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Projection formula for higher direct images and K-theory pushforward

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $f : X \to Y$ be a proper morphism of schemes of finite type over a field $k$ (Proper morphisms, Locally finite type and finite type morphisms). Let $E$ be a finite locally free $O_{Y}$ -module (Locally free sheaves of finite rank) and let $F$ be a coherent $O_{X}$ -module (Coherent module sheaves). Then for every $q \geq 0$ there is a canonical isomorphism of coherent $O_{Y}$ -modules $R^{q} f_{*} (f^{*} E \otimes_{O_{X}} F) ≅ E \otimes_{O_{Y}} R^{q} f_{*} F,$ and consequently, in $K_{0} (Y)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme), $f_{!} (f^{*} [E] \cdot [F]) = [E] \cdot f_{!} [F],$ where $f_{!}$ is the pushforward of Pushforward of coherent sheaves in algebraic K-theory and the products are the $K$ -theory module products.

Facts & Assumptions

Given: the Axiom of Choice; a proper morphism $f : X \to Y$ of finite type $k$ -schemes; a finite locally free $O_{Y}$ -module $E$ ; a coherent $O_{X}$ -module $F$ .

[F1]

The higher direct images $R^{q} f_{*} F$ are coherent and vanish for $q > \dim X$ ; hence the alternating sum $f_{!} [F]$ is a well-defined element of $K_{0} (Y)$ (Pushforward of coherent sheaves in algebraic K-theory, Grothendieck groups of coherent sheaves and of vector bundles on a scheme).

[F2]

Pullback of quasi-coherent sheaves is quasi-coherent, tensor products of quasi-coherent sheaves are quasi-coherent, and pullback of a finite locally free sheaf is finite locally free (Tensor product preserves quasi-coherence, Locally free sheaves of finite rank).

[F3]

For a finitely presented quasi-coherent sheaf $E$ and a quasi-coherent sheaf $G$ on $Y$ , the internal Hom is quasi-coherent and its affine description is the sheaf associated to the module of linear maps; these descriptions agree on overlaps (Internal Hom from a finitely presented sheaf is quasi-coherent). The higher-cohomology comparison below is constructed directly from the tensor maps attached to sections of $E$ .

Proof

technique · direct; reduce to the structure sheaf on a trivializing affine cover, then telescope over the finite algebraic local trivializations

1.1F2F3givenalgebra

Canonical map and the locally free case. For an open $U \subseteq Y$ and a section $e$ of $E ∣_{U}$ , the morphism $F ∣_{f^{- 1} U} \to (f^{*} E \otimes F) ∣_{f^{- 1} U}$ sending $s$ to $f^{*} e \otimes s$ induces a map on every higher direct image. This construction is additive in $e$ , linear over $O_{Y} (U)$ , and compatible with restriction. Tensoring and sheafifying therefore defines the canonical comparison $E \otimes R^{q} f_{*} F \to R^{q} f_{*} (f^{*} E \otimes F)$ . Suppose first that $E = O_{Y}^{d}$ is free. Then $f^{*} E = O_{X}^{d}$ , tensoring with it commutes with the direct image and with cohomology, and the map is the identity componentwise; so the comparison map is an isomorphism in this case.

2.1F2step 1.1algebra

Local trivialization and additivity. Since $E$ is finite locally free, $Y$ is covered by affine opens $U$ on which $E$ is free, and on each such $U$ the restriction of the comparison map is an isomorphism by step 1.1, using that the restrictions of $f$ , $E$ and $F$ compute the restricted higher direct images. Two maps of quasi-coherent sheaves that agree on an open cover agree, so the comparison map is an isomorphism for every finite locally free $E$ : the argument is local on $Y$ and the local triviality makes the free case available, so no further additivity over direct summands is needed.

3.1F1F2step 2.1algebra∎

The K-theory identity. Tensoring a coherent sheaf with a finite locally free sheaf is exact, so on $K_{0}$ the class $[E]$ acts by $[G] \mapsto [E \otimes G]$ , and $f^{*} [E]$ is a finite locally free class by [F2]. Summing the isomorphisms of step 2.1 with alternating signs gives $f_{!} (f^{*} [E] \cdot [F]) = \sum_{q} (- 1)^{q} [E \otimes R^{q} f_{*} F]$ in $K_{0} (Y)$ by [F1], and $\sum_{q} (- 1)^{q} [E \otimes R^{q} f_{*} F] = [E] \cdot \sum_{q} (- 1)^{q} [R^{q} f_{*} F] = [E] \cdot f_{!} [F]$ because the action of $[E]$ is additive. That is the projection formula in $K_{0} (Y)$ .

Projection formula for higher direct images and K-theory pushforward

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources