebraically constructible complexes on separated finite-type complex varieties have bounded finite cochain models with coefficient-extension compatibility, and the local product comparison identifies the pushforward of a pullback along a product projection with the tensor product with the finite fiber model.

[F5]

Locally split descent (Locally split Cech nerves give actual derived Cartesian sheaf descent): for a strict augmented simplicial space the abelian category of simplicial sheaves has explicit coinduced injective resolutions and a bounded derived category; for the Čech nerve of a map admitting local sections, inverse image is an equivalence onto the full derived subcategory with Cartesian cohomology, and for algebraic principal bundles it restricts to the algebraically constructible subcategories. Evaluation at each simplicial degree is exact, its right adjoint is coinduction, and composition of derived right adjoints is computed by the total model.

[F6]

The definition of the equivariant category (Equivariant constructible derived category of complex varieties): $D_{G, c}^{b} (X; k)$ is the full subcategory of the bounded derived category of simplicial sheaves on $G^{∙} \times X$ with Cartesian cohomology and constructible degree-zero restriction, with the ordinary shift; cohomological bounds are read levelwise. Cartesianity identifies each cohomology component with inverse image of its degree-zero component, so restriction detects zero objects.

[F7]

The canonical t-structure (Canonical t structure on a derived category, The canonical pair is a t structure, The derived category inherits a triangulated structure): the bounded derived category of an abelian category is triangulated and carries the canonical t-structure with $Hom (D^{\leq m}, D^{\geq m + 1}) = 0$ ; the aisles are defined by the vanishing of cohomology objects.

[A1]

AC is used exactly through the injective replacements, the finite models and the Serre-spectral-sequence supplier recorded in [F2], [F4] and [F5]; the frame charts, the descent geometry and the image argument below are otherwise choice-free (The Axiom of Choice).

Proof

1.1givenF1F5F6

The frame bundle and its nerve. On a Grassmannian chart determined by an invertible $n \times n$ pivot minor, a frame is uniquely $(\binom{I_{n}}{T}) h$ , with $h \in {GL}_{n} (C)$ and $T$ the graph coordinates. These formulas and their inverses are regular, so they give algebraic principal-bundle charts for $U_{N} \to B_{N}$ . In these charts the diagonal quotient is $V \times X$ , with $((\binom{I}{T}) h, x)$ mapping to $(T, h x)$ , and the principal quotient $q_{N} : P_{N} \to X_{N}$ has section $(T, y) \mapsto ((\binom{I}{T}), y)$ . The finitely many charts glue by the regular change-of-frame maps; they exhibit $X_{N}$ as a separated finite-type variety and $q_{N}$ as a principal $G$ -bundle. Its action nerve is its Čech nerve, so [F5] applies. Projection to $X$ is equivariant for the left action specified in the Statement, and induces the simplicial map $p_{N}$ . In degree $m$ this is a product projection $G^{m} \times U_{N} \times X \to G^{m} \times X$ with fibre $U_{N}$ ; its structure squares are Cartesian, since group action on the frame factor is invertible.

2.1F2F4step 1.1

The fiber is $r_{N}$ -acyclic with finite models. By [F2], $H^{0} (U_{N}; k) ≅ k$ and $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ , and the product of Stiefel models supplies the bounded finite cochain model of the fiber with coefficient-field compatibility. By [F4] the base $G^{m} \times X$ carries the finite conical cochain models, so each levelwise projection of step 1.1 is a locally trivial bundle in the finite constructible local-product setting with $r_{N}$ -acyclic fiber.

3.1F3F5F7step 1.1step 2.1

The simplicial unit bound. For a map of these nerves, the levelwise direct images acquire simplicial structure by the canonical sheaf base-change maps followed by the source structure maps. The levelwise adjunction gives $p_{N}^{*} ⊣ p_{N *}$ , with exact inverse image. Evaluation of a simplicial injective at degree $m$ is injective: evaluation has an exact left adjoint, whose degree- $l$ term is the direct sum over operators $[m] \to [l]$ of their inverse images of the given sheaf. These sums and inverse images are exact. Thus a bounded-below simplicial injective replacement evaluates to an injective replacement at every level, and $(R p_{N *} C)_{m} = R p_{N, m *} C_{m}$ . For $B \geq b$ , [F3] and step 2.1 give that the cone of $B_{m} \to R p_{N, m *} p_{N, m}^{*} B_{m}$ lies in $D^{\geq b + r_{N} + 1}$ at every level. The same bound therefore holds for the simplicial unit cone in the bounded-below derived category; no augmentation of the action nerve to $X$ is used.

4.1F3F5F7step 3.1

The simplicial Hom comparison. The simplicial category has the exact levelwise inverse image $p_{N}^{*}$ left adjoint to the levelwise direct image $R p_{N *}$ and carries the canonical t-structure of [F7], whose aisles are defined cohomology-objectwise and hence are the levelwise aisles. With the cone bound of step 3.1 in place, the argument of [F3] — its Hom comparison, image criterion, full faithfulness and its retract and extension clauses — applies verbatim inside the simplicial bounded-below derived category, with its bounded full subcategory: for $A \in D^{\leq a}$ and $B \in D^{\geq b}$ the long exact Hom sequence of the unit triangle and the orthogonality $Hom (D^{\leq a}, D^{\geq a + 1}) = 0$ identify the pullback map $Hom (A, B [m^{'}]) \to Hom (p_{N}^{*} A, p_{N}^{*} B [m^{'}])$ as an isomorphism for $m^{'} \leq b + r_{N} - a$ ; for $a - b \leq r_{N}$ it is fully faithful on the full subcategory $D^{[b, a]}$ ; and on that interval an object lies in the image exactly when the truncated counit $p_{N}^{*} τ^{\leq a} R p_{N *} C \to C$ is an isomorphism, the image being closed under retracts and under extensions when $r_{N} \geq a - b + 1$ . Every input of that argument is an input already verified here: the adjunction and levelwise model [F5], the cone bound of step 3.1, and the t-structure orthogonality [F7].

5.1F5step 4.1

Descent and the functor $Q_{N}$ . Restricting the conclusion of step 4.1 to the full subcategory of Cartesian objects identified with $D_{c}^{b} (X_{N}; k)$ by [F5], the functor $p_{N}^{*}$ followed by descent is $Q_{N}$ : for $F \in D_{G, c}^{b} (X; k)$ the object $p_{N}^{*} F$ has Cartesian cohomology because $p_{N}$ is a simplicial map and the structure squares of $p_{N}^{*}$ are Cartesian by naturality of inverse image; descent produces $Q_{N} (F) \in D_{c}^{b} (X_{N}; k)$ , and $q_{N}^{*} Q_{N} (F) ≅ p_{N}^{*} F$ canonically. Hence, for $A \leq a$ , $B \geq b$ and $m \leq b + r_{N} - a$ , the isomorphisms of step 4.1 compose with the descent equivalence to give ${Hom}_{G} (A, B [m]) ≅ {Hom}_{X_{N}} (Q_{N} (A), Q_{N} (B) [m])$ , proving claim 1; the same composition proves claim 2 on intervals of width at most $r_{N}$ .

6.1F3F5F7step 3.1step 4.1step 5.1

The image and Cartesian descent. Necessity follows from $q_{N}^{*} Q_{N} (A) = {pr}_{X}^{*} A_{0}$ at degree zero. Conversely suppose $H \in D_{c}^{[b, a]} (X_{N}; k)$ and $q_{N}^{*} H ≅ {pr}_{X}^{*} F$ for $F \in D_{c}^{[b, a]} (X; k)$ . Let $C$ be the Cartesian simplicial pullback of $H$ to the principal-bundle nerve, and put $E = τ^{\leq a} R p_{N *} C$ in the ambient simplicial category. It lies in $[b, a]$ because derived direct image preserves lower bounds. At every level $m$ , Cartesianity identifies $C_{m}$ with the pullback of $C_{0}$ , hence with $p_{N, m}^{*} {pr}_{X, m}^{*} F$ , where ${pr}_{X, m} : G^{m} \times X \to X$ is projection. The levelwise image criterion [F3] therefore makes the truncated counit $p_{N}^{*} E \to C$ an isomorphism in every degree. Its naturality compares the structure maps of $p_{N}^{*} E$ with the Cartesian maps of $C$ . Each $p_{N, m}$ is surjective and its structure squares are Cartesian, so its exact pullback reflects these quasi-isomorphisms; hence $E$ has Cartesian cohomology. Moreover $E_{0} ≅ F$ by the truncated unit and $C_{0} ≅ {pr}_{X}^{*} F$ , so it is algebraically constructible. Thus $E \in D_{G, c}^{b} (X; k)$ and Čech descent gives $Q_{N} (E) ≅ H$ . Retracts and extensions under $r_{N} \geq a - b + 1$ lift by the degree-zero and degree-one Hom comparisons of step 4.1; their cones and summands have bounded Cartesian constructible cohomology, and the same counit argument gives the image. This proves claim 3 without assuming that ordinary descent data automatically strictify.

