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.

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.

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  • . . . . . . . . r = n− 3, (2 terms) e(R/(I+ . . . +Im+ . . . +In )) = 2, e(R/(I1+ . . . +m−2+m+ . . . + n−2)) = 1 r = n− 2, e(R/(I1+ . . . +I + . . . +I −2)) = 2 . Hence e(SymR(I(G))) = m(m− 1) + m(n− 2m) + Herzog, J., Hibi, T.: Monomial Ideals, Springer, Graduate Texts in Mathematics 260 (2011).
  • Herzog, J., Restuccia, G., Tang, Z.: s -sequences and symmetric algebras, Manuscripta Math., 104, 479–501 (2001).
  • Imbesi, M., La Barbiera, M.: Edge ideals and connection problems, Comm. in Appl. and Industr. Maths., 1(2), –134 (2010).
  • K¨uhl, M.: On the symmetric algebra of an ideal, Manuscripta Math., 37, 49–60 (1982).
  • Tang, Z.: On certain monomial sequences, J. of Algebra, 282, 831–842 (2004).
  • Villarreal, R.H.: Monomial Algebras, Dekker, Pure and Applied Mathematics 238 (2001). Maurizio IMBESI
  • Faculty of Sciences, Department of Mathematics, University of Messina Viale Ferdinando Stagno d’Alcontres, 31 I 98166 Messina-ITALY e-mail: imbesim@unime.it Monica LA BARBIERA
  • Faculty of Sciences, Department of Mathematics, University of Messina Viale Ferdinando Stagno d’Alcontres, 31 I 98166 Messina-ITALY e-mail: monicalb@dipmat.unime.it