Arithmetical rank of squarefree monomial ideals of small arithmetic degree
Kyouko Kimura1
, Naoki Terai2
and Ken-ichi Yoshida1
1Nagoya University Graduate School of Mathematics Nagoya 464-8602 Japan
2Saga University Department of Mathematics, Faculty of Culture and Education Saga 840-8502 Japan
2Saga University Department of Mathematics, Faculty of Culture and Education Saga 840-8502 Japan
DOI: 10.1007/s10801-008-0142-3
Abstract
In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/ I in the following cases: (a) I is an almost complete intersection; (b) arithdeg\thinspace I=reg\thinspace I; (c) arithdeg\thinspace I=indeg\thinspace I+1.
We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).
Pages: 389–404
Keywords: keywords arithmetical rank; almost complete intersection; Alexander duality; regularity; arithmetic degree; initial degree
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References
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2. Barile, M.: A note on monomial ideals. Arch. Math. 87, 516-521 (2006)
3. Barile, M.: A note on the edge ideals of Ferrers graphs. Preprint,
4. Barile, M.: On the arithmetical rank of the edge ideals of forests. Preprint,
5. Barile, M.: On the arithmetical rank of certain monomial ideals. Preprint,
6. Barile, M.: Arithmetical ranks of Stanley-Reisner ideals via linear algebra. Preprint,
7. Frühbis-Krüger, A., Terai, N.: Bounds for the regularity of monomial ideals. Matematiche (Catania) 53, 83-97 (1998)
8. Hoa, L.T., Trung, N.V.: On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229, 519-537 (1998)
9. Lyubeznik, G.: On the local cohomology modules H ia(R) for ideals a generated by monomials in an R-sequence. In: Greco, S., Strano, R. (eds.) Complete Intersections, Acireale,
1983. Lecture Notes in Mathematics, vol. 1092, pp. 214-220. Springer, Berlin (1984)
10. Schenzel, P.: Applications of Koszul homology to numbers of generators and syzygies. J. Pure Appl. Algebra 114, 287-303 (1997)
11. Schenzel, P., Vogel, W.: On set-theoretic intersections. J. Algebra 48, 401-408 (1977)
12. Schmitt, T., Vogel, W.: Note on set-theoretic intersections of subvarieties of projective space. Math.
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