7.1F1F3F5step 5.1step 6.1

Coherence and the trivial group. The product $U_{N} \times U_{M}$ is a free smooth $G$ -resolution in its own right. Its projections to $U_{N}$ and $U_{M}$ descend to $π_{N} : X_{N, M} \to X_{N}$ and $π_{M} : X_{N, M} \to X_{M}$ , locally trivial bundles with fibres $U_{M}$ and $U_{N}$ . Their pullbacks are fully faithful on the stated intervals by [F3] and the bounds $r_{M}, r_{N}$ . On the common principal bundle $U_{N} \times U_{M} \times X$ , both pulled-back representatives have the canonical descent identification with the same pullback of $A$ from $X$ ; these identifications are compatible with the principal-bundle nerve maps, so [F5]'s descent equivalence yields a canonical isomorphism $π_{N}^{*} Q_{N} (A) ≃ π_{M}^{*} Q_{M} (A)$ . Naturality is inherited from those same inverse-image and descent maps. On the triple product all comparisons identify the representatives with the same pullback of $A$ , so their two compositions agree and the cocycle law follows. This compares objects and Hom maps in the common category, without equating the two different target categories or identifying the product with a larger frame space. If $G$ is trivial, take the point as its resolution; then $X_{N} = X$ , the action nerve is constant at $X$ , $r_{N} = \infty$ and $Q_{N}$ is the identity. This proves claims 4 and 5.

8.1givenA1step 5.1step 6.1step 7.1∎

Conclusion. Steps 3.1 to 5.1 prove claims 1 and 2, step 6.1 claim 3, and step 7.1 claims 4 and 5. The comparison is used only on the nerve and through the finite local models: no global splitting of $R a_{N *} a_{N}^{*} B$ is asserted, so genuine characteristic-class transgressions are preserved. AC enters only through [A1].

Remarks

The finite resolution bound is $r_{N} = 2 \min_{i} (N_{i} - n_{i})$ : the factor $2$ comes from the real dimension of complex vector spaces in the sphere-fibre calculation, and the minimum is taken because the product of the frame spaces is only as acyclic as its least acyclic factor.
The comparison functor is not asserted to be independent of $N$ as a single functor without passing to products of resolutions; claim 4 records exactly the coherent comparison that the products supply.

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Cartesian simplicial sheaves model actual equivariant derived Hom

Statement

Assume the Axiom of Choice (The Axiom of Choice). The definition Equivariant constructible derived category of complex varieties gives a locally small triangulated category $D_{G, c}^{b} (X; k)$ with conservative restriction to $X$ and the stated ordinary shift, and its morphism groups are computed by the coinduced simplicial injectives of Locally split Cech nerves give actual derived Cartesian sheaf descent. Locally split free-action descent identifies its finite general linear Stiefel resolution with the full bounded resolution-triple subcategory on $X_{N}$ , and the bounded Hom and image comparisons, the degree-zero ring comparison and retract closure of General linear Stiefel approximations stabilize bounded constructible Hom hold for it. The category is idempotent complete. No derived full faithfulness of restriction to $X$ is asserted.

Facts & Assumptions

Given: The field $k$ of characteristic zero, the product $G = \prod_{i} {GL}_{n_{i}} (C)$ acting on the separated finite-type complex algebraic variety $X$ , the action nerve $Z_{∙} = G^{∙} \times X$ and the category $D_{G, c}^{b} (X; k)$ of Equivariant constructible derived category of complex varieties; the Axiom of Choice The Axiom of Choice.

[F1]

The simplicial model (Locally split Cech nerves give actual derived Cartesian sheaf descent): the category ${Sh}_{k} (Z_{∙})$ of simplicial sheaves is abelian with levelwise kernels, cokernels and exactness; under AC it has explicit coinduced injective resolutions $N_{p} I$ , evaluation at each degree is exact with a coinductive right adjoint, and the bounded-below injective model computes $D^{b} ({Sh}_{k} (Z_{∙}))$ , so Hom groups are set-sized. For a locally split Čech nerve, inverse image is an equivalence onto the full derived subcategory with Cartesian cohomology, and composition of derived right adjoints is computed by the total model.

[F2]

The definition and its closure properties (Equivariant constructible derived category of complex varieties, Algebraically constructible complexes of analytic field sheaves, Analytic field sheaves have injective derived models, Complex algebraic constructible complexes have finite conical cochain models): $D_{G, c}^{b} (X; k)$ consists of the bounded complexes with Cartesian cohomology whose degree-zero restriction has algebraically constructible cohomology; bounded algebraically constructible complexes admit one common finite smooth algebraic frontier stratification for cone closure inside the finite constructible setting, and the derived category with shift $[1]$ and cone triangles is triangulated (Derived category of an abelian category); an object of a bounded derived category is zero exactly when all its cohomology objects vanish.

[F3]

The Stiefel resolution (General linear Stiefel approximations stabilize bounded constructible Hom): for $N_{i} \geq n_{i} + 1$ , the finite frame resolutions $U_{N} \to B_{N}$ , the associated varieties $X_{N} = (U_{N} \times X) / G$ and the functor $Q_{N}$ identify the equivariant Hom groups with ordinary Hom groups on $X_{N}$ in the range $m \leq b + r_{N} - a$ , the functor is fully faithful on intervals of width at most $r_{N}$ , and its image is the full resolution-triple subcategory of $D_{c}^{[b, a]} (X_{N}; k)$ with retract closure and extension closure under the strict bound.

[A1]

AC is used exactly through the injective replacements and finite models of [F1] and [F2]; the telescope of step 2.3 uses exact countable coproducts in the ambient simplicial category and is otherwise choice-free (The Axiom of Choice).

Proof

1.1givenF1

The ambient category and its Hom groups. By [F1] the category ${Sh}_{k} (Z_{∙})$ is abelian with levelwise exactness, has enough injectives under AC via the coinduced objects $N_{p} I$ , and its bounded derived category is actual with set-sized Hom groups computed by bounded-below injective replacements. This is the ambient category of the definition, with the ordinary shift and cone triangles of Derived category of an abelian category.

1.2F1F2

Closure under the triangulated operations. Shifts and isomorphisms preserve Cartesian cohomology and the constructibility of the degree-zero restriction because the structure maps of a shifted or isomorphic complex are the shifts or conjugates of the original structure maps. For a cone $C (f)$ of a morphism $f : A \to B$ of $D_{G, c}^{b} (X; k)$ , the structure maps of $C (f)$ are the cones of the levelwise structure maps, and exactness of simplicial inverse image identifies the structural comparison for $C (f)$ with the map of cones of the structural comparisons for $A$ and $B$ . Those two comparisons are quasi-isomorphisms by Cartesianity, so their cone comparison is a quasi-isomorphism by the long exact cohomology sequences. Thus $C (f)$ has Cartesian cohomology. The degree-zero restriction $C (f)_{0}$ is the cone of $f_{0}$ between constructible complexes, and by the common-stratification closure of [F2] it is again constructible. Hence $D_{G, c}^{b} (X; k)$ is a full triangulated subcategory of the ambient bounded derived category.

2.1F1F2step 1.2

Conservative restriction. Let $A \in D_{G, c}^{b} (X; k)$ with $A_{0} ≅ 0$ in $D^{b} ({Sh}_{k} (X))$ . All cohomology sheaves of $A_{0}$ vanish, and by Cartesianity the structure morphism of $H^{j} (A)$ along the projection $d_{0}^{p} : Z_{p} \to Z_{0} = X$ identifies $H^{j} (A)_{p}$ with the pullback of $H^{j} (A_{0})$ for every $p$ , so every cohomology simplicial sheaf of $A$ vanishes. An object of the bounded derived category with vanishing cohomology objects is zero by [F2], so $A ≅ 0$ . Restriction to $X$ is thus conservative.

2.2F1F3step 1.2

The finite resolution triples. Apply [F3] to the frame resolution of $G$ : the simplicial map $p_{N} : G^{∙} \times P_{N} \to G^{∙} \times X$ is levelwise a locally trivial bundle with $r_{N}$ -acyclic fiber, its induced pullback is compatible with the simplicial structure, and the locally split Čech descent of [F1] for the principal bundle $P_{N} \to X_{N}$ identifies the Cartesian-cohomology derived category of the nerve of $P_{N}$ with $D_{c}^{b} (X_{N}; k)$ . Composing with $Q_{N}$ gives the identification of $D_{G, c}^{b} (X; k)$ with the full bounded resolution-triple subcategory of $D_{c}^{[b, a]} (X_{N}; k)$ in the stated range: an equivariant object $A$ corresponds to the triple $(Q_{N} (A), p_{N}^{*} A ≅ q_{N}^{*} Q_{N} (A))$ , and conversely a triple whose frame pullback comes from $X$ arises from an equivariant object. The bounded Hom comparison of [F3] is induced by pullback along $p_{N}$ and the descent equivalence, hence carries composition to composition; at $m = 0$ it is a ring comparison ${End}_{G} (K) ≅ {End}_{X_{N}} (Q_{N} K)$ for objects in the interval, and retract closure is inherited from [F3].

