The Godement resolution of a sheaf is a construction in homological algebra 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. It was discovered by Roger Godement.

Overview

Given a topological space X (more generally, a topos X with enough points), and a sheaf F on X, the Godement construction for F denote the stalk of F at x. Given an open set, define

is a presheaf. One checks the sheaf is flabby can be turned into a functor that sends each section to the 'product' of its germs.

be the set X with the discrete topology be the continuous map induced by the identity. It induces adjoint, 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.

denote the canonical map. The resulting resolution is a flabby resolution of F, and its cohomology is the sheaf cohomology of F.

Definition

(the construction holds more generally in sheaves with more algebraic structure, e.g. sheaves of vector spaces, modules or rings). The Godement resolution

• is a flabby sheaf;

• .

Recall that a sheaf space has the structure of an Abelian group.

is an equivalence of categories.

with the corresponding continuous section of the sheaf space.

A serration that does not need to be continuous of sheaves.

.

Properties

Functoriality

.

are constructed iteratively by the same procedure, these are compositions of functors and are hence themselves functors.

Flabbiness

, it follows then that the higher degree Godement sheaves are also flabby.

Exactness

is an exact functor.

), where the first two complexes are exact, thus the cohomology long exact sequence follows from iterating the same argument.

, and recall the well-known theorem are also exact functors.

Relation to sheaf cohomology

The main use of the Godement resolution is to define sheaf cohomology. In the literature there exist (at least) three methods by which the cohomology of sheaves can be constructed, via

1. Čech cohomology;

2. the Godement resoution;

3. derived functors (injective resolutions).

This list is ordered in terms of increasing generality. Čech cohomology can be defined for any topological space, but it is guaranteed to agree with the other forms of sheaf cohomology only if the space is a paracompact Hausdorff space, the approach via the Godement resolution works on any space and agrees with derived functor cohomology, while the latter can be defined generally also for sheaves on sites.

For sheaves on topological spaces, the Godement resolution has a number of advantages over derived functor cohomology due to the fact that it is canonical and the Godement functors are exact.

As an illustration, recall the fact into an injective sheaf. This is highly non-constructive, i.e. the k is equal to the kHowever, since the injective resolution of the sheaf is not canonical, this definition becomes well-defined only if one shows that the cohomology groups are independent of the choice of injective resolution. Furthermore, one must show that the cohomology long exact sequence exists. Both of these follows from highly general categorical arguments coming from the properties of injective objects.

It is also possible to define an injective resolution which is canonical in the sense that it is functorial in the initial sheaf, but these functors fail to be exact, which means that the existence of the cohomology long exact sequence has to be proven by different means.

, where the latter is the k are exact, the existence of the long exact sequence follows from a simple argument.

Sheaf cohomology axioms

such that the following properties are satisfied:

1. ;

2. of short exact sequences of sheaves induces a corresponding morphism of their cohomology long exact sequences.

Theorem: satisfy the sheaf cohomology axioms.

Proof:.

, consider the commutative diagramwhich establishes the second axiom as well.

Acyclicity and flabby sheaves

is acyclic

is essentially trivial).

of the sequence of global sections vanishes.

.

Consequently,

• flabby sheaves are acyclic, and

• sheaf cohomology can be computed generally by flabby resolutions.

References

External links