Pullback on the cohomology of sheaves of modules #
Let g : Y ⟶ X be a morphism of schemes and M a sheaf of modules on X. This file constructs
the pullback map
g.cohomologyPullback M n : Hⁿ(X, M) →+ Hⁿ(Y, g^* M).
It is the inverse image map on the cohomology of the underlying abelian sheaves
(TopCat.Hom.cohomologyPullback), followed by the map induced on cohomology by the
comparison g⁻¹ M ⟶ g^* M between the inverse image of the underlying abelian sheaf of M and
the underlying abelian sheaf of the module pullback
(AlgebraicGeometry.Scheme.Hom.toSheafPullbackComparison).
The pullback map is natural in M, commutes with the connecting maps of a short exact sequence
whose pullback is still short exact (for instance for flat g and quasi-coherent terms), is
semilinear for the action of global functions along g^♯ : Γ(X, 𝒪_X) ⟶ Γ(Y, 𝒪_Y), and is the
pullback of global sections in degree zero. For the base change Y = X ×_R R' along a ring map
R → R', its R'-linear extension is the flat base change map
R' ⊗_R Hⁿ(X, M) ⟶ Hⁿ(X_{R'}, M_{R'}) (TauCeti.AlgebraicGeometry.cohomologyBaseChange).
Main declarations #
AlgebraicGeometry.Scheme.Hom.cohomologyPullback: the mapHⁿ(X, M) →+ Hⁿ(Y, g^* M);AlgebraicGeometry.Scheme.Hom.cohomologyPullback_cohomologyMap: naturality inM;AlgebraicGeometry.Scheme.Hom.cohomologyPullback_cohomologyδ: compatibility with the connecting maps of long exact cohomology sequences;AlgebraicGeometry.Scheme.Hom.cohomologyZeroEquiv_cohomologyPullback: the degree-zero case;AlgebraicGeometry.Scheme.Hom.cohomologyPullback_smul: semilinearity over global functions.
For an open subset U the analogous map is Scheme.Hom.cohomologyOnPullback, from
Hⁿ(U, M) to Hⁿ(g⁻¹ U, g^* M). It is semilinear over global functions, commutes with
restriction and with the connecting maps for the covers U ∪ V and g⁻¹ U ∪ g⁻¹ V, and
in degree zero is pullback of sections. These comparisons provide the commutative
Mayer–Vietoris ladders used to transport cohomological statements across a morphism of
schemes, without any affine-diagonal hypothesis.
References #
- R. Hartshorne, Algebraic Geometry, Exercise III.8.1 and Proposition III.9.3.
The pullback map Hⁿ(X, M) →+ Hⁿ(Y, g^* M) on the cohomology of a sheaf of modules along a
morphism of schemes g : Y ⟶ X.
Instances For
The pullback map on cohomology is natural in the coefficient sheaf.
The pullback map on cohomology commutes with the connecting maps of a short exact sequence
S of sheaves of modules on X whose pullback to Y is short exact, for instance when g is flat
and the first two terms of S are quasi-coherent
(AlgebraicGeometry.Scheme.Hom.shortExact_map_pullback_of_flat).
In degree zero, under the identification of H⁰ with global sections, the pullback map on
cohomology is the pullback of global sections.
The pullback map on cohomology is semilinear for the action of global functions along
g^♯ : Γ(X, 𝒪_X) ⟶ Γ(Y, 𝒪_Y).
Pullback Hⁿ(U, M) ⟶ Hⁿ(g⁻¹ U, g^* M) on the cohomology of an open subset. It is the
inverse image of extensions from the free abelian sheaf on U, followed by the comparison
from abelian inverse image to module pullback.
Instances For
Pullback over an open is abelian inverse image followed by the map from inverse image to the underlying sheaf of the module pullback.
Pullback on the cohomology of opens is natural in the coefficient module.
Pullback on the cohomology of opens is natural in the coefficient module.
Pullback on the cohomology of opens commutes with restriction.
Pullback on the cohomology of an open is semilinear over global functions.
In degree zero pullback on the cohomology of an open is pullback of sections.
Pullback commutes with the connecting maps of the Mayer–Vietoris sequences for two opens and their inverse images. No flatness or quasi-coherence hypothesis is required.