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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.

Depends on

Used by

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Sources

Joseph Bernstein and Valery Lunts, Equivariant Sheaves and Functors, Lecture Notes in Mathematics 1578, Part I (Derived category and functors) (standard reference, not scraped)
A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Asterisque 100 (1982) (standard reference, not scraped)