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 $$\text { {Proj}}\,(A)$$ associated to a connective $${\mathbb {N}}$$ -graded $${\mathbb {E}}_{\infty }$$ -ring A can be described in terms of $${{\mathbb {Z}}}$$ -graded A-modules.