On graded ${\mathbb{E}}_{\infty }$-rings and projective schemes 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.

30 pages, major revision, a coauthor added