Computing Global Extension Modules for Coherent Sheaves on a Projective   Scheme
  Let X be a projective scheme; let M and N be two coherent O_X-modules. Given an integer m, we present an algorithm for computing the global extension module Ext^m(X;M,N). In particular, this allows one to compute the sheaf cohomology H^m(X,N) and to construct the sheaf corresponding to an element of the module Ext^1(X;M,N). This algorithm can be implemented using only the computation of Grobner bases ans syzygies, and it has been implemented in the computer algebra system Macaulay2. 
Gregory G. Smith