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Proper-pushforward Ext is equivariant Borel–Moore homology of the fiber product

Statement

Assume the Axiom of Choice (The Axiom of Choice) as inherited from the constructible six-functor and Borel-stabilization suppliers. Use the constructible equivariant setting $D_{G, c}^{b} (E; k)$ of Equivariant constructible derived category of complex varieties and proper $G$ -equivariant algebraic maps $f_{i} : X_{i} \to E$ with smooth sources $X_{i}$ of pure complex dimension $d_{i}$ , as in Equivariant Borel–Moore homology and oriented correspondence convolution. Put $L_{i} = f_{i!} k_{X_{i}} [d_{i}]$ . For every integer $r$ and every pair $(i, j)$ there is a natural isomorphism $Φ_{i j}^{r} : {Hom}_{D_{G, c}^{b} (E; k)} (L_{i}, L_{j} [r]) \tilde{\to} H_{d_{i} + d_{j} - r}^{B M, G} (Z_{i j}; k) .$ All groups are degreewise finite-dimensional for these finite-type algebraic varieties and $G$ ; no finite-dimensionality of their direct sum over all degrees is asserted. The isomorphism is equivariant-cohomology linear, commutes with enlargement of the Borel approximation, and is natural in constructible coefficient morphisms before smooth orientation is applied. No semismallness, decomposition theorem, purity, or smoothness of $E$ or $Z_{i j}$ is required.

Facts & Assumptions

Given: the Axiom of Choice; a characteristic-zero field $k$ ; a product $G$ of complex general linear groups; proper $G$ -equivariant algebraic maps $f_{i} : X_{i} \to E$ with $X_{i}$ smooth of pure complex dimension $d_{i}$ and fiber products $Z_{i j} = X_{i} \times_{E} X_{j}$ ; the objects $L_{i} = f_{i!} k_{X_{i}} [d_{i}]$ in $D_{G, c}^{b} (E; k)$ ; and the Borel–Moore normalization of Equivariant Borel–Moore homology and oriented correspondence convolution.

[F1]

Assume AC. The equivariant constructible six-functor theorem Constructible six functors, proper base change and normalized equivariant duality supplies, for $D_{G, c}^{b}$ over characteristic zero, the adjunctions $f_{i!} ⊣ f_{i}^{!}$ and $p^{*} ⊣ p_{*}$ , the extraordinary proper base change $f_{i}^{!} f_{j!} ≅ p_{*} q^{!}$ for the Cartesian square of the two proper maps $f_{i}, f_{j}$ , the dualizing transitivity $q^{!} ω_{X_{j}} = ω_{Z_{i j}}$ transported from the ordinary analytic contract of Coherent six operations on analytic constructible field sheaves, the normalized dualizing object $ω_{X}^{G} ∣_{X} = ω_{X}$ , the smooth complex orientation $ω_{X}^{G} = k_{X} [2 d]$ for smooth $X$ of pure dimension $d$ , and $R f_{!} = R f_{*}$ for proper $f$ .

[F2]

Assume AC. The cartesian-simplicial model lemma Cartesian simplicial sheaves model actual equivariant derived Hom supplies the finite general linear Stiefel resolution comparisons: ${Hom}_{G} (A, B [m]) ≅ {Hom}_{X_{N}} (Q_{N} (A), Q_{N} (B) [m])$ on the stated bounded ranges, the coherent comparison equivalences for products of sufficiently acyclic resolutions. For the trivial group the identity model has unbounded acyclicity range.

[F3]

Assume AC. The definition Equivariant Borel–Moore homology and oriented correspondence convolution fixes $H_{j}^{B M, G} (X; k) = H_{G}^{- j} (X, ω_{X}^{G}) = Hom (k_{X}, ω_{X}^{G} [- j])$ , its stable Borel value $H_{j + 2 b}^{B M} (X \times_{G} U; k)$ for a free smooth approximation with $b = \dim_{C} U - \dim_{C} G$ , and the corner degrees $d_{i} + d_{j} - p$ ; the ambient category is that of Equivariant constructible derived category of complex varieties.

[F4]

Under AC, equivariant Hom groups of bounded algebraically constructible objects are finite-dimensional in every fixed degree, and commute with characteristic-zero field extension (Equivariant constructible Ext is degreewise finite and compatible with characteristic zero scalars).