2.3F1F2step 1.2

Idempotent completeness. Let $e : K \to K$ be an idempotent endomorphism of a bounded object $K \in D_{G, c}^{b} (X; k)$ . Form the telescope in the bounded-below derived category of all simplicial sheaves: let $S = ⨁_{j \geq 0} K$ be the countable sum with the shift map $1 - shift (e) : S \to S$ sending the $j$ -th summand to the $j$ -th summand by $i d$ and to the $(j + 1)$ -st by $- e$ , and let $T = Cone (1 - shift (e))$ . Countable coproducts in ${Sh}_{k} (Z_{∙})$ exist levelwise and are exact, and the cone is levelwise, so $H^{j} (T) = {colim}_{j} (H^{j} (K) \overset{H^{j} (e)}{\to} H^{j} (K) \overset{H^{j} (e)}{\to} \dots) = im H^{j} (e)$ : the colimit of an idempotent is its image, and because $H^{j} (e)$ is an idempotent endomorphism of the simplicial sheaf $H^{j} (K)$ its image is a direct summand. The image of an idempotent of a simplicial sheaf is again a simplicial sheaf, is bounded, is Cartesian because it is a summand of a Cartesian object, and restricts on $X$ to the image of the idempotent $H^{j} (e)_{0}$ , a summand of a constructible sheaf and hence constructible of finite rank. Thus $T$ is an object of the bounded-below ambient derived category with bounded Cartesian constructible cohomology, and by truncation it defines an object of $D_{G, c}^{b} (X; k)$ . The first-summand inclusion $i : K \to T$ satisfies $i e = i$ by the telescope relations, and the map $S \to K$ with every component $e$ annihilates $1 - shift (e)$ , so the triangle yields $p : T \to K$ with $p i = e$ . Then $i p$ is an isomorphism on cohomology, hence by [F2] an isomorphism. Also $(i p)^{2} = i (p i) p = i e p = i p$ , since $i e = i$ . An invertible idempotent is the identity, so $i p = {i d}_{T}$ and $p i = e$ , splitting the idempotent inside $D_{G, c}^{b} (X; k)$ . Hence the category is idempotent complete.

3.1F1F2F3step 1.1step 1.2step 2.1step 2.2

Triangulated, locally small, conservative. Steps 1.1, 1.2 and 2.1 exhibit the ambient category and its Hom groups, the triangulated structure of $D_{G, c}^{b} (X; k)$ , and conservativity of restriction; step 2.2 gives the finite resolution identification and the degree-zero ring and retract comparisons. No derived full faithfulness of restriction to $X$ is asserted or used.

4.1givenA1step 2.2step 3.1step 2.3∎

Conclusion. Steps 1.1, 1.2, 2.1 and 2.2 give the triangulated, locally small model with conservative restriction and the finite resolution comparisons; step 2.3 gives idempotent completeness. AC is consumed exactly as recorded in [A1].

Remarks

The telescope argument splits idempotents inside the actual bounded constructible category without importing a Karoubi completion and without arbitrary coproducts inside the constructible subcategory: the countable sum is formed in the ambient simplicial derived category and the resulting object is shown to lie in the subcategory.
Restriction to $X$ is conservative but is deliberately not claimed to be derived fully faithful; the finite Stiefel comparisons supply the quantitative substitute used by the later perverse and Ext arguments.

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Constructible six functors, proper base change and normalized equivariant duality

Statement

Assume the Axiom of Choice (The Axiom of Choice), that $k$ has characteristic zero, and that a finite product $G = \prod_{i} {GL}_{n_{i}} (C)$ of complex general linear groups acts algebraically on separated finite-type complex varieties (Equivariant constructible derived category of complex varieties). For equivariant algebraic maps between such varieties, the coherent ordinary six-operation contract holds in the equivariant bounded constructible category $D_{G, c}^{b}$ :

the functors $f^{*}, R f_{*}, R f_{!}, f^{!}$ are defined on $D_{G, c}^{b}$ , with $f^{*} ⊣ R f_{*}$ , $R f_{!} ⊣ f^{!}$ , and the tensor and internal-Hom adjunction $\otimes_{k}^{L} ⊣ R H o m$ ;
composition and identity comparisons, support localization, the projection formula and the trace hold, and for a Cartesian square with $f : X \to Y$ , $u : Y^{'} \to Y$ , $v : X^{'} \to X$ and $f^{'} : X^{'} \to Y^{'}$ the exchange $u^{*} R f_{!} ≃ R f_{!}^{'} v^{*}$ and its dual mate $u^{!} R f_{*} ≃ R f_{*}^{'} v^{!}$ obey Cartesian pasting; proper $f$ satisfies $R f_{!} = R f_{*}$ ;
the normalized dualizing object $ω^{G}$ restricts to the ordinary $ω_{X}$ of Constructible Verdier duality and complex orientation, and for a smooth complex $d$ -dimensional $X$ one has $ω_{X}^{G} ≃ k_{X} [2 d]$ ; this is not the unnormalized quotient-stack dualizing shift.

Facts & Assumptions

Given: The field $k$ of characteristic zero; the group $G$ and the separated finite-type complex varieties with their algebraic $G$ -actions; the equivariant category $D_{G, c}^{b}$ and its finite resolutions; the Axiom of Choice The Axiom of Choice; the sheaf-operation conventions of Inverse image presheaf and inverse image sheaf, Direct image of a sheaf along a continuous map, Extension by zero for abelian sheaves on an open subspace, Tensor product of sheaves of modules, The internal Hom sheaf of two module sheaves, Compactly supported global sections and their right derived functors and Derived hom in the bounded setting.

[F1]

The ordinary six operations (Actual analytic constructible sheaves admit coherent six operations, Coherent six operations on analytic constructible field sheaves): on bounded algebraically constructible complexes of analytic sheaves on separated finite-type complex varieties, the operations $f^{*}, R f_{*}, R f_{!}, f^{!}$ , $\otimes_{k}^{L}$ and $R H o m$ restrict to $D_{c}^{b}$ with their three adjunctions, coherent composition, support localization, ordinary and extraordinary proper base change, projection formula and trace; these comparisons commute with Cartesian pasting, with composition of maps, with evaluation and with trace; proper maps satisfy $R f_{!} = R f_{*}$ ; and for a locally trivial product projection the finite product cochain model computes the pushforward of a pullback.

[F2]

Ordinary normalized Verdier duality (Constructible Verdier duality and complex orientation): the dualizing object $ω_{X} = a_{X}^{!} k$ and $D_{X} = R H o m (-, ω_{X})$ preserve $D_{c}^{b}$ , evaluation is an isomorphism, $ω_{X} ≃ k_{X} [2 d]$ on a smooth complex $d$ -fold, and duality exchanges pushforward and compactly supported pushforward and computes $f^{!}$ .

[F3]

The equivariant model (Cartesian simplicial sheaves model actual equivariant derived Hom, Equivariant constructible derived category of complex varieties): $D_{G, c}^{b}$ is a triangulated, locally small category with conservative restriction to $X$ , identified through the finite general linear frame resolutions with the full bounded resolution-triple subcategories on the associated varieties $X_{N}$ ; the comparison functors are fully faithful on cohomological intervals of width at most $r_{N}$ and induce the degree-zero ring comparison.

[F4]

The finite conical supplier Complex algebraic constructible complexes have finite conical cochain models constructs open-projective graph factorizations on quasi-projective pieces and finite affine-open descent for a general separated finite-type map (Proof, steps 6.1–7.1). Combined with the open-inclusion smooth exchange and proper base change of [F1], this supplies smooth base change for arbitrary maps here, by the argument in step 1.1 below.

[F5]

General proper-support exchange (Support localization, base change, projection and trace calculus): for a Cartesian square of locally compact Hausdorff spaces with the uniform finite fibre-dimension bounds, $u^{*} R f_{!} ≃ R f_{!}^{'} v^{*}$ without a properness assumption on $f$ , and these canonical exchange maps respect Cartesian pasting. The analytic spaces and algebraic maps here satisfy these hypotheses: finite-dimensional local algebraic models give locally compact Hausdorff spaces and a uniform finite bound on the cohomological dimensions of all opens in their fibres.

[A1]

AC is used through the injective replacements, the ordinary six-operation and duality suppliers of [F1] and [F2] and the finite resolutions of [F3]; the pasting and normalization comparisons below are canonical (The Axiom of Choice).

