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Cohomology and base-change map

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $f : X \to S$ be a proper morphism of schemes (Proper morphisms), let $F$ be a coherent $O_{X}$ -module (Coherent module sheaves), and let $R^{q} f_{*} F$ be its $q$ -th higher direct image sheaf (Higher direct image of a sheaf), an $O_{S}$ -module.

Fibre map. Let $s \in S$ be a point with residue field $κ (s)$ (The residue field at a point of an affine scheme), let $X_{s} : = X \times_{S} Spec κ (s)$ be the fibre of $f$ over $s$ (Fibre product of schemes) with structure morphism $f_{s} : X_{s} \to Spec κ (s)$ and projection $g_{s}^{'} : X_{s} \to X$ , and let $F_{s} : = g_{s}^{' *} F$ (Pullback of a module along a morphism of ringed spaces). Write $(R^{q} f_{*} F) (s) = (R^{q} f_{*} F)_{s} \otimes_{O_{S, s}} κ (s)$ for the fibre of the $O_{S}$ -module $R^{q} f_{*} F$ at $s$ (Fibre of a module sheaf at a point). The cohomology and base-change map at $s$ is the $κ (s)$ -linear map $φ_{s}^{q} : (R^{q} f_{*} F) (s) ⟶ H^{q} (X_{s}, F_{s})$ induced by pullback of cohomology classes: for every affine open $V = Spec A \subseteq S$ containing $s$ the pullback along $X_{s} \to f^{- 1} V$ and the canonical map $g_{s}^{' - 1} F \to F_{s}$ produce $H^{q} (f^{- 1} V, F) \to H^{q} (X_{s}, F_{s})$ (Variance of sheaf cohomology), these maps are compatible under passage to smaller affine neighbourhoods of $s$ , and they induce the displayed map out of the colimit $(R^{q} f_{*} F)_{s} = {colim}_{V} H^{q} (f^{- 1} V, F)$ (Higher direct images localize over an affine base).

Base-change map. For a morphism $g : S^{'} \to S$ form the Cartesian square with $X^{'} = X \times_{S} S^{'}$ , projections $g^{'} : X^{'} \to X$ and $f^{'} : X^{'} \to S^{'}$ over $S^{'}$ ; as a base change of the proper morphism $f$ , the morphism $f^{'}$ is proper (Properness survives arbitrary base change) and $g^{' *} F$ is quasi-coherent on $X^{'}$ . The base-change map for this square is the $O_{S^{'}}$ -linear morphism $g^{*} R^{q} f_{*} F ⟶ R^{q} f_{*}^{'} g^{' *} F$ obtained as follows: on an affine open $V^{'} = Spec B \subseteq S^{'}$ whose image lies in an affine open $V = Spec A \subseteq S$ one has $g^{*} R^{q} f_{*} F (V^{'}) = B \otimes_{A} H^{q} (f^{- 1} V, F)$ (Scheme pullback preserves quasi-coherence, Module sheaf on an affine scheme), while $R^{q} f_{*}^{'} g^{' *} F (V^{'}) = H^{q} (f^{' - 1} V^{'}, g^{' *} F)$ (Higher direct images localize over an affine base); pullback of cohomology classes along $X_{V^{'}}^{'} : = f^{' - 1} V^{'} \to f^{- 1} V$ composed with the canonical map $g^{' - 1} F \to g^{' *} F$ gives an $A$ -linear map $H^{q} (f^{- 1} V, F) \to H^{q} (f^{' - 1} V^{'}, g^{' *} F)$ , hence by extension of scalars the required $B$ -linear map on sections. These maps are compatible with restriction to smaller affine open pairs, so by Compatible local sheaves glue uniquely up to unique isomorphism they define a unique morphism of sheaves on $S^{'}$ ; it is functorial in the Cartesian square and compatible with composition of base changes.

