Finite diagonalizable torsors and Kummer cohomology over semiring schemes

Abstract

Classical descent treats a finite diagonalizable torsor as a higher-rank locally free object. Over semirings that route is incomplete: fpqc-local freeness is not known to imply Zariski-local freeness in arbitrary rank. We show that diagonalizable symmetry bypasses this obstruction. The coordinate semialgebra splits into character pieces, each piece descends in rank one, and the torsor identity makes their multiplication maps invertible. This gives a natural equivalence between fpqc \(D_X(Q)\)-torsors and Picard-strong \(Q\)-gradings, and represents every torsor by a finite locally free morphism of rank \(|Q|\). The same viewpoint proves that finite-index monomial maps of split tori are finite locally free torsors and yields matrix Kummer classification sequences. A concrete example is the nontrivial Boolean torsor \(\Spec\mathbb B[v^{\pm1}]\to\Spec\mathbb B[u^{\pm1}]\), \(u\mapsto v^m\), whose unit-power classes contribute \(\mathbb Z/m\mathbb Z\) to degree-one cohomology. A two-term resolution of the character group then gives a presentation-independent universal-coefficient filtration for diagonalizable fpqc cohomology. This formalism organizes, but does not compute, \(\mathbf G_{\mathrm m}\)-cohomology. In degree two it identifies the Kummer boundary of a line bundle with its root gerbe. On projective space over the Boolean or real tropical semifield all \(\mu_m\)-torsors vanish, whereas the root gerbe of \(\mathcal O(1)\) has exact order \(m\). Thus torsors and gerbes retain information that can disappear under ordinary ring completion.