Proof

1.1F1F2F3F4F5

Smooth comparison and image preservation. On each quasi-projective source piece over an affine target, [F4] factors the map as an open inclusion into its graph closure in $P^{n} \times Y$ , followed by proper projection. The open-inclusion smooth exchange of [F1] followed by proper base change gives smooth exchange for $R f_{*}$ on this piece. A finite affine source cover and its intersections compute ordinary pushforward by finite open-cover derived descent [F4]; smooth inverse image is exact, so the piece comparisons give the canonical smooth exchange for the general map. The comparison respects composition because its maps are the ordinary units and base-change maps. Proper-support exchange for arbitrary maps is [F5]; smooth exchange for tensor, internal Hom and normalized duality follows locally by adjoining a contractible smooth ball to their finite cochain models. For a fixed frame resolution, write $D_{X}^{N} = R H o m (-, ω_{X_{N}} [- 2 s_{N}])$ . Smooth orientation gives $q_{N}^{*} D_{X}^{N} Q_{N} (A) ≅ {pr}_{X}^{*} D_{X} A_{X}$ , since smooth orientation gives $q_{N}^{*} ω_{X_{N}} = {pr}_{X}^{*} ω_{X} [2 s_{N}]$ and both dualizing shifts cancel the dimensions $u_{N} = g + s_{N}$ . The smooth comparison for the square formed by $q_{N}$ , $f_{N}$ and ${i d}_{U_{N}} \times f$ , followed by the product projection to $f$ , therefore gives $q_{N}^{*} R f_{N *} Q_{N} (A) ≅ {pr}_{Y}^{*} R f_{*} A_{X}$ and likewise for $R f_{N!}$ . Inverse image and tensor have this comparison directly; internal Hom has it by its smooth comparison; $f_{N}^{!} = D_{X}^{N} f_{N}^{*} D_{Y}^{N}$ has it by duality. Thus every ordinary operation on the resolution representatives satisfies the frame-pullback image criterion [F3], with an ordinary bounded constructible object downstairs. Its unshifted cohomological amplitude equals the downstairs amplitude, because both $q_{N}^{*}$ and ${pr}_{X}^{*}$ are exact and conservative. These are bounds independent of $N$ .

2.1F1F3step 1.1

Transport of operations and adjunctions. For each fixed finite collection of inputs, outputs and adjunction maps, choose $N$ whose gap exceeds the widths of their unshifted ordinary amplitudes and the finitely many Hom comparison bounds. By step 1.1 all the ordinary resolution operations lie in the finite-resolution image, so [F3] transports them to $D_{G, c}^{b}$ . On a common product resolution the smooth comparisons identify the two constructions canonically; their coherence gives the cocycle on triple products, hence independence of the choice of $N$ . Every bounded input is treated this way; no decomposition of an object into its cohomology sheaves is assumed. The ordinary adjunction Hom bijections, their units and counits, transport through the fully faithful comparisons. Naturality and the triangle identities are equations of these maps and are reflected by a sufficiently large faithful comparison. This proves $f^{*} ⊣ R f_{*}$ , $R f_{!} ⊣ f^{!}$ and the tensor/internal-Hom adjunction on the actual bounded equivariant category.

3.1F1F3F5step 2.1

Coherence. Composition of maps, identity comparisons, support localization, the projection formula, trace and Cartesian pasting are equations between morphisms built from unit and counit maps of the ordinary adjunctions. By [F1] these equations hold at the level of the resolution varieties, and naturality of the resolution comparisons plus their full faithfulness on the intervals means that an equation holding after transport to $X_{N}, Y_{N}$ holds in $D_{G, c}^{b}$ ; more precisely, the comparison functors are faithful on the relevant Hom groups by [F3], so the two sides of each coherence equation, which agree after transport, agree before it. The general proper-support exchange and its pasting transfer from [F5] in the same way; proper equality $R f_{!} = R f_{*}$ transfers from [F1]. This proves the first part of claim 2.

3.2F2F3step 2.1

Normalization. For a frame resolution $P_{N} = U_{N} \times X \to X_{N}$ with base $B_{N}$ and maps $a_{X_{N}} : X_{N} \to B_{N}$ , $q_{N} : P_{N} \to X_{N}$ and ${pr}_{X} : P_{N} \to X$ , put $u_{N} = \dim_{C} U_{N}$ , $g = \dim_{C} G$ and $s_{N} = \dim_{C} B_{N} = u_{N} - g$ , and define the normalized dualizing object $ω_{X_{N}}^{G} : = ω_{X_{N}} \otimes_{k} a_{X_{N}}^{*} ω_{B_{N}}^{- 1}$ , the inverse taken in the tensor category of bounded constructible sheaves on $B_{N}$ ; here $B_{N}$ is a smooth complex $s_{N}$ -fold, so by [F2] its dualizing object is $ω_{B_{N}} ≃ k_{B_{N}} [2 s_{N}]$ and $a_{X_{N}}^{*} ω_{B_{N}}^{- 1} ≃ k_{X_{N}} [- 2 s_{N}]$ . Transitivity of dualizing objects for the smooth maps ${pr}_{X}$ of relative dimension $u_{N}$ and $q_{N}$ of relative dimension $g$ gives $ω_{P_{N}} ≃ {pr}_{X}^{*} ω_{X} [2 u_{N}] ≃ q_{N}^{*} ω_{X_{N}} [2 g]$ , hence $q_{N}^{*} ω_{X_{N}} ≃ {pr}_{X}^{*} ω_{X} [2 s_{N}]$ and therefore $q_{N}^{*} ω_{X_{N}}^{G} ≃ q_{N}^{*} ω_{X_{N}} [- 2 s_{N}] ≃ {pr}_{X}^{*} ω_{X}$ : the normalization cancels exactly the base contribution, so $ω^{G}$ restricts to the ordinary $ω_{X}$ under the forgetful maps of [F3]. On a smooth complex $d$ -dimensional $X$ , complex orientation gives $ω_{X} ≃ k_{X} [2 d]$ by [F2], and no extra shift survives because the normalization removes exactly the base contribution: this is the exclusion of the unnormalized quotient-stack dualizing shift recorded in the Statement. This proves claim 3.

4.1F2F3F5step 3.1step 3.2

Dual exchange and pasting. On a resolution over the common base $B_{N}$ , let $D_{T}^{N} = R H o m (-, ω_{T_{N}} [- 2 s_{N}])$ for each corner $T$ . The common shift preserves involutivity and the ordinary duality identities [F2]. Applying [F5] to $D_{X}^{N} A$ and then dualizing gives $D_{Y^{'}}^{N} u^{*} R f_{!} D_{X}^{N} A ≃ D_{Y^{'}}^{N} R f_{!}^{'} v^{*} D_{X}^{N} A$ , whose two sides are $u^{!} R f_{*} A$ and $R f_{*}^{'} v^{!} A$ respectively. Thus the extraordinary exchange holds for arbitrary $f$ , rather than only proper $f$ . Transport through the normalized comparison of step 3.2 and the fully faithful resolution comparisons gives the equivariant exchange. Duality carries the pasting identities of [F5] to the dual pasting identities, so these comparisons have the required coherence. This completes claim 2.

5.1givenA1F5step 1.1step 2.1step 3.1step 3.2step 4.1∎

Conclusion. Steps 1.1, 2.1, 3.1, 3.2 and 4.1 prove claims 1 to 3. Every comparison is induced by the ordinary coherent structure of [F1] and [F5] and the normalization of step 3.2; no unnormalized dualizing shift and no unproved base-change identity is introduced. AC enters as recorded in [A1].

Remarks

The normalizing twist is exactly the base-dimension correction $ω_{X_{N}} \otimes a_{X_{N}}^{*} ω_{B_{N}}^{- 1}$ ; on the frame bundle the two base contributions cancel, so the dualizing object restricts to the ordinary $ω_{X}$ and a smooth complex $d$ -fold has $ω_{X}^{G} ≃ k_{X} [2 d]$ .
The statement is an assembly theorem: the load-bearing inputs are the ordinary coherent six-operation and duality theorems and the finite resolution model, and all equivariant statements are obtained by transport through the fully faithful resolution comparisons.

