Blue schemes, semiring schemes, and relative schemes after To\"en and Vaqui\'e

TL;DR

This paper explores the generalization of scheme theory to blue schemes and semiring schemes, establishing Yoneda functors and their limitations, and identifying conditions under which equivalences hold.

Contribution

It constructs Yoneda functors for blue and semiring schemes, analyzes their failures, and proves an equivalence for locally finite type blue schemes over a blue field.

Findings

Yoneda functors for blue and semiring schemes are not equivalences generally.

An inverse to the Yoneda functor is constructed for locally finite type blue schemes.

Compatibility of Yoneda functors with base extensions is verified.

Abstract

It is a classical insight that the Yoneda embedding defines an equivalence of schemes as locally ringed spaces with schemes as sheaves on the big Zariski site. Similarly, the Yoneda embedding identifies monoid schemes (or $\mathbb{F}_1$-schemes in the sense of Deitmar) with schemes relative to sets (in the sense of To\"en and Vaqui\'e). In this paper, we investigate the generalization to blue schemes and to semiring schemes. We establish Yoneda functors for both schemes theories. These functors fail, however, to be equivalences in both situations. The reason for this failure is a divergence in the Grothendieck pretopologies coming from schemes as topological spaces and schemes as sheaves. Restricted to blue schemes that are locally of finite type over a blue field, we construct an inverse to the Yoneda functor, which establishes an equivalence for this subclass of blue schemes. Moreover, we verify the compatibility of the Yoneda functors with the base extension from blue schemes to semiring schemes and with the base extension from semiring schemes to usual schemes.