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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, 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 i ...

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 mak ...

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 ser ...

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 points ...

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