Sheaf Cohomology - Euler Characteristics

Euler Characteristics

The Euler characteristic of a sheaf is defined by

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 for some . 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 exist, 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 H0(X,F) that is of most interest; one way to compute its rank is then by means of a vanishing theorem on the other Hi(X,F). This is a standard indirect method of sheaf theory to produce numerical results.