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

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 $S$.