Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1610.06321

Involutions and stable subalgebras

Karim Johannes Becher·Nicolas Grenier-Boley·Jean-Pierre Tignol
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a central simple algebra with involution over an arbitrary field, étale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre.

Explore related subjects

math.KTmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol. 2017-10-19. Involutions and stable subalgebras. https://doi.org/10.1016/j.jalgebra.2017.09.026

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra

Let $A$ be a finite dimensional algebra and let $\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\calN$ (resp. $G$, $\calG$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\rmHH^*(A)/\calN$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\rmHH^*(A)/G$ is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\calN$; hence $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ is not a finitely generated algebra. Furthermore, we show that $\rmHH^*(A)/\calG\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\rmHH^*(A)/\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.

math.KT↗
arXiv2026

K-theory of Matroids and Monoid Schemes

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

math.KT↗
arXiv2026

Improved injective stability for relative $\mathrm{K_1Sp}$-groups

We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields. As an application, we give a necessary and sufficient condition for the freeness of stably free modules over smooth real $4$-folds with empty real locus.

math.KT↗