The flat base change map on cohomology is an isomorphism #

Let X be a scheme over a commutative ring R, let R → R' be a ring map and let Y = X ×_R R' be the base change, given as a pullback square H with projection g : Y ⟶ X. For a sheaf of modules M on X, the pullback map on cohomology Scheme.Hom.cohomologyPullback g M n : Hⁿ(X, M) →+ Hⁿ(Y, g^* M) is R-linear, so it extends to an R'-linear map

TauCeti.AlgebraicGeometry.cohomologyBaseChange H M n : R' ⊗_R Hⁿ(X, M) →ₗ[R'] Hⁿ(Y, g^* M).

This file proves that this canonical map is an isomorphism in every degree when R → R' is flat, M is quasi-coherent and X is quasi-compact with affine diagonal (for instance quasi-compact and separated): this is flat base change over an affine base (Stacks 02KH, Hartshorne III.9.3). Unlike TauCeti.AlgebraicGeometry.nonempty_linearEquiv_tensor_cohomology_pullback_of_iSup_eq_top, which computes both sides by Čech complexes of a chosen cover, the isomorphism here does not depend on any choice, and it is compatible with the maps induced by morphisms of sheaves and with the connecting maps of short exact sequences. Over a field k and a field extension K / k it shows that the dimension of Hⁿ(X, M) does not change when k is extended to K.

The proof is by dimension shifting, simultaneously for all quasi-coherent sheaves. In degree zero the map is the flat base change of global sections (TauCeti.AlgebraicGeometry.linearEquivTensorCohomologyZeroPullback). Choose a finite affine open cover U of X. Every quasi-coherent M embeds into the quasi-coherent sheaf I = ⨁ᵢ (U i).ι_* (M|_{U i}), which has no higher cohomology on X because each open immersion (U i).ι is affine. By the Čech computation, g^* I has no higher cohomology on Y either. The pullback of 0 ⟶ M ⟶ I ⟶ Q ⟶ 0 is exact because g is flat, and the five lemma, applied to the two long exact sequences, shows that the map in degree n + 1 for M is bijective as soon as those in degree n for I and Q are.

Main declarations #

TauCeti.AlgebraicGeometry.cohomologyBaseChange: the canonical map R' ⊗_R Hⁿ(X, M) →ₗ[R'] Hⁿ(Y, g^* M), characterized by cohomologyBaseChange_tmul;
TauCeti.AlgebraicGeometry.bijective_cohomologyBaseChange: it is bijective for flat R → R', quasi-coherent M and X quasi-compact with affine diagonal;
TauCeti.AlgebraicGeometry.bijective_cohomologyBaseChange_zero: in degree zero it is bijective for every quasi-compact quasi-separated X, being the flat base change of global sections (TauCeti.AlgebraicGeometry.linearEquivTensorCohomologyZeroPullback_toLinearMap);
TauCeti.AlgebraicGeometry.cohomologyMapBaseLinear_comp_cohomologyBaseChange and TauCeti.AlgebraicGeometry.cohomologyδBaseLinear_comp_cohomologyBaseChange: its naturality and its compatibility with connecting maps;
TauCeti.AlgebraicGeometry.cohomologyBaseChangeLinearEquiv: the resulting flat base change isomorphism, whose underlying linear map is cohomologyBaseChange (TauCeti.AlgebraicGeometry.cohomologyBaseChangeLinearEquiv_toLinearMap);
TauCeti.AlgebraicGeometry.finrank_cohomology_pullback_eq_of_isPullback and TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_pullback_of_isPullback: invariance of the dimension and finite-dimensionality of Hⁿ(X, M) under extension of the base field, TauCeti.AlgebraicGeometry.finrank_cohomology_trivial_eq_of_isPullback and TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_trivial_of_isPullback: their specialisation to the structure sheaf, whose pullback is identified with 𝒪_Y, with TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_trivial_of_isPullback_of_isProper the finite-dimensionality for a proper scheme, and TauCeti.AlgebraicGeometry.eulerCharBelow_pullback_eq_of_isPullback: the same invariance of the truncated Euler characteristic Scheme.Modules.eulerCharBelow.

References #

R. Hartshorne, Algebraic Geometry, Proposition III.9.3.
The Stacks Project, Lemma 30.5.2.