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TauCeti.CategoryTheory.Sites.SheafCohomology.Cech.Flat

Čech complexes of presheaves with flat values #

Let α be a meet-semilattice with a top element, for instance the opens of a topological space, let U : ι → α be a finite family and let P be a presheaf of R-modules on α. The degree n term of the Čech complex Č(U, P) is the product, over a : Fin (n + 1) → ι, of the values of P on the intersections U (a 0) ⊓ ⋯ ⊓ U (a n). As ι is finite this is a finite product, so when P takes flat values on the nonempty finite intersections of U, every term of Č(U, P) is a flat R-module (TauCeti.CategoryTheory.flat_cechComplexFunctor_obj_X).

For a quasi-coherent sheaf M on a scheme over R and a cover U with affine finite intersections, take P the presheaf of R-modules of sections of M. When the sections of M over the nonempty finite intersections of U are flat R-modules, for instance when its sections over affine opens are, this gives the complex of flat modules computing the cohomology of every base change of M (TauCeti.AlgebraicGeometry.Cohomology.Cech.BaseChange), as in the proof of Hartshorne's Proposition III.12.2.

Main results #

References #

source
theorem TauCeti.CategoryTheory.flat_cechComplexFunctor_obj_X {R : Type u} [CommRing R] {α : Type w} [SemilatticeInf α] [OrderTop α] {ι : Type v} [Finite ι] (U : ι → α) (P : CategoryTheory.Functor αᵒᵖ (ModuleCat R)) (hP : ∀ (s : Finset ι), s.Nonempty → Module.Flat R ↑(P.obj (Opposite.op (s.inf U)))) (n : ℕ) :
Module.Flat R ↑(((CategoryTheory.cechComplexFunctor U).obj P).X n)

The Čech complex of a presheaf with flat values consists of flat modules. Let P be a presheaf of R-modules on a meet-semilattice with a top element and U a finite family such that the value of P on every nonempty finite intersection of U is a flat R-module. Then every term of the Čech complex of P for U is a flat R-module.