Computational methods for studying sheaves

In this thesis, we focus on the sheaves arising in algebraic geometry and the algebraic study of differential equations. Our goal is to answer geometric questions by applying computational techniques to modules corresponding to these sheaves. In the setting of algebraic geometry, we provide an algorithm for computing the global extension module $ExtmXM,N$, where X is a projective scheme and $M$ and $N$ are coherent $D$-modules. In the context of $D$-modules on $An$, we give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of $D$. This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive.