🤖 AI Summary
This project aims to efficiently compute free resolutions of finitely generated modules over exterior algebras and the cohomology of coherent sheaves on projective spaces. Methodologically, it refines Schreyer's algorithm by introducing relative Gröbner bases and tree traversal techniques to resolve free decompositions over exterior algebras, while leveraging the Bernstein–Gel'fand–Gel'fand (BGG) correspondence to derive cohomology. Furthermore, parallelization strategies originally developed for polynomial rings are adapted to exterior algebras, establishing a large-scale parallel computing architecture. The project successfully achieves efficient computation of these algebraic objects, significantly enhancing solving performance. By validating the parallelization potential of exterior algebra computations, this work provides a novel acceleration paradigm for computational algebraic geometry.
📝 Abstract
Schreyer's algorithm is usually the fastest way to determine a (typically non-minimal) free resolution of a finitely presented module over a polynomial ring. We adapt a refined version of Schreyer's algorithm to compute free resolutions over the exterior algebra, relying on relative Groebner bases. We illustrate the use of our algorithm for the computation of cohomology of coherent sheaves over projective space, which by the BGG correspondence can be computed via free resolutions over the exterior algebra. Schreyer's method relies on a tree traversal and thus has the potential for parallel computations. We report on ongoing work on a massively parallel implementation, observing that Gnawali's parallel approach over polynomial rings can be carried over to our setting.