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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 $\mathrm{Semiring}\cong\mathrm{AffSemiSch}^\circ$?

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  • $\begingroup$ Crossposted: MSE. See Zhen Lin's comment there. $\endgroup$
    – Emily
    Commented Sep 6, 2021 at 7:44
  • $\begingroup$ @ZhenLin's comment referenced above. $\endgroup$
    – LSpice
    Commented Sep 11, 2021 at 15:08

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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."!

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    $\begingroup$ See also Con's cautionary note here regarding Jun's thesis. $\endgroup$
    – Emily
    Commented Sep 11, 2021 at 15:14

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