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hexagon:2609.00112v1

math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere Lnp,fSL_n^{p,f}\mathbb S. For a finite abelian ambient group, subgroups of pp-rank at most n+1n+1 suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava EE-theory and gives a new proof of their chromatic upper bound for the algebraic KK-theory of Lnp,fSL_n^{p,f}\mathbb S-linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Primarily AI-generated textHuman understanding: some partsBurnside ringsSwan inductionchromatic homotopy theory
Sep 27, 2026PDF ↗View work →
hexagon:2609.00111v1

math.AG — Algebraic Geometry

Torsion in K1K_1 and pp-adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime pp, we construct a regular strictly henselian local ring AA whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order pp in SK1(A[1/p])SK_1(A[1/p]), detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues 44 and 88 of ψ2\psi^2 are needed.

Primarily AI-generated textHuman understanding: all partsAdams operationsMilnor K-theoryp-adic vanishing cycles
Sep 27, 2026PDF ↗View work →
hexagon:2609.00110v1

math.AG — Algebraic Geometry

Frobenius orbit maps and rank-four group schemes

Contributed by Akhil Mathew

We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is (x,y)↦(2bxy,2axy)(x,y)\mapsto(2bxy,2axy), and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over Z[a,b]/(a2b−2,a3,b3)\Z[a,b]/(a^2b-2,a^3,b^3). We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies (2a,2b)(2a,2b) as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.

Primarily AI-generated textHuman understanding: some partsHopf algebrasgroup schemes
Sep 27, 2026PDF ↗View work →
hexagon:2609.00105v1

math.AC — Commutative Algebra

Graded absolute integral closures and Picard groups

Contributed by Akhil Mathew

We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over Zp\Z_p, modulo pp, is free with basis degrees smaller than the sum of the weights. A direct pp-adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring RR admits a faithfully flat algebra with seminormal reduction, and every invertible R[T]R[T]-module becomes free after a faithfully flat extension of RR, answering a question of Drinfeld.

Primarily AI-generated textHuman understanding: all partsPicard groupsabsolute integral closureseminormality
Sep 26, 2026PDF ↗View work →