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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 118, Issue 2
  • September 1995, pp. 245-257

On multigraded resolutions

  • Winfried Bruns (a1) and Jürgen Herzog (a2)
  • DOI:
  • Published online: 24 October 2008

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.

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[2]L. L. Avramov and E. Golod . On the homology of the Koszul complex of a local Gorenstein ring. Math. Notes 9 (1971), 3032.

[3]J. Backelin . On the rates of growth of the homologies of Veronese subrings; in J.-E. Roos (ed.). Algebra, Algebraic Topology and Their Interactions. Lecture Notes in Math. No. 1183 (Springer-Verlag, 1986), 79100.

[5]D. A. Buchsbaum and D. Eisenbud . Algebra structures for finite free resolutions, and some structure theorems for ideals in codimension 3. Amer. J. Math. 99 (1977), 447485.

[6]D. A. Buchsbaum and D. Eisenbud . Some structure theorems for finite free resolutions. Adv. in Math. 12 (1974), 84139.

[12]P. Kleinschmidt . Sphären mit wenigen Ecken. Geometriae Dedicata 5 (1976), 307320.

[13]P. Mani . Spheres with a few vertices. J. Combinat. Theory Ser. A 13 (1972), 346352.

[14]R. P. Stanley . Combinatorics and commutative algebra. (Birkhäuser, 1983).

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