Zariski cohomology in second order arithmetic

TL;DR

This paper explores the interpretation of Zariski cohomology for coherent sheaves within second order arithmetic, establishing a finiteness theorem that does not extend to the etale topology.

Contribution

It introduces a finiteness theorem for Zariski cohomology in second order arithmetic and demonstrates its failure in the etale topology.

Findings

Finiteness theorem proven for Zariski cohomology in second order arithmetic.

Finiteness theorem fails for the etale topology on Noetherian schemes.

Provides a logical framework for cohomology theories in arithmetic settings.

Abstract

The cohomology of coherent sheaves and sheaves of Abelian groups on Noetherian schemes are interpreted in second order arithmetic by means of a finiteness theorem. This finiteness theorem provably fails for the etale topology even on Noetherian schemes.