Continuous closure of sheaves

TL;DR

This paper introduces an algebraic method for constructing the continuous closure of finitely generated torsion free modules, demonstrating its compatibility with flat morphisms and field extensions, and establishing its coherence properties.

Contribution

It provides a purely algebraic construction of continuous closure for modules, extending its properties to coherence and compatibility with various morphisms and extensions.

Findings

Continuous closure commutes with flat morphisms in characteristic 0.

The continuous closure of a coherent ideal sheaf remains coherent.

Continuous closure commutes with field extensions.

Abstract

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the \'etale topologies) and it commutes with field extensions.