Witt Groups of Stable Module Categories

Definitions and mathematical statement

Let $k$ be the intended field of characteristic $p>0$, and let $G$ be a finite group. Under the standard finite-dimensional module convention, $kG\text{-}\mathrm{stab}$ has the same objects as the category of finite-dimensional left $kG$-modules, with morphisms \[ \underline{\operatorname{Hom}}_{kG}(M,N) =\operatorname{Hom}_{kG}(M,N)\big/ \{\text{maps factoring through a projective module}\}. \] The group algebra is self-injective, giving this stable category its usual triangulated structure, with suspension represented by inverse syzygy. This is not the ordinary abelian module category or its entire derived category.

A usual duality is $D(M)=\operatorname{Hom}_k(M,k)$, with $(g\cdot f)(m)=f(g^{-1}m)$. A triangular Witt group requires a compatible triangulated duality, its biduality and sign conventions. In that framework $W^i(kG\text{-}\mathrm{stab},D)$ is formed from the appropriate shifted symmetric self-dual objects modulo neutral forms admitting the relevant Lagrangian triangles. It is not simply the Grothendieck group of all modules or an unshifted Witt group of arbitrary bilinear forms.

The question asks whether these Witt groups vanish for every finite $G$. If all shifted triangular Witt groups are intended, its schematic form is \[ \forall G\text{ finite}\ \forall i\in{\mathbb{Z}},\qquad W^i(kG\text{-}\mathrm{stab},D)=0, \] with $k,p$ and the Witt theory taken from the source's missing setup. The record does not specify whether it means all shifts or selected degrees, whether the standard duality above is intended, or whether $k$ is fixed or quantified over all fields of characteristic $p$.

The characteristic restriction needs care: familiar triangular Witt-group constructions often require multiplication by $2$ to be invertible, so characteristic $2$ needs the intended alternative definition or an explicit exclusion. The source says characteristic $p$ without making that distinction; no $p\ne2$ condition is silently added. No algebraic closedness, splitting-field, or divisibility assumption $p\mid|G|$ is supplied either.

The literature triage checked 2026-08-21 calls this a fragment about arbitrary $G$ with insufficient hypotheses and says the cited framework does not resolve the general vanishing question. The open label and editorial verification note remain source metadata, not independently checked status. Missing duality and characteristic conventions must be resolved before asserting one universal vanishing theorem.

Short English statement

Do the intended triangular Witt groups of every finite group's stable module category vanish in characteristic $p$? The stable category and a standard duality are described, but the source does not specify shifts, field restrictions or a characteristic-two theory. Those choices are essential to a definite vanishing statement.

Sources

• Local UnsolvedMath registry, permanent ID 30000520, source code catalogue entry, "Witt Groups of Stable Module Categories," including original and cleaned vanishing question and verification metadata.
• Quadratic Forms and Linear Algebraic Groups (2007), Oberwolfach Reports, pp. 1743--1794, DOI 10.4171/owr/2006/29: https://doi.org/10.4171/owr/2006/29. The title uses 2007, while the DOI and proposed-year metadata use 2006.
• Supplied reference "Balmer, triangular Witt groups": https://www.math.ucla.edu/~balmer/Pubfile/TWG-I.pdf. The URL is cited as supplied; no PDF has been opened.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.