Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T04:05:31.431Z Has data issue: false hasContentIssue false

On multigraded resolutions

Published online by Cambridge University Press:  24 October 2008

Winfried Bruns
Affiliation:
Universität Osnabrück, Standort Vechta. D-49364 Vechta, Germany
Jürgen Herzog
Affiliation:
Universität Essen, FB Mathematik und Informatik, D-45117 Essen, Germany

Extract

This paper was initiated by a question of Eisenbud who asked whether the entries of the matrices in a minimal free resolution of a monomial ideal (which, after a suitable choice of bases, are monomials) divide the least common multiple of the generators of the ideal. We will see that this is indeed the case, and prove it by lifting the multigraded resolution of an ideal, or more generally of a multigraded module, keeping track of how the shifts ‘deform’' in such a lifting; see Theorem 2·1 and Corollary 2·2.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aramova, A., Barcanescu, S. and Herzog, J.. On the rate of relative Veronese submodules, preprint.Google Scholar
[2]Avramov, L. L. and Golod, E.. On the homology of the Koszul complex of a local Gorenstein ring. Math. Notes 9 (1971), 3032.CrossRefGoogle Scholar
[3]Backelin, J.. On the rates of growth of the homologies of Veronese subrings; in Roos, J.-E. (ed.). Algebra, Algebraic Topology and Their Interactions. Lecture Notes in Math. No. 1183 (Springer-Verlag, 1986), 79100.CrossRefGoogle Scholar
[4]Bruns, W. and Herzog, J.. Cohen-Macaulay Rings. (Cambridge University Press, 1993).Google Scholar
[5]Buchsbaum, D. A. and Eisenbud, D.. Algebra structures for finite free resolutions, and some structure theorems for ideals in codimension 3. Amer. J. Math. 99 (1977), 447485.CrossRefGoogle Scholar
[6]Buchsbaum, D. A. and Eisenbud, D.. Some structure theorems for finite free resolutions. Adv. in Math. 12 (1974), 84139.CrossRefGoogle Scholar
[7]Cartan, H. and Eilenberg, S.. Homological algebra. (Oxford University Press, 1973).Google Scholar
[8]Eisenbud, D.. Commutative algebra with a view towards algebraic geometry. A book in preparation, to be published by Springer.Google Scholar
[9]Eisenbud, D., Reeves, A. and Totaro, B.. Initial ideals, Veronese subrings, and rates, preprint.Google Scholar
[10]Hibi, T.. Betti number sequences of simplicial complexes, Cohen—Macaulay types and Möbius functions of partially ordered sets, and related topics, preprint.Google Scholar
[11]Hochster, M.. Cohen-Macaulay rings, combinatorics, and simplicial complexes; in Mcdonald, B. R. and Morris, R. A. (eds.), Ring theory II. Lecture Notes in Pure and Applied Math. 26 (M. Dekker, 1977), pp. 171223.Google Scholar
[12]Kleinschmidt, P.. Sphären mit wenigen Ecken. Geometriae Dedicata 5 (1976), 307320.CrossRefGoogle Scholar
[13]Mani, P.. Spheres with a few vertices. J. Combinat. Theory Ser. A 13 (1972), 346352.CrossRefGoogle Scholar
[14]Stanley, R. P.. Combinatorics and commutative algebra. (Birkhäuser, 1983).CrossRefGoogle Scholar
[15]Stückrad, J. and Simon, S.. Gorenstein ideals with a matroidal property, preprint.Google Scholar
[16]Taylor, D.. Ideals generated by monomials in an R-sequence. PhD Thesis (University of Chicago, 1966).Google Scholar