Graded algebras, projective spectra and blow-ups in derived algebraic geometry

TL;DR

This paper develops a derived algebraic geometry framework for graded algebras, projective spectra, and blow-ups, extending classical concepts to the derived setting with new constructions like the derived Rees algebra.

Contribution

It introduces a derived version of graded algebras, projective spectra, and blow-ups, generalizing classical algebraic geometry tools to the derived context.

Findings

Derived $ ext{Proj}$ is representable by a derived scheme.

Extended Rees algebra is generalized to the derived setting.

Deformation to the normal cone is extended to derived schemes.

Abstract

We define graded, quasi-coherent $\mathcal{O}_S$-algebras over a given base derived scheme $S$, and show that these are equivalent to derived $\mathbb{G}_{m,S}$-schemes which are affine over $S$. We then use this $\mathbb{G}_{m,S}$-action to define the projective spectrum $\mathrm{Proj} (\mathcal{B})$ of a graded algebra $\mathcal{B}$ as a quotient stack, show that $\mathrm{Proj} (\mathcal{B})$ is representable by a derived scheme over $S$, and describe the functor of points of $\mathrm{Proj} (\mathcal{B})$ in terms of line bundles. The theory of graded algebras and projective spectra is then used to define the blow-up of a closed immersion of derived schemes. Our construction will coincide with the existing one for the quasi-smooth case. The construction is done by generalizing the extended Rees algebra to the derived setting, using Weil restrictions. We close by also generalizing the deformation to the normal cone to the derived setting.