On isomorphism conditions for algebra functors with applications to Leavitt Path Algebras

Abstract

We introduce certain functors from the category of commu- tative rings (and related categories) to that of Z-algebras (not neces- sarily 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 condi- tions an isomorphism F (K ) ∼ = G (K ) of K -algebras implies the exis- tence of an isomorphism F (K) ∼ = G (K) of K-algebras? We find some positive answers to that problem for the so-called “extension invari- ant 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 exten- sion 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.