Kummer Torsors Classify Polyadic Line Products Over Semirings

Polyadic algebra over semirings has a basic descent problem: an n-ary product on a line object can look locally like iterated multiplication, while failing to come from any global binary product. This paper gives that failure a precise cohomological form. For a semiring scheme X, a line bundle L, and m = n−1, the authors classify multilinear n-ary products on L and show that the nondegenerate ones are controlled by trivializations of L^m.

Core Contribution

The main contribution is a Kummer-style classification that works without additive inverses. The paper identifies the symmetric monoidal groupoid of nondegenerate n-ary line objects with the groupoid of fpqc μ_m-torsors. In concrete terms, an n-ary product is not treated as an arbitrary operation; it is a section of the appropriate inverse tensor power of the line. Nondegeneracy means that this section is invertible, so the product records an m-th power trivialization.

That turns arity into a torsion invariant. A line class with exact Picard order m naturally supports a first nondegenerate product of arity m+1, and the paper proves that no smaller nondegenerate arity exists on the same line. This is the cleanest part of the result: the algebraic arity is detected by Picard torsion rather than by a choice of coordinates.

Technical Approach

The semiring step is the paper's main technical hinge. In ordinary algebra, the Kummer sequence depends on adjoining roots and using fpqc-local surjectivity of the power map on the multiplicative group. The authors prove the corresponding root construction for arbitrary commutative semirings: for every unit u of A, adjoining an m-th root produces a finite free faithfully flat A-algebra. This supplies the missing descent mechanism in the absence of subtraction.

With that root construction in place, the paper establishes a Kummer exact sequence for semiring schemes and uses it to package the normalized-frame torsor. Locally, after choosing frames for L, a nondegenerate product has unit coefficients whose transition rule involves (n−1)-st powers. The resulting Čech cocycle is μ_m-valued, and its cohomology class is the obstruction to gluing local binary envelopes.

The later sections make this obstruction explicit. A binary envelope is an R-bilinear operation from L × L to L whose iterated n-fold product equals the original n-ary product. Lemma 6.1 shows that such a binary operation is determined by a unique parameter β in the dual line, and its n-fold iterate corresponds to β^(n−1). Theorem 6.1 then gives the exact criterion: a product with parameter θ has a binary envelope exactly when θ = β^(n−1); for a labelled family, the same β must work for every label.

Results and Analysis

The evidence is formal rather than experimental. The paper proves two obstruction layers for binary reducibility. First, if the Picard class of L is nonzero, a globally nondegenerate product cannot have a binary envelope, because any such envelope would trivialize L. Second, even when L is trivial, a scalar coefficient λ must lie in the (n−1)-st powers of the unit group. This separates geometric torsion from the residual unit-power obstruction.

The arithmetic examples are useful because they show the obstruction is not an artifact of language. Over R = ℤ[s] with s² = −5, the ideal I = (2, 1+s) is invertible and nonprincipal, with I² = (2). The paper constructs the ternary product [x,y,z] = xyz/2 on the additive group of I. Its linearization is an isomorphism, so the product is globally nondegenerate, but it admits no R-bilinear binary envelope because that would trivialize the nonprincipal ideal.

More generally, for a Dedekind domain and a nonprincipal invertible fractional ideal of exact order d > 1, the product μ(x1,...,x(d+1)) = x1⋯x(d+1)/a, with I^d = aR, is a nondegenerate (d+1)-ary product with no binary envelope and no smaller nondegenerate arity on I. The paper also checks the trivial-Picard boundary. When L = R, the invariants collapse to scalar coefficients: the Picard layer disappears, but the Kummer unit-power class can remain. The examples over ℕ[q±1] and 𝔹[q±1] show that q x1⋯xn has exact obstruction order m and becomes binary only after the explicit root torsor q = z^m.

Limitations

This is a proof paper, so the support comes from constructions, equivalences, and examples rather than computation. The authors leave several geometric questions open: representability and properties of Bμ_m over idempotent and tropical bases, calculations of H¹_fpqc(X, μ_m) for tropical curves, compactifications of the nondegenerate torsor locus, and higher-rank polyadic structures where associativity is no longer automatic. Those are real boundaries of the present result, but they also make the scope clear: the paper settles the rank-one line-object classification and the binary-descent obstruction, not the full moduli theory of polyadic semimodules.