[A1]

The Axiom of Choice is consumed exactly through [F1], [F2] and [F4], whose constructions (injective replacements, equivariant descent, Stiefel resolution comparisons) declare it; the adjunction chain, the orientation computation and the degree bookkeeping below are choice-free (The Axiom of Choice).

Proof

technique · direct

1.1F1given

The adjunction chain. Write $p : Z_{i j} \to X_{i}$ and $q : Z_{i j} \to X_{j}$ for the projections and put $t = r + d_{j} - d_{i}$ . Removing the source shift by $[- d_{i}]$ and using the adjunction $f_{i!} ⊣ f_{i}^{!}$ of [F1] gives ${Hom}_{E} (f_{i!} k_{i}, f_{j!} k_{j} [t]) = {Hom}_{X_{i}} (k_{i}, f_{i}^{!} f_{j!} k_{j} [t])$ . In the Cartesian square with lower arrow $f_{j}$ and right arrow $f_{i}$ , the extraordinary proper base change of [F1] gives $f_{i}^{!} f_{j!} k_{j} = p_{*} q^{!} k_{j}$ , because both maps are proper and the square is Cartesian; the adjunction $p^{*} ⊣ p_{*}$ then identifies the group with ${Hom}_{Z_{i j}} (k_{Z}, q^{!} k_{j} [t])$ . This is the complete cornerwise chain and it is valid for every integer $r$ , including negative $r$ .

2.1F1F3step 1.1

Orientation and the degree. The smooth orientation of [F1] on the smooth pure $d_{j}$ -dimensional source $X_{j}$ says $k_{X_{j}} = ω_{X_{j}} [- 2 d_{j}]$ , and dualizing transitivity for $q$ gives $q^{!} k_{X_{j}} = q^{!} ω_{X_{j}} [- 2 d_{j}] = ω_{Z_{i j}} [- 2 d_{j}]$ . Substituting $t = r + d_{j} - d_{i}$ , the group of step 1.1 is $Hom (k_{Z}, ω_{Z} [r - d_{i} - d_{j}])$ , which is exactly $H_{d_{i} + d_{j} - r}^{B M, G} (Z_{i j}; k)$ in the normalization of [F3]. At a finite free smooth approximation of dimension offset $b = \dim_{C} U - \dim_{C} G$ , the space $X_{j} \times_{G} U$ is smooth of complex dimension $d_{j} + b$ while the descended complex retains the shift $d_{j}$ , so the identical calculation gives ordinary Borel–Moore degree $d_{i} + d_{j} + 2 b - r$ ; removing $2 b$ gives the stated equivariant degree, and no stack-dimension shift is inserted into $ω_{X}^{G}$ .

3.1A1F1F2F3F4step 1.1step 2.1∎

Naturality, linearity and finiteness. Every map in the chain of steps 1.1 and 2.1 is an adjunction, base-change or orientation map supplied naturally by [F1], so the composite is natural in constructible coefficient morphisms before the smooth orientation is applied and is compatible with composition of the $f_{i}$ . By [F2] these maps commute with the approximation transition pullbacks on the bounded ranges, so the composite induces an isomorphism in each stable degree; coherent scalar pullback from the base point together with the projection formula of [F1] gives equivariant-cohomology linearity. Degreewise finiteness is exactly the conclusion of [F4] for bounded algebraically constructible objects, and it is asserted nowhere that the direct sum over all degrees is finite-dimensional. Smoothness of the sources is used only for the orientation in step 2.1, and properness only in the base change and in the identification $f_{i!} = f_{i *}$ ; no semismallness, decomposition theorem, purity or smoothness of $E$ or of the fiber products $Z_{i j}$ is used.

Depends on

Used by

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Sources

V. Ginzburg, Geometric methods in the representation theory of Hecke algebras and quantum groups, arXiv:math/9802004v3; 1997 Montreal lectures (57-page PDF) (standard reference, not scraped)
M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, arXiv:0901.3992 (complete article) (standard reference, not scraped)
Y. Varshavsky, Lefschetz-Verdier trace formula, arXiv:math/0505564 (complete article) (standard reference, not scraped)