The linear two-open computation of first cohomology #
For a cover X = U ∪ V, the Mayer–Vietoris boundary from sections on U ∩ V to
H¹(X, M) is linear over global functions. If first cohomology vanishes on both
members, it identifies H¹(X, M) with the module cokernel of the difference of
restrictions from the two members, whose range is the sum of the ranges of the two
restrictions. No acyclicity on the overlap is needed.
This upgrades the additive computation in SectionsQuotient.Basic to a linear one,
so it can be used for dimensions and extension of scalars. For quasi-coherent coefficients,
affine acyclicity supplies the hypotheses on any two affine opens of any scheme. The
quotient uses the sign s| - t| of the Mayer–Vietoris sequence.
The construction reuses mayerVietorisSectionClass, its kernel calculation and
surjectivity, and Mathlib's linear first isomorphism theorem. It does not assert a comparison of
Čech complexes or flat base change.
If H¹(X, M) itself vanishes, the difference of restrictions is surjective, so the global
sections H⁰(X, M) are the kernel of a surjection Γ(M, U) × Γ(M, V) → Γ(M, U ∩ V). For a scheme
over a commutative ring R whose sections over U, V and U ∩ V are flat R-modules, this
makes H⁰(X, M) flat over R (flat_cohomology_zero_of_subsingleton_cohomology_one).
References #
- R. Hartshorne, Algebraic Geometry, Chapter III, Section 4.
The difference of restrictions from two opens, linear over global functions.
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The differences of restrictions from two opens are the sums of restrictions from the two opens: the range of the difference map is the sum of the ranges of the two restrictions.
Restricting a global section to the two members of an open cover, linear over global functions.
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The two components are the usual restrictions to the cover members.
A section on a two-open cover is determined by its two restrictions.
The sections on a two-open cover that agree on the overlap are precisely restrictions of global sections.
On a two-open cover X = U ∪ V, global sections, through H⁰(X, M) ≅ Γ(M, ⊤), are the pairs
of sections over U and V that agree on U ∩ V.
On a two-open cover, a degree-zero class is determined by its restrictions to the members.
The Mayer–Vietoris connecting class of a section on the overlap, linear over global functions. Its sign is fixed by the existing Mayer–Vietoris boundary.
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The linear boundary is the site-level connecting class followed by the terminal-object comparison with cohomology of the whole scheme.
The boundary is natural in the coefficient module.
The kernel of the boundary consists exactly of differences of restricted sections.
If H¹(X, M) vanishes, every section over the overlap of a two-open cover is a difference of
restrictions of sections over the two members.
When first cohomology vanishes on both covering members, the boundary is surjective.
First cohomology of a two-open cover with acyclic members is the module cokernel of restriction differences. Only vanishing in degree one on each member is required.
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The quotient comparison sends a represented section to its connecting class.
The inverse comparison returns the quotient class of an overlap section from its boundary.
Flatness of H⁰ when H¹ vanishes. Let X be a scheme over R covered by two opens U
and V such that the sections of M over U, V and U ∩ V are flat R-modules. If
H¹(X, M) = 0, then H⁰(X, M) is a flat R-module: it is the kernel of the surjective
difference of restrictions Γ(M, U) × Γ(M, V) → Γ(M, U ∩ V).