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 . This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants. Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory.

=Definitions=

The first version of sheaf cohomology to be defined was that based on Cech cohomology, in which the relatively small change was made of attributing to an open set U of a topological space X an element in F ( U ), an abelian group that varies with U , rather than an abelian group A that is fixed ahead of time. This means that cochains are easy to write down rather concretely; in fact the model applications, such as the Cousin problems on meromorphic functions, stay within fairly familiar mathematical territory. From the sheaf point of view, the Cech theory is the restriction to the theory of sheaves of locally constant functions with values in A . Within sheaf theory it is easy to see that twisted versions, with local coefficients on which the fundamental group acts, are also subsumed — along with some very different sorts of more general coefficients.

One problem with that theory was that Cech cohomology itself fails to have good properties, unless X itself is proposed a more abstract definition that would build in the long exact sequence.

The Grothendieck definition clarified (at a cost) the status of sheaf cohomology as a s; what that means is that in theory calculations can be done with injective resolutions, though in practice short and long exact sequences may be a better idea.

=Applications=

Subsequently there were further technical extensions (for example in Godement s book), and areas of application. For example, sheaves were applied to transformation groups; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces; to representation theory in the Borel-Bott-Weil theorem; as well as becoming standard in algebraic geometry and complex manifolds.

The particular needs of étale cohomology were more about reinterpreting sheaf in sheaf cohomology , than cohomology , given that the derived functor approach applied. Flat cohomology, crystalline cohomology and successors are also applications of the basic model.

=Euler characteristics=

The Euler characteristic of a sheaf F is by definition

:χ( F ) = Σ (−1) i rank( H i ( X , F ))

with the sum taken over all integers i ≥ 0. To make sense of this expression, which generalises the Euler characteristic as alternating sum of Betti numbers, two conditions must be fulfilled. Firstly the summands must be almost all zero, i.e. zero for i ≥ N for some N . Further, rank must be some well-defined function from module theory, such as rank of an abelian group or vector space dimension, that yields finite values on the cohomology groups in question. Therefore finiteness theorems of two kinds are required.

In theories such as coherent cohomology, where such theorems obtain, the value of χ( F ) is typically easier to compute, from other considerations (for example the Hirzebruch-Riemann-Roch theorem or Grothendieck-Riemann-Roch theorem), than the individual ranks separately. In practice it is often H 0( X , F ) that is of most interest; one way to compute its rank is then by means of a vanishing theorem on the other H i ( X , F ). This is a standard indirect method of sheaf theory to produce numerical results.