Base change for the Čech cohomology of quasi-coherent sheaves #
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.
Let M be a quasi-coherent sheaf on X and U a finite family of opens covering X whose
nonempty finite intersections are affine; for instance an affine open cover of a quasi-compact
scheme with affine diagonal. This file shows that the Čech complex Č(U, M) of R-modules of
sections computes the cohomology of every base change: Hⁿ(Y, g^* M) is the n-th cohomology of
R' ⊗_R Č(U, M), with no flatness of R → R'. When the sections of M over the finite
intersections of U are flat R-modules, for instance when its sections over affine opens are,
every term of Č(U, M) is flat (TauCeti.CategoryTheory.flat_cechComplexFunctor_obj_X).
Together, for such M, these are the first step of the theory of cohomology and base change: the
cohomology of all the base changes of M, in particular of all its fibres X ×_R κ(p), is
computed by a single complex of flat R-modules.
When R → R' is flat, extension of scalars commutes with taking cohomology, which gives flat base
change Hⁿ(Y, g^* M) ≅ R' ⊗_R Hⁿ(X, M) in every degree n.
The proof computes both sides by Čech complexes. The inverse images g⁻¹ U form a cover of Y of
the same kind, because g is affine. By Leray's theorem, Hⁿ(X, M) is the cohomology of
Č(U, M), and likewise on Y (AlgebraicGeometry.Scheme.Modules.homologyCechComplexLinearEquiv).
On an affine open W of X, the sections of g^* M over g⁻¹ W are the base change of the
sections of M over W (AlgebraicGeometry.Scheme.Hom.isBaseChange_pullbackSections), with no
flatness hypothesis. As the finite intersections of U are affine, R' ⊗_R Č(U, M) is therefore
the Čech complex of g^* M for g⁻¹ U.
The base change of the Čech complex is an explicit isomorphism of complexes
(TauCeti.AlgebraicGeometry.cechComplexBaseChangeIso), while the isomorphisms on cohomology are
stated as Nonempty linear equivalences: their construction depends on the chosen cover, and no
naturality is claimed. For a quasi-compact scheme with affine diagonal, the
canonical map R' ⊗_R Hⁿ(X, M) ⟶ Hⁿ(Y, g^* M) induced by pullback is an isomorphism when R → R'
is flat (TauCeti.AlgebraicGeometry.bijective_cohomologyBaseChange); its proof uses flat base
change here to see that g^* preserves the vanishing of higher cohomology. Flat base change in
degrees zero and one on a two-affine cover, along any base-ring map in degree one, is in
TauCeti.AlgebraicGeometry.Cohomology.BaseChange.
Main definitions #
TauCeti.AlgebraicGeometry.sectionsPullbackExtend: the map of presheaves ofR'-modulesR' ⊗_R Γ(M, -) ⟶ Γ(g^* M, g⁻¹ -),s ⊗ m ↦ s • g^* m, adjoint to pulling back sections (extendRestrictScalarsAdj_homEquiv_sectionsPullbackExtend_app_hom_apply).TauCeti.AlgebraicGeometry.cechComplexBaseChangeIso: the isomorphism of complexesR' ⊗_R Č(U, M) ≅ Č(g⁻¹ U, g^* M), along any ring mapR → R'. Its factors are the components ofsectionsPullbackExtend(cechComplexBaseChangeIso_hom_f_π).
Main results #
All in the namespace TauCeti.AlgebraicGeometry:
nonempty_linearEquiv_homology_extendScalars_cechComplex_cohomology_pullback_of_iSup_eq_top:Hⁿ(Y, g^* M)is the cohomology ofR' ⊗_R Č(U, M), along any ring mapR → R'.nonempty_linearEquiv_tensor_cohomology_pullback_of_iSup_eq_top: flat base change for a finite cover with affine intersections.
References #
- R. Hartshorne, Algebraic Geometry, Proposition III.9.3 (flat base change), and the Čech complex in the proof of Proposition III.12.2 (cohomology of all base changes).
- D. Mumford, Abelian Varieties, Section 5 (the Čech complex in the proof of the main lemma).
- The Stacks Project, Lemma 30.5.2, flat base change.
The morphism of presheaves of R'-modules R' ⊗[R] Γ(M, -) ⟶ Γ(g^* M, g⁻¹ -) on X,
s ⊗ m ↦ s • g^* m: the adjoint, under extension and restriction of scalars along R → R', of
pulling back sections along g (AlgebraicGeometry.Scheme.Hom.pullbackSectionsBaseLinear).
Instances For
sectionsPullbackExtend is adjoint to pulling back sections along g: its adjoint under
extension and restriction of scalars along R → R' sends a section m of M to g^* m.
On an affine open, sections of g^* M are the base change of sections of M, so the
adjoint of the pullback of sections is an isomorphism.
The Čech complex commutes with base change. Extending scalars from R to R' in the
Čech complex of the sections of M for U gives the Čech complex of the sections of g^* M for
g⁻¹ U. No flatness of R → R' is needed, as U has affine finite intersections.
Instances For
The factor indexed by a of the degree n component of cechComplexBaseChangeIso is the
component R' ⊗_R Γ(M, W) ⟶ Γ(g^* M, g⁻¹ W) of sectionsPullbackExtend at the intersection
W = U (a 0) ∩ ⋯ ∩ U (a n), followed by the identification of g⁻¹ W with the intersection of
the g⁻¹ U (a j).
The factor indexed by a of the degree n component of cechComplexBaseChangeIso is the
component R' ⊗_R Γ(M, W) ⟶ Γ(g^* M, g⁻¹ W) of sectionsPullbackExtend at the intersection
W = U (a 0) ∩ ⋯ ∩ U (a n), followed by the identification of g⁻¹ W with the intersection of
the g⁻¹ U (a j).
The Čech complex computes the cohomology of every base change. Let X be a scheme over
R, let R → R' be any ring map and let Y = X ×_R R', given as a pullback square. Let M be a
quasi-coherent sheaf on X and U a finite family of opens covering X whose nonempty finite
intersections are affine. Then Hⁿ(Y, g^* M) is the n-th cohomology of the complex
R' ⊗_R Č(U, M) obtained by extending scalars in the Čech complex of the sections of M, through
the isomorphism of complexes cechComplexBaseChangeIso.
Flat base change for the cohomology of quasi-coherent sheaves, for a finite cover with
affine intersections. Let X be a scheme over R, let R → R' be a flat ring map and let
Y = X ×_R R', given as a pullback square. Let M be a
quasi-coherent sheaf on X and suppose that X is covered by finitely many opens whose nonempty
finite intersections are affine. Then Hⁿ(Y, g^* M) ≅ R' ⊗_R Hⁿ(X, M) for every n.