First cohomology as a quotient of sections on a two-open cover #
For a sheaf of modules M and a cover X = U ∪ V with vanishing H¹(U, M) and
H¹(V, M), this file identifies H¹(X, M) additively with
Γ(M, U ∩ V) / {s|_{U ∩ V} - t|_{U ∩ V} : s ∈ Γ(M, U), t ∈ Γ(M, V)}.
The equivalence is induced by the Mayer–Vietoris connecting map. The coefficients need not be quasi-coherent, and no vanishing on the overlap is required. Thus the same computation applies whenever acyclicity of the two members is available. This computes first cohomology directly from the Mayer–Vietoris sequence; it does not identify a Čech complex with derived cohomology.
Use TauCeti.AlgebraicGeometry.sectionsQuotientEquivCohomologyOne M U V hUV hU hV
for the comparison, where hUV states that the opens cover and hU, hV state
vanishing of first cohomology on the two opens.
References #
- R. Hartshorne, Algebraic Geometry, Chapter III, §4.
The difference of restrictions to the overlap of two open subsets, with sign s| - t|
matching the Mayer–Vietoris sequence. This is the negative of the standard Čech differential.
Instances For
The two-open quotient description in degree one: if the members cover the scheme and have vanishing first cohomology, the quotient of overlap sections by coboundaries is first cohomology of the scheme.
Instances For
The quotient comparison sends an overlap section to its Mayer–Vietoris connecting class, transported from the union to the whole scheme.