Flat base change for cohomology #

Let X be a scheme over a commutative ring R, let R → R' be a flat ring map, and let Y = X ×_R R' be the base change, given as a pullback square H over Spec R' ⟶ Spec R. This file proves base change in degree zero for quasicoherent sheaves on a quasi-compact quasi-separated scheme, and in degree one on a two-affine cover, and records the structure-sheaf consequences:

H⁰(Y, g^* M) ≅ R' ⊗_R H⁰(X, M) for flat R → R', when X is quasi-compact and quasi-separated, using the canonical pullback of global sections;
H¹(Y, g^* M) ≅ R' ⊗_R H¹(X, M) for a quasicoherent M and an arbitrary base-ring map, when X has a two-affine cover with affine intersection;
H⁰(Y, 𝒪_Y) ≅ R' ⊗_R H⁰(X, 𝒪_X) when X is quasi-compact and quasi-separated;
H¹(Y, 𝒪_Y) ≅ R' ⊗_R H¹(X, 𝒪_X) for flat R → R', when X is quasi-separated and covered by two affine opens.

Over a field k and a field extension K / k, these say that dim_k H⁰(X, 𝒪_X) and, for X quasi-separated and covered by two affine opens, dim_k H¹(X, 𝒪_X) do not change under base change to K; when moreover H¹(X, 𝒪_X) is finite-dimensional, so is H¹ after base change, and the genus does not change either. Smooth proper curves are covered by two affine opens, so this applies to them (TauCeti.AlgebraicGeometry.Curves.Genus.BaseChange).

The structure sheaf is written as the trivial line bundle (InvertibleSheaf.trivial X).obj, as for the genus Scheme.genus. The degree-one isomorphisms are stated as Nonempty linear equivalences: their construction depends on the chosen cover, and no naturality is claimed. The degree-zero coefficient equivalence is canonical and has the expected section equation on 1 ⊗ x. Coefficient comparisons use the canonical pullback (Scheme.Modules.pullback g).obj M.

On sections, flat base change is AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right: for a quasi-compact quasi-separated open W ⊆ X, Γ(Y, g⁻¹ W) = R' ⊗_R Γ(X, W), restated over the base ring as TauCeti.AlgebraicGeometry.isBaseChange_sectionsBaseChange. For quasicoherent coefficients it is AlgebraicGeometry.Scheme.Hom.isBaseChange_pullbackSections_of_isQuasiSeparated. For a cover X = U ∪ V whose members have vanishing first cohomology, the Mayer–Vietoris sequence presents H¹(X, 𝒪_X) as the cokernel of Γ(U) × Γ(V) → Γ(U ∩ V) (TauCeti.AlgebraicGeometry.ker_mayerVietorisBoundaryLinear), and tensor products preserve cokernels (IsBaseChange.of_right_exact). Affine opens have vanishing higher cohomology by Serre's vanishing theorem, before and after base change.

Main results #

TauCeti.AlgebraicGeometry.linearEquivTensorCohomologyZeroPullback: canonical flat base change for H⁰ of quasicoherent coefficients on a quasi-compact quasi-separated scheme.
TauCeti.AlgebraicGeometry.nonempty_linearEquiv_tensor_cohomology_zero_trivial, TauCeti.AlgebraicGeometry.nonempty_linearEquiv_tensor_cohomology_one_trivial: flat base change for H⁰ and H¹ of the structure sheaf.
TauCeti.AlgebraicGeometry.nonempty_linearEquiv_tensor_cohomology_one_pullback: base change for H¹ of a quasicoherent sheaf on a two-affine cover, along any base-ring map.
TauCeti.AlgebraicGeometry.subsingleton_cohomology_one_pullback: on such a cover, vanishing of H¹ of a quasicoherent sheaf persists after any base change.
TauCeti.AlgebraicGeometry.finrank_cohomology_zero_pullback_eq_of_isPullback, TauCeti.AlgebraicGeometry.finrank_cohomology_one_pullback_eq_of_isPullback and TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_one_pullback_of_isPullback: dimension and finiteness consequences over field extensions.
TauCeti.AlgebraicGeometry.finrank_cohomology_zero_trivial_eq_of_isPullback, TauCeti.AlgebraicGeometry.finrank_cohomology_one_trivial_eq_of_isPullback, TauCeti.AlgebraicGeometry.genus_eq_of_isPullback: invariance of h⁰(𝒪_X), h¹(𝒪_X) and of the genus under extension of the base field.

References #

R. Hartshorne, Algebraic Geometry, Proposition III.9.3 (flat base change).
The Stacks Project, Lemma 30.5.2, flat base change.