Some computational aspects of spectral sequences in Čech cohomology

Abstract

Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily challenges the capacities of modern computers. We describe an algorithm and its implementation to compute a spectral sequence converging to the higher direct images of a bounded complex of sheaves on a product of projective spaces $\mathbb{P}={\mathbb{P}}^{r_1}\times \cdots \times {\mathbb{P}}^{r_m}$ over an arbitrary affine base $\mathrm{Spec}R$. We assume the ring $R$ to be computable and the complex of sheaves to be represented by an actual complex of (multi-)graded modules.

17 pages, 1 figure