Generalized Koszul properties for augmented algebras

Abstract

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.Comment: 14 page

Quotients of Koszul Algebras and 2-d-Determined Algebras

“…This induces a filtration F on A, Tor A (k, k) and E(A). (See, for example, [10, Chapter IV] and [9]. )…”

A∞-algebra structures associated to K2 algebras

“…The algorithm can be used to show the algebra T (V )/ R is K 2 . Proposition 2.1 implies that R is an essential Gröbner basis for I (see [15]). By Theorem 3.13 in [15], the fact that T (V )/ R is K 2 implies B is K 2 .…”

“…Proposition 2.1 implies that R is an essential Gröbner basis for I (see [15]). By Theorem 3.13 in [15], the fact that T (V )/ R is K 2 implies B is K 2 . Though this method of proving B is K 2 is quite easy, we do not present the details here.…”

A Generalized Koszul Property for Skew PBW Extensions

“…In [8], it was defined the notion of generalized Koszul modules and Koszul algebras in a similar way to the classical case. Phan in [10] and [11] defined Koszul algebras for augmented algebras and R-augmented algebras.…”

“…In [22] we study the Koszul property (in the direction of Priddy) for skew PBW extensions over fields. Since the skew PBW extensions are rings, our interest in this paper is to study the generalized Koszul property for skew extensions, according to the definitions given by Li and Phan in [8] and [10], respectively.…”