TY - JOUR

T1 - Homogeneous algebras via homogeneous triples

AU - Marcos, E.

AU - Volkov, Y.

N1 - Publisher Copyright: © 2020 Elsevier Inc.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - To study s-homogeneous algebras, we introduce the category of quivers with s-homogeneous corelations and the category of s-homogeneous triples. We show that both of these categories are equivalent to the category of s-homogeneous algebras. We prove some properties of the elements of s-homogeneous triples and give some consequences for s-Koszul algebras. Then we discuss the relations between the s-Koszulity and the Hilbert series of s-homogeneous triples. We give some application of the obtained results to s-homogeneous algebras with simple zero component. We describe all s-Koszul algebras with one relation recovering the result of Berger and all s-Koszul algebras with one dimensional s-th component. We show that if the s-th Veronese ring of an s-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all s-homogeneous algebras with s-th Veronese rings k〈x,y〉/(xy,yx) and k〈x,y〉/(x2,y2). In particular, we show that all of these algebras are not s-Koszul while their s-homogeneous duals are s-Koszul.

AB - To study s-homogeneous algebras, we introduce the category of quivers with s-homogeneous corelations and the category of s-homogeneous triples. We show that both of these categories are equivalent to the category of s-homogeneous algebras. We prove some properties of the elements of s-homogeneous triples and give some consequences for s-Koszul algebras. Then we discuss the relations between the s-Koszulity and the Hilbert series of s-homogeneous triples. We give some application of the obtained results to s-homogeneous algebras with simple zero component. We describe all s-Koszul algebras with one relation recovering the result of Berger and all s-Koszul algebras with one dimensional s-th component. We show that if the s-th Veronese ring of an s-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all s-homogeneous algebras with s-th Veronese rings k〈x,y〉/(xy,yx) and k〈x,y〉/(x2,y2). In particular, we show that all of these algebras are not s-Koszul while their s-homogeneous duals are s-Koszul.

KW - s-homogeneous algebra

KW - s-Koszul algebras

KW - Veronese ring

UR - http://www.scopus.com/inward/record.url?scp=85091066685&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2020.09.012

DO - 10.1016/j.jalgebra.2020.09.012

M3 - Article

AN - SCOPUS:85091066685

VL - 566

SP - 259

EP - 282

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -