Category equivalences involving graded modules over weighted path algebras and weighted monomial algebras (Q401931)

Let \(k\) be any field, \(A\) an \(\mathbb N\)-graded \(k\)-algebra. Let \(\text{Gr}A\) denote the category with objects and morphisms the graded \(A\)-modules and (degree-preserving) homomorphisms respectively. Let \(\text{Fdim}A\subseteq\text{Gr}A\) be the localizing subcategory of modules that are the direct sum of their finite dimensional submodules, and let \(\text{QGR}A\) denote the quotient with \(\pi^\ast:\text{GR}A\rightarrow\text{QGR}A\) the canonical quotient functor. The right adjoint of \(\pi^\ast\) is denoted \(\pi_\ast\). The shifting functor determines an auto-equivalence \((1)\) on \(\text{Gr}A\) which descends to an auto-equivalence on \(\text{QGr}A\).NEWLINENEWLINEThis article investigates, generalizes and give answers to the following question: Let \(F=k\langle x_1,\dots,x_n\rangle\). What does \(\text{QGr}F\) look like?NEWLINENEWLINEThe authors prove that \(\text{QGr}F\equiv\text{QGr}kQ\) where \(kQ\) is the path algebra of a finite quiver \(Q\) with each arrow of degree \(1\) and and defined by: Start by viewing \(F\) as the path algebra of the weighted quiver with one vertex and \(n\) loops of degrees \(\deg(x_i)\). If \(\deg(x_i)>1\), replace the loop for \(x_i\) by \(\deg(x_i)-1\) vertices and a cycle thorugh them consisting of \(\deg(x_i)\) arrows each of degree \(1\). Then the answer to the main question and a result of S. P. Smith, says that \(\text{QGr}F\equiv\text{Mod}S\) where \(S\) is ultramatricial, hence a von Neumann regular, algebra.NEWLINENEWLINEThe generalizations is given by looking to the following classes of finitely presented \(\mathbb N\)-graded \(k\)-algebras: \textbf{PA1}: Path algebras of finite quivers with grading induced by declaring that all arrows have degree 1. \textbf{WPA}: Weighted path algebras of finite quivers (each arrow is given a grading \(\geq 1\)). \textbf{MA}: Monomial algebras, i.e. algebras of the form \(kQ/I\) where \(kQ\) is a weighted path algebra of a finite quiver and \(I\) is an ideal generated by a finite set of paths. \textbf{CMA}:Connected monomial algebras, monomial algebras \(kQ/I\) in which \(Q\) has only one vertex. \textbf{CMA1}: Connected monomial algebras that are generated by elements of degree \(1\).NEWLINENEWLINEA main result of the article is the following Theorem, enabling the frequent use of the shift equivalence: If \(C\) and \(C^\prime\) are two of the five classes above, and if \(A\) is an algebra in \(C\), then there is an algebra \(A^\prime\) in \(C^\prime\) and an equivalence \(F:\text{QGr}A\rightarrow\text{QGr}A^\prime\) such that \(F(M(1))\simeq F(M)(1)\) for all \(M\in \text{QGr}A\).NEWLINENEWLINEThe five classes of algebras are chosen because free algebras are universal objects in the category of commutative \(k\)-algebras. Monomial algebras can be understood through the combinatorics of words, and monomial relations are the simples relations. Monomial algebras often have infinite global dimension, so are not so nice for homological arguments. Path algebras have global dimension \(\leq 1\) so although \(\text{Gr}A\) is homologically wild, \(\text{QGr}A\) is not. Finally, many technicalities can be avoided by assuming the algebra is generated by elements of degree \(1\).NEWLINENEWLINENew contributions of this article is that \(\text{QGr}(CMA)\subset\text{QGr}(WPA)\) and that \(\text{QGr}(WPA)\subset\text{QGr}(PA1)\) completing the proof of the main theorem above.NEWLINENEWLINEIt should be noted that the proof that \(\text{QGR}(CMA)\subset\text{QGR}(WPA)\) follows the proof of Holdaway and Smith, associating to \(A\) in CMA a weighted quiver \(Q=Q(A)\). This quiver is the weighted Ufnarovskii graph of \(A\), and it shows there is a homomorphism of graded algebras \(A\rightarrow kQ\). The construction of the Ufnarovskii graph is given a detailed construction in the article. We also notice that the authors identifies the categories \(\text{QGR}kQ\) and \(\text{GrRep}kQ\) as these categories are equivalent.NEWLINENEWLINEThe article is very well written, easy to read, and rather self contained. It has interesting results, and is a nice illustration of category theory combined with representation theory.