The Ext-algebra and associated monomial algebra of a finite dimensional \(K\)-algebra (Q6081711)

Let \(K\) denote an algebraically closed field with characteristic 0, \(Q\) denote a finite quiver, (I) denote an admissible homogeneous ideal of the path algebra \(KQ\), \(A=KQ/I\) denote a graded finite dimensional \(K\)-algebra, and \(E(A)\) denote the Ext-algebra of \(A\). It is well-known that \(E(A)\) is a finite dimensional \(K\)-algebra if and only if the global dimension of \(A\) is finite. However, the inquiry into the conditions under which \(E(A)\) is finitely generated as a \(K\)-algebra remains unresolved. When \(A\) is a monomial algebra, \textit{G. Davis} [J. Algebra 310, No. 2, 526--568 (2007; Zbl 1165.16005)] for cyclic algebras and \textit{A. Conner} et al. [J. Pure Appl. Algebra 218, No. 1, 52--64 (2014; Zbl 1309.16009)] for local algebras have addressed this question. It is known that \(A\) is D-Koszul if and only if \(E(A)\) is generated in degrees less than or equal to 2. If the defining ideal is generated by elements of length 2 and d, \(A\) is referred to as a 2-d-determined algebra. When \(E(A)\) is generated in degrees 0, 1, and 2, \(A\) is referred to as a \(\mathcal{K}_2\) algebra. 2-d-determined algebras and \(\mathcal{K}_2\) algebras are generalizations of D-Koszul algebras. There is another unresolved question as to when 2-d-determined algebras are \(\mathcal{K}_2\). In the paper under review, the author has partially addressed this question. A family of elements in \(KQ\) that yields a projective resolution of the top of \(A\) is referred to as the AGS resolution. In the main theorem of this paper, the author has shown that when the AGS resolution is minimal, if \(E(A_{MON})\) is generated in degrees \(0, 1, \ldots, m\) for some \(m\), then \(E(A)\) is also generated in degrees \(0, 1,\dots, m\). Specifically, if \(E(A_{MON})\) is finitely generated, then \(E(A)\) is finitely generated. By using this result, the author has also demonstrated that if \(A\) is a 2-d-determined algebra such that the AGS resolution of the top of \(A\) is minimal, then \(E(A)\) is generated in degrees \(0, 1, 2\). Through the provision of an example, the author has shown that the minimality of the AGS resolution cannot be removed.