Flat quasi-coherent sheaves of finite cotorsion dimension

TL;DR

This paper investigates the properties of quasi-coherent sheaves on certain schemes, establishing conditions for finite cotorsion and pure injective dimensions, and demonstrating an equivalence between specific homotopy and derived categories.

Contribution

It proves that schemes with finite cotorsion dimension sheaves are n-perfect and establishes an equivalence between homotopy and derived categories of pure injective sheaves.

Findings

Schemes with finite cotorsion dimension are n-perfect.

On coherent schemes, all quasi-coherent sheaves have finite pure injective dimension.

An equivalence between the homotopy category of pure injective sheaves and the pure derived category is established.

Abstract

Let X be e quasi-compact and semi-separated scheme. If every at quasi- coherent sheaf has finite cotorsion dimension, we prove that X is n-perfect for some n > 0. If X is coherent and n-perfect(not necessarily of finite krull dimension), we prove that every at quasi-coherent sheaf has finite pure injective dimension. Also, we show that there is an equivalence K(PinfX)---> D(FlatX) of homotopy categories, whenever K(PinfX) is the homotopy category of pure injective at quasi-coherent sheaves and D(FlatX) is the pure derived category of at quasi-coherent sheaves.