Some computational aspects of spectral sequences in \v{C}ech 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 $\mathbb P = \mathbb P^{r_1}\times \dots \times \mathbb P^{r_m}$ 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.