Distributivity, affineness, and the structure sheaf

TL;DR

This paper explores the conditions under which the structure sheaf generates perfect complexes on schemes, linking lattice distributivity, affinization maps, and scheme properties with new examples.

Contribution

It establishes equivalences between lattice distributivity, 0-affineness of the affinization map, and generation of perfect complexes for certain schemes, including new examples.

Findings

Structure sheaf generates perfect complexes iff lattice of thick subcategories is distributive.

Equivalence between these conditions and the affinization map being 0-affine.

Identification of a quasi-projective scheme that is not quasi-affine but satisfies these conditions.

Abstract

We observe that for a quasi-compact and quasi-separated scheme the structure sheaf generates the perfect complexes if and only if the lattice of thick subcategories is distributive if and only if the affinization map is 0-affine. Examples are discussed, including an example of a quasi-projective scheme which is not quasi-affine but for which these equivalent conditions hold.