Not asserted to be isomorphisms. The maps $φ_{s}^{q}$ and $g^{*} R^{q} f_{*} F \to R^{q} f_{*}^{'} g^{' *} F$ are not isomorphisms without hypotheses: injectivity may fail when $F$ is not flat over $S$ , and surjectivity may fail when the fibre dimensions jump. Criteria under which they are isomorphisms are the content of the cohomology-and-base-change theorems, and the failure of automatic base change is recorded separately. In degree $q = 0$ the fibre map is the evaluation of local sections, $(f_{*} F)_{s} \otimes_{O_{S, s}} κ (s) \to H^{0} (X_{s}, F_{s})$ ; the global restriction $H^{0} (X, F) \to H^{0} (X_{s}, F_{s})$ factors through it. The base-change map is the canonical map $g^{*} f_{*} F \to f_{*}^{'} g^{' *} F$ .

Facts & Assumptions

Given: The Axiom of Choice, a proper morphism $f : X \to S$ , a coherent $O_{X}$ -module $F$ , a point $s \in S$ , and a morphism $g : S^{'} \to S$ with Cartesian square $X^{'} = X \times_{S} S^{'}$ .

[F1]

Higher direct images: $R^{q} f_{*} F$ is a specific $O_{S}$ -module depending on the fixed functorial injective resolution datum, functorially in $F$ , with $R^{q} f_{*} F = 0$ for $q < 0$ and $R^{0} f_{*} F ≅ f_{*} F$ . For $f$ quasi-compact and separated and $F$ quasi-coherent, each $R^{q} f_{*} F$ is quasi-coherent, and on an affine open $V = Spec A \subseteq S$ it is the associated sheaf of the $A$ -module $H^{q} (f^{- 1} V, F)$ , so in particular $(R^{q} f_{*} F) (V) = H^{q} (f^{- 1} V, F)$ (Higher direct image of a sheaf, Higher direct images localize over an affine base). A proper morphism is separated, of finite type and quasi-compact (Proper morphisms), and a coherent module is quasi-coherent (Quasi-coherent module on a scheme, Coherent module sheaves), so these descriptions apply to $f$ and $F$ ; the base-changed morphism $f^{'}$ is again proper (Properness survives arbitrary base change) and $g^{' *} F$ is quasi-coherent (Scheme pullback preserves quasi-coherence), so the descriptions apply to $f^{'}$ and $g^{' *} F$ as well.

[F2]

Contravariance of sheaf cohomology in the space: a continuous map $u : Y \to Z$ and a morphism of sheaves $ψ : u^{- 1} G \to H$ on $Y$ induce maps $H^{q} (Z, G) \to H^{q} (Y, H)$ , natural in the data and compatible with composition; in degree $0$ they are the section pullback with $ψ$ . In particular pullback along a morphism of schemes $X_{V^{'}}^{'} \to f^{- 1} V$ combined with the canonical map $g^{' - 1} F \to g^{' *} F$ gives $H^{q} (f^{- 1} V, F) \to H^{q} (f^{' - 1} V^{'}, g^{' *} F)$ . (Variance of sheaf cohomology, Sheaf cohomology as right derived global sections, A sheaf on a topological space).

[F3]

Pullback of quasi-coherent modules: $g^{' *} F$ is quasi-coherent. For affine opens $V^{'} = Spec B \subseteq S^{'}$ and $V = Spec A \subseteq S$ with $g (V^{'}) \subseteq V$ , if the quasi-coherent module $R^{q} f_{*} F$ restricts to $\tilde{M}$ on $V$ , then $g^{*} R^{q} f_{*} F$ restricts to the associated sheaf of $B \otimes_{A} M$ on $V^{'}$ . There is a canonical morphism $g^{' - 1} F \to g^{' *} F$ (Scheme pullback preserves quasi-coherence, Pullback of a module along a morphism of ringed spaces, Module sheaf on an affine scheme, Modules on a ringed space).

[F4]

Morphisms of sheaves glue: a collection of morphisms of abelian groups on the members of a basis of a topological space, compatible under restriction to smaller basis members, defines a unique morphism of the associated sheaves (Compatible local sheaves glue uniquely up to unique isomorphism).

