Abstract
For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.
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References
N. Deo,Graph Theory with Applications to Engineering and Computer Science, Prentice Hall series in Automatic Computation, 1974
J. A. Eagon and M. Hochster, R-sequences and indeterminates, Quart. J. Math. Oxford25 (1974), 61–71
R. Fröberg, A study of graded extremal rings and of monomial rings, Math. Scand.51 (1982), 22–34
F. Harary,Graph Theory, Addison-Wesley, Reading, MA, 1972
J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing up rings, Proc. Trento Commutative Algebra Conf., Lectures Notes in Pure and Applied Math., vol.84, Dekker, New York, 1983, 79–169
J. Herzog, A. Simis and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc.283 (1984), 661–683
C. Huneke and M. E. Rossi, The dimension and components of symmetric algebras, J. Algebra98 (1986), 200–210
I. Kaplansky,Commutative Rings, University of Chicago Press, Chicago, 1974
H. Matsumura,Commutative Algebra, Benjamin/Cummings, Reading, MA, 1980
L. Mirsky and H. Perfect, Systems of representatives, J. Math. Anal. Applic.15 (1966), 520–568
M. D. Plummer, On a family of line-critical graphs. Monatsh. Math.71 (1967), 40–48
G. Relsner, Cohen-Macaulay quotients of polynomial rings, Adv. Math.21 (1976), 31–49
A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math.103 (1981), 203–224
A. Simis and W. V. Vasconcelos, Krull dimension and integrality of symmetric algebras, Manuscripta Math.61 (1988), 63–75
R. Stanley,Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983
D. Taylor, Ideals generated by monomials in an R-sequence, Thesis, University of Chicago, 1966
W. V. Vasconcelos, Koszul homology and the structure of low codimension Cohen-Macaulay ideals, Trans. Amer. Math. Soc.301 (1987), 591–613
W. V. Vasconcelos, Symmetric Algebras, Preprint, 1988
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Partially supported by SNI-SEP, COFAA-IPN and CONACyT.
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Villarreal, R.H. Cohen-macaulay graphs. Manuscripta Math 66, 277–293 (1990). https://doi.org/10.1007/BF02568497
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DOI: https://doi.org/10.1007/BF02568497