The Godement resolution of 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 ...
is a construction in
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing
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 ...
. It was discovered by
Roger Godement
Roger Godement (; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books.
Biography
Godement started as a student at the École normale supérieure in 1940, where he ...
.
Godement construction
Given a topological space ''X'' (more generally, a
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
X with enough points), and a sheaf ''F'' on X, the Godement construction for ''F'' gives a sheaf
constructed as follows. For each point
, let
denote the stalk of ''F'' at ''x''. Given an open set
, define
:
An open subset
clearly induces a restriction map
, so
is a
presheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
. One checks the
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 ...
axiom easily. One also proves easily that
is
flabby, meaning each restriction map is surjective. The map
can be turned into a functor because a map between two sheaves induces maps between their stalks. Finally, there is a canonical map of sheaves
that sends each section to the 'product' of its
germs. This canonical map is a natural transformation between the identity functor and
.
Another way to view
is as follows. Let
be the set ''X'' with the discrete topology. Let
be the continuous map induced by the identity. It induces adjoint direct and inverse image functors
and
. Then
, and the unit of this adjunction is the natural transformation described above.
Because of this adjunction, there is an associated monad on the category of sheaves on ''X''. Using this monad there is a way to turn a sheaf ''F'' into a coaugmented cosimplicial sheaf. This coaugmented cosimplicial sheaf gives rise to an augmented cochain complex that is defined to be the Godement resolution of ''F''.
In more down-to-earth terms, let
, and let
denote the canonical map. For each
, let
denote
, and let
denote the canonical map. The resulting
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 ...
is a flabby resolution of ''F'', and its cohomology is the
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 ...
of ''F''.
References
*
* {{Citation , last1=Weibel , first1=Charles A. , title=An introduction to homological algebra , publisher=
Cambridge University Press , isbn=978-0-521-55987-4 , mr=1269324 , year=1994 , doi=10.1017/CBO9781139644136
Sheaf theory
Algebraic topology
Homological algebra