Constructing spectra using cone injectivity

TL;DR

This paper generalizes the construction of spectra from commutative algebra to cone injectivity classes in various categories, analyzing conditions for the spectrum functor's faithfulness and relating schemes to functors of points.

Contribution

It introduces a generalized spectrum construction for cone injectivity classes and explores conditions for the spectrum functor's full faithfulness, connecting schemes with functors of points.

Findings

Spectrum functor's full faithfulness depends on specific conditions.

Generalized spectrum construction applies beyond commutative rings.

Equivalence established between schemes and functors of points.

Abstract

We provide a generalization of the construction of a spectrum of a commutative ring as a locally ringed space, applicable to cone injectivity classes in general contexts, especially in locally finitely presentable categories. In its full generality, the spectrum functor fails to be fully faithful and we study reasonable sufficient conditions under which it is. Further, assuming the full faithfulness, we introduce a generalization of another concept from algebraic geometry -- the functor of points -- and prove equivalence of the two resulting notions of schemes.