Affine Schemes in Commutative Semiring Theory

• Affine schemes for commutative semirings are generalized geometric structures that replace rings with semirings, employing prime ideals, subtractive kernels, and congruences.
• They adapt classical Zariski topologies and structure sheaf constructions to support localization and gluing, enabling direct developments in tropical and idempotent geometry.
• Applications span non-archimedean geometry and F1-geometry, with tools like universal valuations and lattice-theoretic techniques unifying various spectrum approaches.

Affine schemes for commutative semirings generalize classical affine scheme theory by replacing the ring structure with that of a commutative semiring, and adapting the underlying geometric and algebraic data accordingly. This theory supports multiple generalizations—including those using prime ideals, prime (subtractive) kernels, and prime congruences—and enables direct development of tropical and idempotent schemes, with applications in non-archimedean geometry and “$\mathbb F_1$-geometry.” The subject has diverse foundations tied to congruence theory, lattice-theoretic and order-theoretic techniques, and the theory of idempotent semirings.

1. Commutative Semirings and Prime Objects

A commutative semiring $A$ consists of a set $A$ with two binary operations $+$ and $\cdot$ such that $(A, +)$ is a commutative monoid with identity $0_A$, $(A, \cdot)$ is a commutative monoid with identity $1_A$, distributivity holds, and $a \cdot 0_A = 0_A$ for all $a \in A$. Additively idempotent semirings (where $a+a = a$ for all $a$) play a central role in tropical geometry and idempotentification procedures (Boudreau et al., 2023, Gualdi et al., 20 Jan 2026).

Several prime-like objects are key for defining spectra:

• Prime ideals: $I \subseteq A$ is a prime ideal if $ab \in I$ implies $a \in I$ or $b \in I$.
• Subtractive (prime kernel) ideals: $I$ is subtractive if, whenever $a+b=c$ with $b,c\in I$, then $a \in I$. Prime subtractive ideals are called prime kernels. For rings, all ideals are subtractive, but not for general semirings (Gualdi et al., 20 Jan 2026).
• Prime congruences: A congruence $p \subseteq A \times A$ is an equivalence relation that is a subsemiring. The spectrum of prime congruences, $Spec^c(A)$, plays a parallel role to $Spec(A)$ in the classical case (Qiu, 2015).

2. Spectra and Zariski-Type Topologies

Multiple spectrum constructions underpin affine semiring geometry:

• Prime Ideal Spectrum: $Spec(A) = \{\mathfrak p \subseteq A : \mathfrak p$ is a prime ideal$\}$, with closed sets $$V(S) = \{\mathfrak p \,|\, S \subseteq \mathfrak p\}$$. The corresponding Zariski topology uses these closed sets or their basic open complements $D(f)$ (Gualdi et al., 20 Jan 2026).
• Prime Kernel Spectrum: $Sp(A) = \{\mathfrak p \subseteq A : \mathfrak p$ is a prime subtractive (kernel) ideal$\}$, with topology inherited from $Spec(A)$. The basic opens are $D(f)\cap Sp(A)$ (Gualdi et al., 20 Jan 2026).
• Prime Congruence Spectrum: $Spec^c(A) = \{p \subseteq A \times A : p$ is a prime congruence$\}$; the closed sets are $$V^c(\sigma) = \{p \in Spec^c(A): \sigma \subset p\}$$ for congruences $\sigma$ (Qiu, 2015).

For idempotent semirings, spectrum theories often yield topologies with superior dimension-theoretic behavior and better connections to tropicalization (Gualdi et al., 20 Jan 2026).

3. Structure Sheaves, Gluing, and Localization

The structure sheaf formalism for affine schemes over semirings adapts classical gluing, localization, and stalk arguments:

• Spec(A)-schemes: The structure sheaf $\mathcal O_{Spec(A)}$ is defined uniquely so that $\mathcal O(D(f)) \cong A_f$ (localization), and stalks are $A_{\mathfrak p}$ (Gualdi et al., 20 Jan 2026).
• Spec$^c$(A)-schemes: The congruence-based construction employs basic opens $D(a,b) = \{p \in Spec^c(A) : (a,b)\notin p\}$ with structure sheaf $\mathcal O_{Spec^c(A)}$ given by localizations at suitable multiplicative systems. This construction ensures that gluing and stalk properties directly mirror those of classical affine schemes (Qiu, 2015).
• Idempotent and kernel spectra: For $Sp(A)$, one defines the sheaf by “kernel-localization,” i.e., inverting the saturated multiplicative system associated to an open in $Sp(A)$, yielding sheaf $\mathcal O_{Sp(A)}$ (Gualdi et al., 20 Jan 2026). The idempotentization process also produces sheaves with stalks at $p$ given by localizations $T_p$, where $T = fg\,Id(A)$ is the semiring of finitely generated ideals (Boudreau et al., 2023).

These constructions are functorial, compatible with morphisms between semirings, and behave well under base change and localization.

