Quotients of Koszul Algebras and 2-d-Determined Algebras

Abstract

Abstract. Vatne [12] and Green & Marcos [8] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green & Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.

The Ext algebra of a Brauer graph algebra

“…Moreover, for a 2-d-homogeneous Brauer graph algebra A Γ with Brauer graph Γ, we show in Theorem 9.6, that the following four conditions are equivalent: (1) Γ has no truncated edges, (2) A Γ is 2-d-determined, (3) A Γ is 2-d-Koszul, and (4) the Ext algebra of A Γ is generated in degrees 0, 1 and 2 (that is, A Γ is K 2 ). It should be noted that these properties are not, in general, equivalent, as is demonstrated by Cassidy and Phan in [6].…”

“…In contrast to Theorem 9.6, a negative answer was given by Cassidy and Phan to the first two questions posed by Green and Marcos in [10]. In [6], Cassidy and Phan give specific infinite-dimensional algebras A and B such that A is 2-4-determined but E(A) is not finitely generated, and B is 2-4-determined of infinite global dimension, E(B) is finitely generated, but E(B) is not generated in degrees 0, 1 and 2. Another generalization of Koszul is given by Herscovich and Rey in [14], where they study multi-Koszul algebras.…”

“…It is well-known that all the Brauer graph algebras in Proposition 2.2(1), (2), ( 5), (6) and in Proposition 2.3 are d-Koszul (see for instance [8,9,21] and the references therein). However, it is easy to verify that the algebras of Proposition 2.2(3) and (4) are not Koszul, since the resolution of the simple module at the vertex 1 is not linear in either of these cases.…”