algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...

, the De Rham–Weil theorem allows computation of

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when ...

using an acyclic

resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual m ...

of the sheaf in question. Let

\mathcal F

be a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper s ...

on a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

X

and

\mathcal F^\bullet

a resolution of

\mathcal F

by acyclic sheaves. Then :

H^q(X,\mathcal F) \cong H^q(\mathcal F^\bullet(X)),

where

H^q(X,\mathcal F)

denotes the

q

-th

group of

X

with coefficients in

\mathcal F.

The De Rham–Weil theorem follows from the more general fact that derived functors may be computed using acyclic resolutions instead of simply injective resolutions.

References

* * * * {{DEFAULTSORT:De Rham-Weil theorem Homological algebra Sheaf theory