s-homogeneous algebras via s-homogeneous triples

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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.