[F5]

Fibre of a module at a point: for an $O_{S}$ -module $G$ and $s \in S$ , the fibre $G (s) = G_{s} \otimes_{O_{S, s}} κ (s)$ is a $κ (s)$ -vector space, and for $G$ quasi-coherent it is the pullback of $G$ to $Spec κ (s)$ evaluated there (Fibre of a module sheaf at a point, Scheme pullback preserves quasi-coherence).

Proof

technique · direct: build the fibre map as the colimit of restrictions of cohomology classes to the fibre over affine neighbourhoods, and build the base-change map affine-locally by tensoring the pullback map on cohomology over the base ring, then glue over a basis

1.1F1F2

The fibre map of the definition is well defined: for affine open neighbourhoods $V \subseteq V_{0}$ of $s$ the pullback maps $H^{q} (f^{- 1} V_{0}, F) \to H^{q} (X_{s}, F_{s})$ and $H^{q} (f^{- 1} V, F) \to H^{q} (X_{s}, F_{s})$ agree, because the pullback squares $X_{s} \to f^{- 1} V \to f^{- 1} V_{0}$ compose and the maps of [F2] are compatible with composition; hence the colimit description $(R^{q} f_{*} F)_{s} = {colim}_{V} H^{q} (f^{- 1} V, F)$ of [F1] yields a well-defined $O_{S, s}$ -linear map $(R^{q} f_{*} F)_{s} \to H^{q} (X_{s}, F_{s})$ .

1.2F1F2F5

The map of 1.1 is $O_{S, s}$ -linear and its target carries the $O_{S, s}$ -action through $κ (s)$ , so it factors uniquely through $(R^{q} f_{*} F) (s) = (R^{q} f_{*} F)_{s} \otimes_{O_{S, s}} κ (s)$ : this is the $κ (s)$ -linear fibre map $φ_{s}^{q}$ of the definition.

1.3F2F3

The affine-local recipe for the base-change map is well defined: for affine $V^{'} = Spec B$ over $V = Spec A$ the extension-of-scalars map $B \otimes_{A} H^{q} (f^{- 1} V, F) \to H^{q} (f^{' - 1} V^{'}, g^{' *} F)$ is defined by the $A$ -linear pullback map of [F2] and the identifications of [F3], and it is $B$ -linear.

1.4F2F3F4

The recipes of 1.3 are compatible with restriction: replacing $(V, V^{'})$ by a smaller pair $(V_{0}, V_{0}^{'})$ restricts both sides and the pullback maps compose, by the compatibility clause of [F2]; the affine pairs cover $S^{'}$ , and every intersection of two such pairs is covered by smaller affine pairs over affine opens in $S$ ; compatibility on these common refinements lets [F4] glue the maps to a unique $O_{S^{'}}$ -linear morphism $g^{*} R^{q} f_{*} F \to R^{q} f_{*}^{'} g^{' *} F$ .

2.1F1F2F5∎

Boundaries and conventions. For $q < 0$ both sides vanish by [F1] and the map is the zero map; for $q = 0$ the fibre map is evaluation of germs of local sections on the fibre (and global restriction factors through it), while the base-change map reduces to the canonical $g^{*} f_{*} F \to f_{*}^{'} g^{' *} F$ given by adjunction; if $S^{'} = \emptyset$ or $F = 0$ both sides are zero; if $S$ (and hence $X$ ) is empty there are no points $s$ and no fibre maps, and the base-change map is the zero map between zero sheaves on $S^{'}$ . The Axiom of Choice is a hypothesis, consumed through the injective resolution datum defining the higher direct images [F1] and through the cohomology functoriality of [F2]. No choice is made in the constructions of 1.1-1.4.

Depends on

Used by

Dependency tree · two levels

99 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, Chapter 30, Sections 30.2-30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2 (standard reference, not scraped)