Some computational aspects of spectral sequences in \v{C}ech cohomology

TL;DR

This paper presents an algorithm and implementation for computing spectral sequences that converge to higher direct images of sheaves on products of projective spaces, addressing computational challenges in algebraic geometry.

Contribution

It introduces a novel algorithm and software implementation for calculating spectral sequences of sheaves on complex algebraic varieties, enhancing computational tools in algebraic geometry.

Findings

Successfully computes spectral sequences for sheaves on product of projective spaces.

Efficiently handles complexes of multi-graded modules over computable rings.

Provides a practical method for higher direct image calculations in algebraic geometry.

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.