4. Congruence Schemes, Algebraic Varieties, and the Congruence Nullstellensatz

Prime congruence schemes and their associated Zariski topologies extend classical algebraic geometry:

• Given $A \subseteq B$ (semirings), $S = A[x_1,\ldots,x_n]$, and a congruence $\rho$ on $B$, one defines, for $T \subseteq S \times S$, the $\rho$-vanishing locus $$Z_\rho(T)(B) = \{P \in B^n : (f(P),g(P)) \in \rho,\, \forall (f,g)\in T\}$$ (Qiu, 2015).
• If $\rho$ is a prime congruence, the collection of $Z_\rho(T)$ satisfies the axioms of the closed subsets of a topology, with explicit union and intersection formulas involving the twist-product of pairs in $T$.
• The theory establishes a Galois correspondence between congruences on $S$ and $\rho$-closed sets in $B^n$, including a version of the Nullstellensatz for congruences: the congruence of vanishing on $Y$, $\mathfrak B_\rho(Y)$, and the $\rho$-radical of $T$ interact as expected, and in favorable conditions (e.g., $\rho = id_B$ or $B$ a $\rho$-semifield), strict equalities as in the classical Nullstellensatz are obtained.
• There is an interpretation akin to Hilbert's Nullstellensatz in terms of morphisms between suitably quotiented $A$-algebras and $B$-semirings: $$Z_\rho(\sigma)(B) / \rho \simeq \operatorname{Hom}_{A\text{-alg}}(S/\sigma^c, B/\rho)$$ (Qiu, 2015). Irreducible $\rho$-varieties correspond to prime congruences on $S$ containing $\sigma$.

5. Idempotentization, Subtractive Ideals, and Lattice-Theoretic Structures

Idempotentization (or “tropicalization”) of affine schemes replaces the ring by its idempotent semiring of finitely generated ideals $A^{id} = fg\,Id(A)$; addition and multiplication correspond to sum and product of ideals. The global sections of the idempotentized structure sheaf are identified with $A^{id}$. On a Noetherian ring $A$, this yields a homeomorphism between the spectrum of subtractive $k$-prime ideals of $A^{id}$ and the usual $\operatorname{Spec}A$ (Boudreau et al., 2023).

Key lattice-theoretic correspondences arise:

• For an $A$-module $M$, the poset of $A$-submodules is isomorphic to the poset of $k$-ideals of the semiring of finitely generated submodules of $M$.
• The set of subtractive ideals $Id^k(S)$ embeds as a topological retract of the space of all congruences $Cong(S)$. With the coarse-lower topology, $c: Id^k(S) \to Cong(S)$ (sending $I$ to the generated congruence) and $t: Cong(S) \to Id^k(S)$ (sending a congruence $Y$ to $\{a \mid (a,0)\in Y\}$) satisfy $t\circ c = Id$ (Boudreau et al., 2023).
• Similarly, subtractive-closure yields a retraction $j: Id(S) \to Id^k(S)$, identifying $Id^k(S)$ as a closed subspace of $Id(S)$ with the coarse-upper topology.

This structure underpins the well-behaved nature of spectra defined using subtractive ideals and relates to both topological and order-theoretic aspects of tropical schemes.

6. Universal Valuations and Unification of Spectra

Universal valuations provide a natural bridge between spectrum constructions:

• For an $R$-algebra $A$, the canonical $G$-valuation $v_R: A \to M_R(A)$ (where $M_R(A)$ is the semiring of finitely generated $R$-subsemimodules of $A$) has the property that any $G$-valuation $v: A \to S$ factors through $v_R$ uniquely (Gualdi et al., 20 Jan 2026).
• The induced map $v_R^* : Sp M_R(A) \xrightarrow{\sim} Spec(A)$ is a homeomorphism, so the ideal-theoretic and kernel-theoretic (i.e., subtractive) approaches coincide after idempotentization of the coordinate algebra.
• This suggests that, in the presence of idempotentization, geometric objects parametrized by affine schemes for commutative semirings can be functorially interpreted in terms of $G$-valuations and tropical points.

The universal-valuation framework unifies the disparate approaches to semiring schemes and clarifies the categorical relations between them.

7. Examples and Applications

Selected examples illustrate the range of these theories:

• For $A = \mathbb N$, $Spec\mathbb N$ includes all $p\mathbb N$ (for $p$ prime), $\{0\}$, and $\mathbb N\setminus\{1\}$, but $Sp \mathbb N$ is just the former two, reflecting more geometric behavior (Gualdi et al., 20 Jan 2026).
• For the Boolean semiring $\mathbb B[x]$, $Sp \mathbb B[x] = \{\{0\}, (x)\}$ is a two-point Sierpiński space; $Spec\mathbb B[x]$ is infinite.
• For tropical semirings ($\mathbb T$), both $Spec \mathbb T$ and $Sp \mathbb T$ are $\{\{0\}\}$.
• For $A = \mathbb N[x]$, $Spec\mathbb N[x]$ contains both arithmetic and geometric primes, while $Sp \mathbb N[x] \simeq Spec \mathbb Z[x]$ via hardening (Gualdi et al., 20 Jan 2026).
• In idempotentization, for $A = \mathbb Z[x]$, $A^{id}$ is the semiring of finitely generated ideals of $\mathbb Z[x]$, with localizations over distinguished opens matching the localization of ideals in $\mathbb Z[x]_{f}$ (Boudreau et al., 2023).

These results highlight both the versatility of affine schemes over semirings and nuances regarding the spectra, sheaf theory, and categorical structures compared to classical algebraic geometry.