4

One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, cf. Lorscheid 2012. The theory follows the same set up of scheme theory, but we use semirings instead of rings.

Given a semiring $R$, we have a semiringed space $\mathrm{Spec}(R)$, defined by mimicking the usual definition for rings. This gives a functor $\mathrm{Spec}$ from the opposite of the category semirings to that of affine semiring schemes.

Conversely, there's also a global sections functor $\Gamma:\mathrm{AffSemiSch}^\circ\to\mathrm{Semiring}$ sending a semiringed space $(X,\mathcal{O}_X)$ to $\Gamma(X,\mathcal{O}_X)$.

Does the pair $(\mathrm{Spec},\Gamma)$ give as in ordinary algebraic geometry a contravariant equivalence of categories $\mathrm{Semiring}\cong\mathrm{AffSemiSch}^\circ$?

Emily
  • 1,331
  • 3
    I did some calculations related to this question last year. If I recall correctly you can carry over a lot of the proofs but you have to be careful with some definitions. I don't think the ring definition of prime ideal works well for semirings (or, as I prefer to call them, rigs). It seems to me the correct definition involves prime filters instead. – Zhen Lin Aug 14 '21 at 04:31
  • Crossposted: MO. – Emily Sep 06 '21 at 07:45

1 Answers1

2

The thesis Algebraic geometry over semi-structures and hyper-structures of characteristic one by Jaiung Jun accessible here gives half of the proof in proposition 2.2.6, the remaining half being easy.

It states that $\mathrm{Spec}:\mathrm{Semiring}\to\mathrm{AffSemiSch}^\circ$ is fully faithful. Since it is essentially surjective by construction, it is an equivalence of categories.

Also $\Gamma(\mathrm{Spec}(R),\mathscr{O}_{\mathrm{Spec}(R)})\cong R$ by construction, and $\mathrm{Spec}$ is left adjoint to $\Gamma$ by the usual proof (here is one). So since $\Gamma$ is right adjoint to an equivalence of categories, it follows that $\Gamma$ itself must be an equivalence, and therefore $\mathrm{Spec}$ and $\Gamma$ are mutually inverse.

Edit: a better reference is Čech cohomology of semiring schemes, item (3) of proposition 2.1, which explicitly states "The opposite category of affine semiring schemes is equivalent to the category of semirings."!

Emily
  • 1,331
  • 4
    While this might not be relevant in this case, be cautious with this thesis in general. A friend of mine that works on $\mathbb{F}_1$-geometry once told me that the author stated somewhere that there were quite some mistakes in the thesis. To my knowledge this concerns the algebraic geometry over hyperrings part though. – Con Sep 11 '21 at 15:03
  • 2
    @Con Thank you so much for the warning, I'll be cautious with it! This made me went to look for a published reference (not that this guarantees any correctness, but it's at least less likely to be errors), and during that I found a paper by Jun with exactly this statement! – Emily Sep 11 '21 at 15:12