De Rham-Weil theorem
Encyclopedia
In algebraic topology
, the De Rham-Weil theorem allows computation of sheaf cohomology
using an acyclic resolution of the sheaf in question.
Let
be a sheaf
on a topological space
and 
a resolution of 
by acyclic sheaves. Then
where
denotes the 
-th sheaf cohomology
group
of
with coefficients in 
----
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
, the De Rham-Weil theorem allows computation of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
using an acyclic resolution of the sheaf in question.
Let
be a sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
on a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and a resolution of by acyclic sheaves. Thenwhere
denotes the -th sheaf cohomologySheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
group
Inverse
Inverse may refer to:* Inverse , a type of immediate inference from a conditional sentence* Inverse , a program for solving inverse and optimization problems...
of
with coefficients in ----