Symmetric quasi-coherent sheaves

TL;DR

This paper introduces and studies a new category of symmetric quasi-coherent sheaves using stable homotopy theory, revealing their structure and connections to classical projective schemes.

Contribution

It defines symmetric quasi-coherent sheaves for non-commutative graded algebras and demonstrates their properties and relation to classical schemes.

Findings

The category is a closed symmetric monoidal Grothendieck category.

Quasi-coherent sheaves on a projective scheme are recovered from symmetric quasi-coherent sheaves.

Classical projective schemes are recovered from symmetric projective schemes.

Abstract

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric monoidal Grothendieck category with invertible generators. It is proven that the category of quasi-coherent sheaves on a projective scheme is recovered out of symmetric quasi-coherent sheaves. As an application, symmetric projective schemes associated to such algebras are introduced and studied. It is shown that classical projective schemes are recovered from symmetric ones.