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General linear Stiefel perverse normalization is exact and conservative

Statement

Assume the Axiom of Choice (The Axiom of Choice) and the finite general linear Stiefel resolution $P_{N} = U_{N} \times X \to X_{N} = P_{N} / G$ of Cartesian simplicial sheaves model actual equivariant derived Hom for $G = \prod_{i} {GL}_{n_{i}} (C)$ acting on the separated finite-type complex variety $X$ , with base $B_{N}$ , frame space $U_{N}$ , $u_{N} = \dim_{C} U_{N}$ , $g = \dim_{C} G$ and $s_{N} = \dim_{C} B_{N} = u_{N} - g$ . Write $Q_{N}$ for the finite-resolution functor and let ${pr}_{X} : P_{N} \to X$ , $q_{N} : P_{N} \to X_{N}$ be the two projections. Then:

the middle-perverse t-structure on $D_{G, c}^{b} (X; k)$ is detected by restriction to $X$ : for every sufficiently large $N$ and every $a$ , $A \in^{p} D_{G}^{\leq a}$ holds if and only if the restriction of $A$ to $X$ lies in $^{p} D^{\leq a} (X; k)$ , if and only if $Q_{N} (A) [s_{N}]$ lies in $^{p} D^{\leq a} (X_{N}; k)$ , and independently of the sufficiently large $N$ ;
the functor $A \mapsto Q_{N} (A) [s_{N}]$ is exact for these middle-perverse t-structures and conservative in the sense that $Q_{N} (A) = 0$ if and only if $A = 0$ ;
perverse truncations lift: for $A \in D_{G, c}^{b} (X; k)$ and all sufficiently large $N$ there is an equivariant perverse truncation $^{p} τ_{G}^{\leq a} A$ with $Q_{N} (^{p} τ_{G}^{\leq a} A) [s_{N}] ≅^{p} τ^{\leq a} (Q_{N} (A) [s_{N}])$ , canonical up to unique isomorphism, and the family is independent of $N$ ;
at the degree-zero Hom stabilization bound $r_{N} \geq a - b$ for objects concentrated in ordinary degrees $[b, a]$ , the ring comparison ${End}_{G} (K) ≅ {End}_{X_{N}} (Q_{N} K)$ lifts finite families of orthogonal idempotents, and after the common shift $s_{N}$ the normalized functor preserves and reflects perverse concentration and simplicity.

Facts & Assumptions

Given: The group $G$ , the variety $X$ , the resolution data $U_{N}, B_{N}, P_{N}, X_{N}$ and the functor $Q_{N}$ of Cartesian simplicial sheaves model actual equivariant derived Hom; the ordinary middle-perverse t-structure of Ordinary middle perverse t structure, recollement and intersection complexes on the ordinary constructible derived categories; the Axiom of Choice The Axiom of Choice; the shift conventions of Derived category of an abelian category and Derived hom in the bounded setting.

[F1]

Smooth pullback and perverse exactness (Shifted smooth complex pullback is exact for the middle perverse t structure): for a smooth algebraic map $a : X \to Y$ of pure relative dimension $d$ the shifted pullback $a^{*} [d]$ is exact for the ordinary middle-perverse t-structures and commutes with truncation and cohomology; complex orientation gives $a^{!} = a^{*} [2 d]$ ; a surjective $a$ has conservative normalized pullback reflecting the perverse support and cosupport inequalities.

[F2]

The ordinary perverse core (Ordinary middle perverse t structure, recollement and intersection complexes): the middle-perverse structure is a bounded t-structure on $D_{c}^{b} (Y; k)$ independent of compatible refinement, its truncations and cohomology functors are defined and unique, and the heart is abelian of finite length with intermediate-extension simple objects.

[F3]

The equivariant model (Cartesian simplicial sheaves model actual equivariant derived Hom): $D_{G, c}^{b} (X; k)$ is triangulated and locally small, restriction to $X$ is conservative, and $Q_{N}$ is fully faithful on ordinary intervals of width at most $r_{N}$ with a ring comparison at $m = 0$ and retract closure; the resolution triples satisfy ${pr}_{X}^{*} A_{X} ≅ q_{N}^{*} Q_{N} (A)$ .

[F4]

The actual finite-resolution image and Hom comparisons, including interval $[0, 0]$ , are General linear Stiefel approximations stabilize bounded constructible Hom. Equivariant localization and ordinary restriction are Constructible six functors, proper base change and normalized equivariant duality, and smooth local product comparison for open-inclusion $R j_{*}$ is Coherent six operations on analytic constructible field sheaves.

[A1]

AC is used through the ordinary perverse suppliers of [F1], [F2] and the resolution model of [F3]; the transport and uniqueness arguments below are canonical (The Axiom of Choice).

Proof

1.1F1F3given

The dimension correction. The projections ${pr}_{X} : P_{N} \to X$ and $q_{N} : P_{N} \to X_{N}$ are smooth surjective algebraic maps of pure relative complex dimensions $u_{N}$ and $g$ , and ${pr}_{X}^{*} A_{X} ≅ q_{N}^{*} Q_{N} (A)$ by [F3]. Since $s_{N} + g = u_{N}$ , this gives $q_{N}^{*} (Q_{N} (A) [s_{N}]) [g] ≅ {pr}_{X}^{*} A_{X} [u_{N}]$ . The left side is the normalized object under the perverse-exact functor $q_{N}^{*} [g]$ , and the right side is $A_{X}$ under the perverse-exact functor ${pr}_{X}^{*} [u_{N}]$ . Thus the comparison uses the unshifted base object $A_{X}$ , without an extra shift by $u_{N}$ on $X$ .

1.2F3F4given

Connected fibres and local systems. Each ${GL}_{n_{i}} (C)$ is path connected: Gaussian elimination expresses an invertible matrix as a product of elementary matrices and nonzero diagonal matrices, each joined to the identity by a path. Each frame space $Inj (C^{n_{i}}, C^{N_{i}})$ is the continuous image of connected ${GL}_{N_{i}} (C)$ under its first- $n_{i}$ -columns map, so $U_{N}$ is connected. A locally trivial bundle with connected fibre induces a surjection on fundamental groups: lift a base loop locally and join its endpoint to its initial point inside the fibre. Thus on a connected smooth invariant stratum $S$ both $q_{N} : U_{N} \times S \to S_{N}$ and ${pr}_{S} : U_{N} \times S \to S$ induce surjective fundamental-group maps. Local trivializations define parallel transport by subdivision along paths; subdivision of a homotopy square proves homotopy invariance, and a sub-local-system is exactly a subspace of one stalk invariant under this loop transport, extended to all stalks by transport. Thus these pullbacks preserve and reflect irreducibility, since the full monodromy image is unchanged. If $L$ is an equivariant local system and $L_{N} = Q_{N} L$ , any ordinary sub-local-system $M_{N} \subset L_{N}$ pulls back to a sub-local-system of ${pr}_{S}^{*} L$ . On the product the $U_{N}$ factor acts trivially on this monodromy representation and its subrepresentations, so $q_{N}^{*} M_{N} = {pr}_{S}^{*} M$ for a local system $M$ on $S$ . The actual resolution-image criterion in ordinary interval $[0, 0]$ descends $M_{N}$ to an equivariant local system and the degree-zero Hom comparison lifts its injection. Hence $L$ is simple equivariantly if and only if $L_{N}$ is irreducible ordinarily, and by the connected-fibre pullbacks if and only if its underlying local system on $S$ is irreducible.

2.1F1F3step 1.1

Detection via restriction to $X$ . By [F1] the shifted pullbacks ${pr}_{X}^{*} [u_{N}]$ and $q_{N}^{*} [g]$ are exact for the ordinary middle-perverse t-structures and commute with truncation; hence, for every $a$ , $q_{N}^{*} (Q_{N} (A) [s_{N}]) [g] = {pr}_{X}^{*} (A_{X} [u_{N}])$ lies in the perverse aisle $^{p} D^{\leq a}$ of $P_{N}$ if and only if $A_{X}$ lies in $^{p} D^{\leq a}$ of $X$ , and by exactness of $q_{N}^{*} [g]$ this is equivalent to $Q_{N} (A) [s_{N}] \in^{p} D^{\leq a} (X_{N})$ . The same computation for the dual aisle identifies the two inequalities, so the structure is detected by restriction to $X$ and by the normalized resolution simultaneously. Independence of the sufficiently large $N$ follows because the conditions are those of the fixed ordinary object $A_{X}$ . This proves claim 1.

3.1F1F2F3step 1.1step 2.1

Lift the truncation triangle. Fix an arbitrary perverse cutoff $c$ , and write $A_{X}$ for restriction of $A$ . Let $T_{N} =^{p} τ^{\leq c} (Q_{N} (A) [s_{N}])$ and $V_{N} =^{p} τ^{\geq c + 1} (Q_{N} (A) [s_{N}])$ . The ordinary perverse-exact smooth pullbacks [F1] give $q_{N}^{*} T_{N} ≅ {pr}_{X}^{*} (^{p} τ^{\leq c} A_{X}) [s_{N}]$ and $q_{N}^{*} V_{N} ≅ {pr}_{X}^{*} (^{p} τ^{\geq c + 1} A_{X}) [s_{N}]$ . Since the two unshifted pullbacks are exact and conservative, the ordinary amplitudes of $T_{N} [- s_{N}]$ and $V_{N} [- s_{N}]$ equal those of the two ordinary truncations on $X$ , independently of $N$ . Choose one interval containing these amplitudes and that of $A$ , enlarge it to include cones of the triangle arrows, and choose $r_{N}$ larger than its width. The image criterion [F3] lifts the two unshifted truncations, and degree-zero full faithfulness lifts their arrows to and from $A$ . Complete the lifted first arrow to a distinguished triangle; exactness of $Q_{N}$ and full faithfulness identify its cone and connecting map with $V_{N} [- s_{N}]$ and its connecting map. Thus the whole truncation triangle lifts. On $X$ its outer objects lie in the desired two perverse aisles, by step 2.1. No arbitrary ordinary perverse object is assumed to descend.

