Global functions acting on the cohomology of open subsets #
For a sheaf of modules M on a scheme X and an open subset U, this file equips the cohomology
Hⁿ(U, M) (Scheme.Modules.cohomologyOn M n U) with its module structure over the ring
Γ(X, ⊤) of global functions: a global function r acts through the map on cohomology induced by
multiplication by r on M. This is the structure over which the cohomology of a quasi-coherent
sheaf on an open subset is a localization of the cohomology on a larger one.
The maps induced by morphisms of coefficient sheaves (Scheme.Modules.cohomologyOnMapLinear), the
restriction maps (Scheme.Modules.cohomologyOnResLinear) and the connecting maps of the
Mayer-Vietoris sequence (Scheme.Modules.mayerVietorisδLinear) are linear. In degree zero the
action is the action of the restriction of r to U on the sections of M over U
(Scheme.Modules.cohomologyOnZeroEquiv_smul, bundled as
Scheme.Modules.cohomologyOnZeroSemilinearMap), under the identification
Scheme.Modules.cohomologyOnZeroEquiv of H⁰(U, M) with the sections of M over U from
TauCeti.AlgebraicGeometry.Cohomology.Basic. At U = ⊤ the module Hⁿ(⊤, M) is the module
Hⁿ(X, M) of TauCeti.AlgebraicGeometry.Cohomology.Module.Basic
(Scheme.Modules.cohomologyOnTopLinearEquiv).
The construction mirrors the action of TauCeti.AlgebraicGeometry.Cohomology.Module.Basic on the
cohomology of the whole scheme, with the cohomology presheaf U ↦ Hⁿ(U, M) in place of the
cohomology functor.
The action of global functions on the cohomology Hⁿ(U, M) of an open subset, bundled as a
ring homomorphism into additive endomorphisms: a global function r acts through the map induced
by multiplication by r on M.
Instances For
A global function r acts on Hⁿ(U, M) through the map induced by multiplication by r on
M.
The cohomology Hⁿ(U, M) of an open subset is a module over the ring of global functions.
A global function r acts on Hⁿ(U, M) through the map induced by multiplication by r on
M.
On Hⁿ(U, M) viewed as an Ext group, a global function r acts by postcomposition with
multiplication by r.
The connecting map Hⁿ⁰(U ⊓ V, M) ⟶ Hⁿ¹(U ⊔ V, M) of the Mayer-Vietoris sequence, as a linear
map over global functions.
Instances For
In degree zero, a global function r acts on H⁰(U, M), identified with the sections of M
over U, through its restriction to U.
In degree zero, the identification cohomologyOnZeroEquiv of H⁰(U, M) with the sections of
M over U, as a map that is semilinear along the restriction Γ(X, ⊤) ⟶ Γ(X, U).
Instances For
The identification cohomologyOnTopIso of the cohomology Hⁿ(⊤, M) of the open subset ⊤
with the cohomology Hⁿ(X, M) of the scheme is linear over global functions.