arXiv subjects

Abhay Soman

Publications and source records attributed to Abhay Soman.

3 recordsLinked to original sources

arXiv2026

Totally positive field extensions and the pythagorean index

For a formally real field $F$, we study totally positive field extensions $K$ over $F$. We show that, if $K /F$ is Galois and totally positive then so is the corresponding extension of their pythagorean closures $K_{\rm py}$ over $F_{\rm py}$. We also study the behaviour of weak isotropy and weak hyperbolicity of central simple algebras with an orthogonal involution over totally positive field extensions. We use some of these results to prove new cases for which a conjecture due to Becher holds.

math.NT↗

arXiv2025

Weak isotropy of central simple algebras with orthogonal involutions over totally positive field extensions

In this paper, we explore the behavior of orthogonal involutions in the context of totally positive field extensions. Let $K/F$ be a totally positive extension of formally real fields. By Becher's result, if a quadratic form $q$ over $F$ becomes isotropic over $K$, then $q$ is weakly isotropic over $F$. We present an example in which, despite $K/F$ being totally positive, a central simple algebra $(A,\sigma)$ over $F$ with an orthogonal involution becomes isotropic over $K$, while remaining strongly isotropic over $F$. However, when $K/F$ is assumed to be a Galois totally positive $2$-extension of formally real fields, we show that an analogue of Becher's result for quadratic forms holds for orthogonal involutions. Furthermore, for a totally positive Galois field extension $K/F$, we verify Becher's conjecture for central division algebras of index $2^n$ and exponent $2$ containing a subfield of $F_{py}$ of degree $2^{n-2}$ over $F$.

math.RA↗

arXiv2025

The Serre-Swan Theorem in supergeometry

We show the analogue of the Serre-Swan theorem in a context of supergeometry. This theorem gives an equivalence of the category of locally free supersheaves of bounded rank over locally ringed superspace with the category of finitely generated super projective modules over its coordinate superring, under the assumptions that every locally free supersheaf is generated by global sections and it is acyclic.

math.AG↗