On isomorphism conditions for algebra functors with applications to Leavitt path algebras (Q6095349)

For a commutative unital ring \(K\), the category of associative and commutative \(K\)-algebras with non-zero unit, \(\mathrm{alg}_K\); and the category of all \(K\)-algebras \(\mathrm{Alg}_K\); a \(K\)-algebra functor \(F\) is defined as \(F :\mathrm{alg}_K \rightarrow \mathrm{Alg}_K\) such that \(F(R)\) is an \(R\)-algebra for any \(R \in \mathrm{alg}_K\). This paper under review tries to answer the question: if \(F, G\) are algebra functors and \(K \subset K'\) a field extension, under what conditions an isomorphism \(F(K') \cong G (K')\) of \(K'\)-algebras implies the existence of an isomorphism \(F(K) \cong G (K)\) of \(K\)-algebras? (So called, ``descending'' isomorphism condition.) \(F\) is said to be extension invariant if for every \(R \in \mathrm{alg}_K\) there is an \(R\)-algebra isomorphism \(\tau_R : F(R) \rightarrow F(K) \otimes_K R \in \mathrm{Alg}_K\) which is natural in \(R\). In Definition 2.1, for a given \(K\)-algebra \(A \in \mathrm{Alg}_K\), a special algebra functor \(\underline{A}\) is defined as mapping \(R\) into \(A \otimes_K R\). Proposition 2.3 and Remark 2.4 points out that the algebra functor \(\underline{A}\) is extension invariant, furthermore, all extension invariant functors are precisely of the form \(\underline{A}\) for some \(A \in\mathrm{Alg}_K\). Corollary 2.9 in this section gives the main result related to the isomorphism of Leavitt path algebras: For any two graphs \(E_1\), \(E_2\) and any \(K\)-algebra \(H\) endowed with an augmentation, then \(L_H(E_1)\cong L_H(E_2)\) if and only if \(L_K(E_1) \cong L_K(E_2)\). Section 2 also include the functors associated with Leavitt path algebras, Steinberg algebras, path algebras, group algebras, evolution algebras and similar results. In Section 3, the paper introduces the technique of extending the Hilbert's Nullstellensatz Theorem for polynomials in possibly infinitely many variables as one of the main tools to generalize the extension invariant functor to other algebraic structures. Theorems in Section 4 and 5 give the result that if \(K \subset F\) is a field extension with \(K\) algebraically closed, then for any \(F, G\) two extension invariant algebra functors, \(F(K) \cong G (K)\) if and only if \(F(F)\cong G (F)\). (Theorem 4.8 assumes \(F(K)\) and \(G (K)\) are finite-dimensional as \(K\)-algebras, Theorem 5.6 assumes infinite dimensional with cardinality of \(K\) greater than the dimension of algebras \(F(K)\) and \(G (K)\)).