On graded $\mathbb{E}_{\infty}$-rings and projective schemes in spectral algebraic geometry

TL;DR

This paper develops the theory of graded $ ext{E}_ ext{infty}$-rings and modules within spectral algebraic geometry, constructing spectral projective schemes and describing their sheaves in terms of graded modules.

Contribution

It introduces graded $ ext{E}_ ext{infty}$-rings and modules, and constructs spectral projective schemes, linking their sheaves to graded modules under finiteness conditions.

Findings

Construction of spectral projective schemes from graded $ ext{E}_ ext{infty}$-rings.

Equivalence of sheaves over spectral projective schemes and graded modules.

Framework for studying quasi-coherent sheaves in spectral algebraic geometry.

Abstract

We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the $\infty$-category of almost perfect quasi-coherent sheaves over a spectral projective scheme $\mathrm{Proj}\,(A)$ associated to a connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-ring $A$ can be described in terms of $\mathbb{Z}$-graded $A$-modules.