4.1F1F2F3step 2.1step 3.1

The perverse structure, uniqueness and conservativity. Define the two equivariant aisles by ordinary restriction. Their shift closures are inherited from [F2]. For objects in opposite aisles, choose a single sufficiently large resolution for their ordinary amplitudes; step 2.1 places their normalized images in opposite ordinary perverse aisles, and the faithful degree-zero Hom comparison [F3] gives orthogonality. Step 3.1 supplies truncation triangles, so these aisles form a bounded t-structure. The usual truncation uniqueness follows from orthogonality: a map from an object in the lower aisle to $A$ factors uniquely through the lower truncation, as the Hom groups to the upper term and its $[- 1]$ shift vanish. Consequently the lifts are canonical up to unique isomorphism and agree for different sufficiently large $N$ . Step 2.1 makes $Q_{N} (-) [s_{N}]$ perverse exact. Finally, $Q_{N} (A) = 0$ implies ${pr}_{X}^{*} A_{X} = q_{N}^{*} Q_{N} (A) = 0$ ; surjective exact ${pr}_{X}^{*}$ gives $A_{X} = 0$ , and conservative restriction [F3] gives $A = 0$ . This proves claims 1–3, without applying a Hom bound to an arbitrary fixed small $N$ .

4.2F1F2F3step 2.1step 3.1

Projectors, concentration and simplicity. Let $K \in D_{G, c}^{b} (X; k)$ have ordinary cohomology in $[b, a]$ with $a - b \leq r_{N}$ , so that the degree-zero ring comparison ${End}_{G} (K) ≅ {End}_{X_{N}} (Q_{N} K)$ of [F3] is an isomorphism. A finite family of mutually orthogonal idempotents in ${End}_{X_{N}} (Q_{N} K)$ summing to the identity pulls back to a finite family in ${End}_{G} (K)$ with the same relations, and the idempotent splitting of [F3] splits each of them; the summands remain in $[b, a]$ because ordinary cohomology of a direct summand is a direct summand. By claims 1 and 2 the normalized functor is exact and conservative, so a nonzero proper equivariant subobject of a lift would give a nonzero proper ordinary subobject of the corresponding normalized object; this proves reflection of simplicity, which is the direction consumed by the semisimplicity application, as well as detection of perverse concentration. The preservation direction is proved below using the actual connected general linear group hypothesis and local-system descent. This proves claim 4.

5.1F2F3F4step 2.1step 3.1step 4.1step 1.2

Underlying ordinary simplicity. The equivariant truncations of step 3.1 and the localization adjunctions of [F4] define intermediate extension by the image of $^{p} H^{0} j_{!} \to^{p} H^{0} R j_{*}$ . A simple equivariant heart object $A$ has irreducible invariant support $Z$ : the finite support components are permuted by connected $G$ , hence individually invariant, and localization on a dense component would otherwise yield a nonzero proper subobject. Choose the invariant dense smooth open where its ordinary cohomology is locally constant; equivariance preserves this locus and algebraic constructibility gives a dense algebraic such open. The perverse inequalities force restriction there to be a local system $L [\dim Z]$ . A proper equivariant sub-local-system yields a proper subobject by localization adjunction and perverse image, so $L$ is simple equivariantly. Absence of boundary subobjects and quotients identifies $A$ with $j_{! *} L [\dim Z]$ by the image and adjunction characterization of [F2]. Restriction is perverse exact by step 2.1 and commutes with localization in [F4], hence with that image. Step 1.2 makes $L$ irreducible ordinarily, so $A_{X}$ is the ordinary simple intersection complex of [F2].

6.1F1F2F3F4step 1.1step 2.1step 4.2step 5.1

Preservation after normalization. Smooth dimension-shifted pullback along a locally trivial map with connected smooth fibres preserves ordinary simple intersection complexes: the fundamental-group surjection preserves irreducibility on the dense smooth open, smooth local product comparison commutes with $j_{!}$ and $R j_{*}$ on boundary charts, and perverse exactness carries their intermediate-extension image to the same image upstairs. Here the fibres are irreducible frame spaces, so the inverse image of an irreducible support is irreducible. Apply this to ${pr}_{X}$ , with fibre $U_{N}$ . For simple equivariant $A$ , step 5.1 makes $A_{X}$ simple, so ${pr}_{X}^{*} A_{X} [u_{N}]$ is simple. By step 1.1 this is $q_{N}^{*} (Q_{N} (A) [s_{N}]) [g]$ . Any nonzero proper subobject of $Q_{N} (A) [s_{N}]$ would give one upstairs through exact conservative $q_{N}^{*} [g]$ , a contradiction. Thus normalization preserves simplicity; reflection is step 4.2 and concentration is step 2.1. Both directions use the same $s_{N} = u_{N} - g$ and stated ring-comparison threshold, with the actual product of connected general linear groups.

7.1givenA1step 1.2step 2.1step 3.1step 4.1step 4.2step 5.1step 6.1∎

Conclusion. Steps 1.1, 2.1, 3.1 and 4.1 prove claims 1–3, and steps 4.2, 5.1 and 6.1 claim 4. The dimension shift is $s_{N} = u_{N} - g = \dim_{C} B_{N}$ , not $u_{N}$ and not $g$ ; no arbitrary perverse object on $X_{N}$ is asserted to descend, and no derived full faithfulness of restriction to $X$ is used. AC enters as recorded in [A1].

Remarks

The binding identity is $s_{N} = u_{N} - g$ : the frame space has complex dimension $u_{N}$ , the group has complex dimension $g$ , and the quotient base has dimension $s_{N}$ , so the common shift of the normalized finite-resolution comparison is exactly the base dimension.
The lifting argument is finite: one sufficiently large resolution handles the finitely many truncations and perverse cohomology objects of a fixed bounded object, and uniqueness of truncation makes the lift independent of the choice.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Middle perverse t structure glues and has simple intersection complexes

Statement

Assume the Axiom of Choice (The Axiom of Choice) and let $k$ be a field of characteristic zero. On the equivariant constructible derived category $D_{G, c}^{b} (X; k)$ of Equivariant constructible derived category of complex varieties, middle perversity is tested on the ordinary restriction $A_{X}$ and its smooth complex constructibility strata $S$ by the two inequalities $H^{q} (i_{S}^{*} A_{X}) = 0$ for $q > - \dim_{C} S$ and $H^{q} (i_{S}^{!} A_{X}) = 0$ for $q < - \dim_{C} S$ , where $i_{S}$ is the inclusion of a stratum. These inequalities give a bounded t-structure independent of compatible refinement and gluing across $G$ -invariant algebraic open/closed decompositions, and the perverse aisles are detected by restriction to $X$ . Its heart is abelian of finite length, and its simple objects are the intermediate extensions ${IC}_{Z} (L) = i_{Z *} j_{! *} (L [\dim_{C} Z])$ of simple finite-rank local systems $L$ on dense smooth opens of irreducible supports $Z$ , where $Z$ and the dense open are $G$ -invariant, $i_{Z} : Z ↪ X$ is the closed inclusion, $j : U ↪ Z$ is the dense open inclusion, and $L$ has equivariant data. If $A, B$ lie in the heart, then $Hom (A, B [m]) = 0$ for $m < 0$ . A simple object's endomorphism ring is a division $k$ -algebra, not asserted to equal $k$ .

Facts & Assumptions

Given: The field $k$ of characteristic zero; the group $G$ and variety $X$ with the equivariant category $D_{G, c}^{b} (X; k)$ ; the ordinary middle-perverse t-structure of Ordinary middle perverse t structure, recollement and intersection complexes; the Axiom of Choice The Axiom of Choice; the conventions of Derived category of an abelian category, Derived hom in the bounded setting and Canonical t structure on a derived category.

[F1]

The normalized bridge (General linear Stiefel perverse normalization is exact and conservative): the middle-perverse structure on $D_{G, c}^{b} (X; k)$ is detected by restriction to $X$ , the functor $A \mapsto Q_{N} (A) [s_{N}]$ is perverse exact and conservative for all sufficiently large $N$ , and perverse truncations lift uniquely; at the degree-zero Hom stabilization bound finite rings of projectors lift and perverse concentration and simplicity are preserved and reflected after the shift $s_{N}$ .

[F2]

The equivariant six operations (Constructible six functors, proper base change and normalized equivariant duality): $D_{G, c}^{b}$ carries the coherent operations $f^{*}, R f_{*}, R f_{!}, f^{!}, \otimes_{k}^{L}, R H o m$ with their adjunctions, support localization and Cartesian pasting for equivariant algebraic maps, and the normalized dualizing object restricts to the ordinary one.

