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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 $K_{0} (Y)$ .

Depends on

Used by

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58 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Cohomology of Schemes, Section 30.6 (projection formula) (standard reference, not scraped)
Borel and Serre, Le theoreme de Riemann-Roch (1958), §5 (c) (standard reference, not scraped)