Gregory S. Smith - Computing global extension modules

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Gregory S. Smith - Computing global extension modules

GREGORY S. SMITH, Department of Mathematics, University of California at Berkeley, Berkeley, California 94720-3840, USA

Computing global extension modules

Let X be a projective scheme; let ${\cal M}$ and ${\cal N}$ be two coherent ${\cal O}_{X}$ -modules. Given an integer m, we present an algorithm for computing the global extension module $\textrm{Ext}^{m}(X;{\cal M},{\cal N})$ . In particular, this allows one to calculate the sheaf cohomology $H^{m}(X,{\cal N})$ and to construct the sheaf corresponding to an element of the module $\textrm{Ext}^{1}(X;{\cal M},{\cal N})$ . This algorithm can be implemented using only the computation of Gröbner bases and syzygies, and it has been implemented in the computer algebra system Macaulay2.