[F3]

The ordinary perverse core (Ordinary middle perverse t structure, recollement and intersection complexes): on $D_{c}^{b} (X; k)$ the middle-perverse inequalities define a bounded t-structure independent of compatible refinement and gluing across open/closed decompositions; its heart is abelian of finite length; its simples are the intermediate extensions $j_{! *} L [\dim Z]$ of simple finite-rank local systems on dense smooth opens of irreducible supports; heart objects have no negative Hom; and the endomorphism ring of a simple object is a division algebra.

[A1]

AC is used through the ordinary perverse core and the equivariant six-operation and normalization suppliers; the recollement and intermediate-extension arguments below are otherwise choice-free (The Axiom of Choice).

Proof

1.1F1F3given

The equivariant t-structure. For $A \in D_{G, c}^{b} (X; k)$ define $A \in^{p} D_{G}^{\leq a}$ if and only if the restriction $A_{X}$ lies in the ordinary aisle $^{p} D^{\leq a} (X; k)$ , and $^{p} D_{G}^{\geq a}$ dually. By [F1] this is equivalent to $Q_{N} (A) [s_{N}] \in^{p} D^{\leq a} (X_{N}; k)$ for every sufficiently large $N$ , so the definition is independent of the resolution. The pair $(^{p} D_{G}^{\leq 0},^{p} D_{G}^{\geq 0})$ satisfies the t-structure axioms because the defining conditions are those of a t-structure pointwise: closure under shifts and extensions follows from the corresponding ordinary closures and the conservativity of restriction, orthogonality $Hom (^{p} D_{G}^{\leq 0},^{p} D_{G}^{\geq 1}) = 0$ follows from the ordinary orthogonality transported through the fully faithful Hom comparison of [F1], and truncation triangles exist by the lifting clause of [F1]. The structure is bounded because the ordinary one is and restriction to $X$ is conservative. This proves the t-structure and detection assertions.

2.1F2F3step 1.1

Gluing and independence of refinement. The inequalities of the Statement are checked stratumwise and are therefore independent of compatible refinement by the ordinary clause of [F3]; gluing across an algebraic open/closed decomposition uses the localization triangles of [F2] and the ordinary gluing theorem: for a $G$ -invariant open immersion $j$ and its closed complement $i$ , the Čech-style recollement conditions $j^{*} i_{*} = 0$ , the triangles $j_{!} j^{*} \to i d \to i_{*} i^{*}$ and $i_{*} i^{!} \to i d \to j_{*} j^{*}$ hold in $D_{G, c}^{b}$ by [F2], and the transport of step 1.1 reduces the glued t-structure statement to the ordinary one.

2.2F1F3step 1.1

Heart and finite length. Kernel and cokernel of a morphism of the heart are the perverse cohomology objects in degrees $- 1$ and $0$ of its cone, computed by the truncation functors of step 1.1; hence the heart is abelian. Restriction to $X$ is perverse exact and conservative on the heart by [F1], so a strictly increasing or decreasing chain of subobjects in the equivariant heart gives a strictly increasing or decreasing chain in the ordinary heart; the ordinary heart has finite length by [F3], so the equivariant heart does. For heart objects $A, B$ and $m < 0$ , $B [m]$ lies in $^{p} D_{G}^{\geq 1}$ while $A$ lies in $^{p} D_{G}^{\leq 0}$ , so orthogonality gives $Hom (A, B [m]) = 0$ .

3.1givenF1F2F3step 1.1step 2.1step 2.2

Local systems and intermediate extension. The group $G$ is path connected: elementary matrices and nonzero diagonal matrices are joined to the identity, and Gaussian elimination expresses every invertible complex matrix as their product. Let $L$ be an equivariant local system on an invariant smooth open $U$ , with action isomorphism $ϕ : {pr}_{U}^{*} L ≃ a^{*} L$ on $G \times U$ . For any ordinary sub-local-system $M \subset L$ , the sub-local-systems $ϕ ({pr}_{U}^{*} M)$ and $a^{*} M$ agree on ${e} \times U$ by the identity condition. Every point $(g, x)$ is joined to $(e, x)$ by a path in $G \times U$ ; parallel transport therefore makes these sub-local-systems agree everywhere. Restricting $ϕ$ supplies equivariant data on $M$ , and the identity and cocycle equations restrict as well. In the action-nerve model this is a Cartesian sub-sheaf, hence an equivariant local system. Thus $L$ is simple equivariantly if and only if it is simple ordinarily. For irreducible invariant $Z$ and invariant dense smooth $j : U ↪ Z$ of dimension $d$ , a simple $L$ gives the simple ordinary perverse object $L [d]$ by [F3], and exact conservative restriction [F1] makes it simple in the equivariant heart too. Define $j_{! *} (L [d]) = im (^{p} H^{0} j_{!} L [d] \to^{p} H^{0} R j_{*} L [d])$ . The first term has no boundary quotient and the second no boundary subobject, by the localization adjunctions [F2], so the image has neither. Any extension with neither property is this image: its adjunction maps from the first term and into the second have boundary cokernel and kernel, hence are epic and monic. A subobject of this image restricts either to zero or to all of $L [d]$ ; the boundary subobject or resulting boundary quotient must then vanish. Closed pushforward $i_{Z *}$ is exact and fully faithful on hearts by the same localization and detection, so ${IC}_{Z} (L) = i_{Z *} j_{! *} (L [d])$ is simple.

4.1givenF1F2F3step 2.1step 2.2step 3.1

Classification of simple objects. Let $A$ be a nonzero simple equivariant heart object, and let $T$ be the closed support of $A_{X}$ . Localization identifies $A$ with closed pushforward of a heart object on $T$ , which we again call $A$ . Equivariance preserves $T$ and its smooth local-system locus. Its finitely many irreducible components are individually invariant because connected $G$ cannot permute them nontrivially. In the smooth locus of each component away from all component intersections, algebraic constructibility gives a dense algebraic open on which the cohomology of $A_{X}$ is locally constant. Its $G$ -saturation is still an algebraic open in this locus, and equivariance makes the cohomology locally constant there. These saturations form disjoint invariant smooth opens $W_{i}$ ; set $W = ⨆_{i} W_{i}$ . Write $h : W ↪ T$ . The perverse inequalities give $h^{*} A = ⨁_{i} L_{i} [\dim W_{i}]$ , with every $L_{i}$ nonzero. The left heart adjunction supplies a map $^{p} H^{0} h_{!} (L_{i} [\dim W_{i}]) \to A$ whose restriction is the inclusion of this summand. Its image is nonzero, hence all of $A$ by simplicity. Exact open restriction then forces this summand to equal $h^{*} A$ , excluding every other support component. Thus $T = Z$ is irreducible. On its invariant dense smooth open $U$ , write $j^{*} A = L [\dim Z]$ . A nonzero proper equivariant sub-local-system $M \subset L$ would similarly give an image subobject of $A$ restricting to $M [\dim Z]$ , contradicting simplicity. Hence $L$ is simple equivariantly, and therefore ordinarily by step 3.1. Simplicity of $A$ and $j^{*} A \neq 0$ exclude boundary subobjects and quotients. The characterization of step 3.1 identifies $A$ with the intermediate extension, and closed pushforward gives precisely ${IC}_{Z} (L)$ on $X$ .

5.1F3step 2.2step 3.1step 4.1

Endomorphism rings. Let $S$ be a simple heart object and $f : S \to S$ a nonzero endomorphism. Its kernel is a subobject of $S$ and hence zero, and its image is a nonzero subobject and hence all of $S$ ; thus $f$ is invertible. Therefore $End (S)$ is a division $k$ -algebra, and it is not asserted to equal $k$ . This completes the proof.

6.1givenA1step 1.1step 2.1step 2.2step 3.1step 4.1step 5.1∎

Conclusion. Steps 1.1 and 2.1 give the t-structure, its independence of refinement and gluing; steps 2.2, 3.1, 4.1 and 5.1 give the heart, its finite length, the negative-Hom vanishing and the intersection-complex classification with division endomorphism rings. AC enters as recorded in [A1], and no semisimplicity or characteristic restriction beyond characteristic zero is used.

Remarks

The equivariant t-structure is detected by restriction to $X$ ; the normalized finite-resolution bridge is used only to transport truncations, orthogonality and the intermediate-extension arguments, not to assert that arbitrary ordinary perverse objects descend.
The endomorphism ring of a simple equivariant perverse sheaf is a division $k$ -algebra; for coefficients in fields where the equivariant local system has larger endomorphism algebra it is genuinely larger than $k$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Equivariant constructible Ext is degreewise finite and compatible with characteristic zero scalars

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $G = \prod_{i} {GL}_{n_{i}} (C)$ be a finite product of complex general linear groups acting algebraically on a separated finite-type complex algebraic variety $X$ , and let $k$ be a field of characteristic zero. For bounded algebraically constructible $A, B$ in the equivariant category $D_{G, c}^{b} (X; k)$ of Equivariant constructible derived category of complex varieties:

every ${Ext}_{G}^{m} (A, B) = Hom (A, B [m])$ is finite-dimensional over $k$ and vanishes for all sufficiently negative $m$ ;
for every field extension $k \to K$ of characteristic-zero fields, the canonical map ${Ext}_{G}^{m} (A, B) \otimes_{k} K \to {Ext}_{G}^{m} (A_{K}, B_{K})$ is an isomorphism, natural in $A$ and $B$ and compatible with shifts and with Yoneda composition;
coefficient extension commutes with the coherent six operations, normalized Verdier duality, perverse truncation and intermediate extension on these bounded constructible objects.

