TAILIEUCHUNG - Invariants of symmetric algebras associated to graphs

In this work we deal with the symmetric algebra of monomial ideals that arise from graphs, the edge ideals. The notion of s-sequence is explored for such ideals in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs. | Turk J Math 36 (2012) , 345 – 358. ¨ ITAK ˙ c TUB doi: Invariants of symmetric algebras associated to graphs Maurizio Imbesi, Monica La Barbiera Abstract In this work we deal with the symmetric algebra of monomial ideals that arise from graphs, the edge ideals. The notion of s -sequence is explored for such ideals in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs. Key Words: Edge ideals, symmetric algebra, s-sequence 1. Introduction In this article we study the symmetric algebra of monomial ideals ([1], [4]), in particular of some ideals arising from graphs. In order to compute standard invariants of such symmetric algebra, we investigate some cases for which the monomial ideals are generated by s-sequences. In [2] the notion of s-sequence is employed to compute the invariants of the symmetric algebra of finitely generated modules. Our proposal is to compute standard invariants of the symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graph. This computation can be obtained for finitely generated modules generated by an s-sequence. Let G be a graph with no cycles. An algebraic object attached to G is the edge ideal I(G) that is a monomial ideal of R = K[X1 , . . . , Xn ] , K a field, n the number of vertices of G . I(G) is generated by square-free monomials of degree two in the polynomial ring R, I(G) = {Xi Xj

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