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

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.