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Scheme theory for commutative semirings

Roberto Gualdi, Arne Kuhrs, Mayo Mayo Garcia, Xavier Xarles

TL;DR

The paper develops a two-pronged framework for affine schemes over commutative semirings by pairing Spec, built from prime ideals, with Sp, built from prime subtractive kernels, highlighting their divergent topologies and their unifying role through universal valuations. It then constructs a robust structure sheaf on Spec via three equivalent localization-based approaches and extends the theory to Sp, establishing the category of affine semischemes as the opposite of the category of semirings and defining semischemes as locally affine. Central to linking the two theories is the introduction of G-valuations and the universal G-valuation, which induces a canonical bridge between Spec and the k-spectrum, and yields presheaf constructions that encode valuation-theoretic data on semiring spaces. The work further develops global semirings and the globalization functor, studying when global sections recover the original semiring and how hardness interacts with globality and spectrum calculus. Together, these results lay a foundational, valuation-driven framework for semiring-based algebraic geometry, including tropical and idempotent-geometric perspectives, and open pathways to further globalization of valuations on semischemes.

Abstract

In this survey, we describe two different approaches to constructing affine schemes for commutative semirings: one based on prime ideals, and another based on prime kernels (also called subtractive ideals). We then explain how these two approaches are related through the theory of universal valuations.

Scheme theory for commutative semirings

TL;DR

The paper develops a two-pronged framework for affine schemes over commutative semirings by pairing Spec, built from prime ideals, with Sp, built from prime subtractive kernels, highlighting their divergent topologies and their unifying role through universal valuations. It then constructs a robust structure sheaf on Spec via three equivalent localization-based approaches and extends the theory to Sp, establishing the category of affine semischemes as the opposite of the category of semirings and defining semischemes as locally affine. Central to linking the two theories is the introduction of G-valuations and the universal G-valuation, which induces a canonical bridge between Spec and the k-spectrum, and yields presheaf constructions that encode valuation-theoretic data on semiring spaces. The work further develops global semirings and the globalization functor, studying when global sections recover the original semiring and how hardness interacts with globality and spectrum calculus. Together, these results lay a foundational, valuation-driven framework for semiring-based algebraic geometry, including tropical and idempotent-geometric perspectives, and open pathways to further globalization of valuations on semischemes.

Abstract

In this survey, we describe two different approaches to constructing affine schemes for commutative semirings: one based on prime ideals, and another based on prime kernels (also called subtractive ideals). We then explain how these two approaches are related through the theory of universal valuations.
Paper Structure (22 sections, 35 theorems, 159 equations)

This paper contains 22 sections, 35 theorems, 159 equations.

Table of Contents

  1. Semirings and their ideals
  2. Semirings and localizations
  3. Ideals and prime ideals
  4. Subtractive ideals and kernels
  5. Hard semirings
  6. Spectrum constructions
  7. Spectrum and $k$-spectrum
  8. The Zariski topology
  9. Induced morphisms
  10. The structure sheaf for
  11. Three constructions of $\mathcal{O}_X$
  12. Equivalence of definitions
  13. The category of affine semischemes
  14. The category of semischemes
  15. The structure sheaf for
  16. ...and 7 more sections

Key Result

Lemma 1.4

Let $S\subseteq A$ be a multiplicative submonoid. Then $S^{\mathrm{sat}}$ is the smallest saturated multiplicative submonoid of $A$ that contains $S$. Moreover, and the natural morphism $(S^{\mathrm{sat}})^{-1}A \to S^{-1}A$ obtained from the universal property of the localization at $S^{\mathrm{sat}}$ is an isomorphism.

Theorems & Definitions (125)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Example 1.8
  • Remark 1.9
  • ...and 115 more