Computing Global Extension Modules

Let X be a projective scheme and let M and N be two coherentOX -modules. Given an integer m, we present an algorithm for computing the global extension module ExtXm(M, N). As a consequence, one may calculate the sheaf cohomologyHm(X, N) and construct the sheaf corresponding to an element of the module ExtX1(M, N). This algorithm depends only on the computation of Gröbner bases and syzygies and has been implemented in the computer algebra system Macaulay2.