On isomorphism conditions for algebra functors with applications to Leavitt path algebras

TL;DR

This paper investigates conditions under which algebra isomorphisms over field extensions imply isomorphisms over smaller fields, focusing on functors related to Leavitt path algebras and employing an extended Nullstellensatz.

Contribution

It establishes criteria for descending algebra isomorphisms from extended fields to base fields for extension invariant functors, including Leavitt path algebras.

Findings

Positive results for extension invariant functors

Extension of Nullstellensatz for infinitely many variables

Isomorphism over Hopf algebra implies universal isomorphism

Abstract

We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor $R\mapsto L_R(E)$ for a given graph $E$. Our goal is to find "descending" isomorphism results of the type: if $\mathfrak{F},\mathcal{G}$ are algebra functors and $K\subset K'$ a field extension, under what conditions an isomorphism $\mathfrak{F}(K')\cong \mathcal{G}(K')$ of $K'$-algebras implies the existence of an isomorphism $\mathfrak{F}(K)\cong\mathcal{G}(K)$ of $K$-algebras? We find some positive answers to that problem for the so-called "extension invariant functors" which include the functors associated to Leavitt path algebras, Steinberg algebras, path algebras, group algebras, evolution algebras and others. For our purposes, we employ an extension of the Hilbert's Nullstellensatz Theorem for polynomials in possibly infinitely many variables, as one of our main tools. We also remark that for extension invariant functors $\mathfrak{F},\mathcal{G}$, an isomorphism $\mathfrak{F}(H)\cong\mathcal{G}(H)$, for some Hopf $K$-algebra $H$, implies the existence of an isomorphism $\mathfrak{F}(S)\cong\mathcal{G}(S)$ for any commutative and unital $K$-algebra $S$.