The total Ext space $⨁_{m} {Ext}_{G}^{m} (A, B)$ need not be finite-dimensional, and simplicity is not asserted to be preserved under coefficient extension.

Facts & Assumptions

Given: The group $G$ and variety $X$ with the equivariant category $D_{G, c}^{b} (X; k)$ over the characteristic-zero field $k$ ; the field extension $k \to K$ ; the Axiom of Choice The Axiom of Choice; the conventions of Derived category of an abelian category, Derived hom in the bounded setting, Tensor product of sheaves of modules and The internal Hom sheaf of two module sheaves.

[F1]

The finite Stiefel comparison (General linear Stiefel approximations stabilize bounded constructible Hom, General linear frame spaces have the Stiefel cohomology gap): for $A \leq a$ and $B \geq b$ and $m \leq b + r_{N} - a$ there is a natural isomorphism ${Hom}_{G} (A, B [m]) ≅ {Hom}_{X_{N}} (Q_{N} (A), Q_{N} (B) [m])$ ; the frame spaces have the cohomology gap $H^{j} (U_{N}; k) = 0$ for $1 \leq j \leq r_{N}$ , and for a fixed bounded amplitude one can choose $N$ with $r_{N}$ larger than any prescribed finite bound.

[F2]

Finite constructible models (Finite simplicial constructible sheaves have an actual derived incidence model, Complex algebraic constructible complexes have finite conical cochain models, which globalizes the incidence models by finite affine-open descent): bounded algebraically constructible complexes on separated finite-type complex varieties have bounded finite-dimensional cochain models for ordinary Hom, and the models commute with every coefficient field extension.

[F3]

The equivariant operations and perverse structure (Constructible six functors, proper base change and normalized equivariant duality, Middle perverse t structure glues and has simple intersection complexes, General linear Stiefel perverse normalization is exact and conservative): $D_{G, c}^{b}$ carries the coherent equivariant six operations, normalized Verdier duality, the middle-perverse t-structure with its truncations, and the intermediate extension $j_{! *}$ constructed as the image $^{p} H^{0} j_{!} \to^{p} H^{0} j_{*}$ .

[F4]

Coefficient extension (Tensor product of sheaves of modules): for a field extension $k \to K$ the functor $- \otimes_{k} K$ is exact, termwise, and commutes with inverse image, with finite tensor products and with finite-dimensional duals; on the fixed finite models of [F2] it is a termwise base change of finite-dimensional vector spaces.

[F5]

The canonical derived t-structure and its precise Hom vanishing bound are proved by The canonical pair is a t structure: if $A \leq a$ and $B \geq b$ , then $Hom (A, B [m]) = 0$ for $m < b - a$ .

[A1]

AC is used through the finite-resolution comparison [F1] and the constructible models [F2]; the finite-dimensional Hom, tensor and truncation comparisons themselves are choice-free (The Axiom of Choice).

Proof

1.1F1F2F5

Finiteness and vanishing. Let $A, B$ be bounded constructible with $A \leq a$ and $B \geq b$ , fix $m$ , and let $I$ be an ordinary cohomological interval containing the amplitudes of $A$ and of $B [m]$ . Choose $N_{i} \geq n_{i} + 1$ with $r_{N} \geq length (I)$ ; such a choice is possible because $r_{N} = 2 \min_{i} (N_{i} - n_{i})$ grows without bound with the $N_{i}$ . By [F1] there is a natural isomorphism ${Ext}_{G}^{m} (A, B) ≅ {Hom}_{X_{N}} (Q_{N} (A), Q_{N} (B) [m])$ , and by [F2] the right-hand side is computed by a bounded finite-dimensional cochain model on the finite-type variety $X_{N}$ , hence is finite-dimensional. If $b - a > m$ , then $B [m] \geq b - m > a$ and the ordinary t-structure orthogonality makes ${Hom}_{G} (A, B [m]) = 0$ for $m < b - a$ , so the groups vanish for sufficiently negative $m$ . This proves claim 1.

1.2F1F2F4

Scalar extension. Coefficient extension on the simplicial category is termwise tensor with $K$ , hence exact by [F4], and it commutes with inverse image and with the free-quotient descent used to define $Q_{N}$ because those constructions are given locally by exact inverse image along algebraic sections together with gluing. Therefore $Q_{N} (A_{K}) ≅ (Q_{N} (A))_{K}$ and $Q_{N} (B_{K}) ≅ (Q_{N} (B))_{K}$ , and the same $N$ works over $k$ and over $K$ : the fiber-gap vanishing of [F1] is field-independent and the finite models of [F2] commute with coefficient extension, so the interval bookkeeping is unchanged. In the resulting commutative square of Hom comparisons the vertical arrows are the full-faithfulness isomorphisms of [F1] over $k$ and over $K$ , and the bottom horizontal arrow is the ordinary bounded constructible Hom coefficient-extension isomorphism of [F2]; hence the top horizontal arrow, which is the canonical map ${Ext}_{G}^{m} (A, B) \otimes_{k} K \to {Ext}_{G}^{m} (A_{K}, B_{K})$ , is an isomorphism. Naturality in $A$ and $B$ and compatibility with shifts hold because all maps are induced by functors and by the termwise base change, and compatibility with Yoneda composition holds because the comparisons are induced by functors and one sufficiently large $N$ serves the finitely many composable classes of a Yoneda product. No infinite totalization or infinite product of injectives is tensored with $K$ . This proves claim 2.

2.1F2F3F4step 1.2

Commutation with the operations and the perverse structure. The ordinary coherent six operations and normalized Verdier duality are determined on the finite constructible models of [F2], and the ordinary coefficient-extension clause of Actual analytic constructible sheaves admit coherent six operations shows that termwise exact coefficient extension commutes with inverse image, finite cochain pushforward, tensor, internal Hom and the finite-dimensional duality comparisons of [F3]; transporting through the finite-resolution comparisons as in step 1.2 gives the commutation with the equivariant operations and with normalized duality. For the perverse structure, exact coefficient extension preserves the upper support inequalities on ordinary restriction. It also preserves the lower inequalities, since it commutes with ordinary Verdier duality and that duality exchanges the two aisles. Hence it is perverse t-exact in the equivariant category detected by restriction [F3]. Applying it to a truncation triangle gives the unique truncation triangle over $K$ , so perverse truncations and cohomology commute with extension. Its induced functor on the abelian hearts is exact and therefore preserves the image defining $j_{! *}$ ; the already proved commutation with $j_{!}$ and $R j_{*}$ gives intermediate-extension commutation. This proves claim 3. Simplicity under extension is deliberately not asserted: a simple local system may fail to remain simple after extending scalars.

3.1givenA1step 1.1step 1.2step 2.1∎

Conclusion. Steps 1.1, 1.2 and 2.1 prove claims 1 to 3. The proof is finite-stage: one sufficiently large resolution handles a fixed bounded object and its finitely many shifted targets, and no infinite-dimensional total Ext space is used. The total Ext space is not claimed to be finite, and simplicity is not claimed to be preserved. AC enters as recorded in [A1].

Remarks

The finiteness statement is degreewise; the total Ext space $⨁_{m} {Ext}_{G}^{m} (A, B)$ can be infinite-dimensional, as the point example of this page's companion shows.
Coefficient extension is exact on the heart but need not preserve simplicity of objects; the theorem asserts only the Ext and operation compatibilities.

5 · Examples, counterexamples and false statements

None yet.

Equivariant Constructible Perverse Foundations

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Equivariant constructible derived category of complex varieties

Definition

Acyclic product fibers stabilize bounded constructible pullback Hom

Statement

Facts & Assumptions

Proof

Remarks

General linear frame spaces have the Stiefel cohomology gap

Statement

Facts & Assumptions

Proof

Remarks

General linear Stiefel approximations stabilize bounded constructible Hom

Statement

Facts & Assumptions

Proof

Remarks

Cartesian simplicial sheaves model actual equivariant derived Hom

Statement

Facts & Assumptions

Proof

Remarks

Constructible six functors, proper base change and normalized equivariant duality

Statement

Facts & Assumptions

Proof

Remarks

General linear Stiefel perverse normalization is exact and conservative

Statement

Facts & Assumptions

Proof

Remarks

Middle perverse t structure glues and has simple intersection complexes

Statement

Facts & Assumptions

Proof

Remarks

Equivariant constructible Ext is degreewise finite and compatible with characteristic zero scalars

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, counterexamples and false statements

Sources