On Isomorphism Conditions for Algebra Functors with Applications to Leavitt Path Algebras

• Published in: Mediterranean journal of mathematics 2023-10, Vol.20 (5), Article 273
• Main Authors: Gil Canto, Cristóbal, Martín Barquero, Dolores, Martín González, Cándido, Ruiz Campos, Iván
• Format: Article
• Language: English
• Subjects: Algebra; Commutativity; Invariants; Isomorphism; Mathematical analysis; Mathematics; Mathematics and Statistics; Polynomials; Rings (mathematics)
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-023-02475-2

We introduce certain functors from the category of commutative rings (and related categories) to that of Z -algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor R ↦ L R ( E ) for a given graph E . Our goal is to find “descending” isomorphism results of the type: if F , G are algebra functors and K ⊂ K ′ a field extension, under what conditions an isomorphism F ( K ′ ) ≅ G ( K ′ ) of K ′ -algebras implies the existence of an isomorphism F ( K ) ≅ 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 with 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 F , G , an isomorphism F ( H ) ≅ G ( H ) , for some K -algebra H endowed with an augmentation, implies the existence of an isomorphism F ( S ) ≅ G ( S ) for any commutative and unital K -algebra S .