Coherent sheaves and cohesive sheaves

TL;DR

This paper explores the relationship between coherent and cohesive sheaves of modules over open sets in complex space, establishing conditions under which they are equivalent and proving the flatness of the sheaf of Banach space valued holomorphic germs.

Contribution

It demonstrates that coherent sheaves are cohesive and vice versa under certain conditions, and proves the flatness of the sheaf of Banach space valued holomorphic germs.

Findings

Coherent sheaves are cohesive over open sets in olds.

Certain sheaves derived from cohesive sheaves are coherent.

The sheaf of Banach space valued holomorphic germs is flat.

Abstract

We consider coherent and cohesive sheaves of $\cO$--modules over open sets $\Omega\subset\bC^n$. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat.