One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, cf. [Lorscheid 2012](https://arxiv.org/abs/1212.3261). 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 $\mathrm{Semiring}\cong\mathrm{AffSemiSch}^\circ$?



Are affine semiring schemes equivalent to semirings?
The thesis Algebraic geometry over semi-structures and hyper-structures of characteristic one by Jaiung Jun [accessible here](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/37850/JUN-DISSERTATION-2015.pdf?sequence=1&isAllowed=y) 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](https://math.stackexchange.com/a/1526426)). 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.




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Edit: a better reference is [Čech cohomology of semiring schemes](https://www.sciencedirect.com/science/article/pii/S0021869317302132), item (3) of proposition 2.1, which explicitly states "The opposite category of affine semiring schemes is equivalent to the category of semirings."!


