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Modules with pure resolutions. (English) Zbl 1407.13001

Let \(R\) be a standard graded \(K\)-algebra (\(K\) an infinite field) and \(M\) a graded \(R\)-module of finite projective dimension \(p\). Let \(\beta_{i,j}\) be the \((i,j)\)-th graded Betti number of \(M\) over \(R\). The authors call \(M\) to satisfy the Herzog-Kühl equations if for \(l = 0,\dots,p-1\), \[ \sum (-1)^ij^l\beta_{i,j}=0. \] In Theorem 3.6, the authors prove that if (in the above setting) \(R\) is Cohen-Macaulay, then \(M\) is Cohen-Macaulay if and only if \(M\) satisfies the Herzog-Kühl equations.
This result extends a result by Boij-Söderberg result [M. Boij and J. Söderberg, J. Lond. Math. Soc., II. Ser. 78, No. 1, 85–106 (2008; Zbl 1189.13008)] wherein \(R\) is a polynomial ring.
Another interesting fact in this paper is Theorem 3.9 where the authors show that: If (in the above setting) \(M\) is pure of type \((d_0,\dots,d_p)\), then \(M\) satisfies the Herzog-Kühl equations if and only if codim\((M)=p\).
As applications, the authors show that the betti numbers of a prefect ideal of codimension \(2\) is the same as a power of a regular sequence of length \(2\). As well in Theorem 4.6 they give a criterion for Gorensteinness of ideals with pure resolution, when \(R\) is a polynomial ring.

MSC:

13A02 Graded rings
13C14 Cohen-Macaulay modules
13D02 Syzygies, resolutions, complexes and commutative rings

Citations:

Zbl 1189.13008

Software:

BoijSoederberg
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Full Text: DOI arXiv

References:

[1] Boij, M. Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, J. Lond. Math. Soc., 78, 1, 85-106, (2008) · Zbl 1189.13008
[2] Boij, M.; Söderberg, J., Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case, Algebra Number Theory, 6, 3, 437-454, (2012) · Zbl 1259.13009
[3] Bruns, W.; Herzog, J., Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, 39, (1999), Cambridge University Press, Cambridge
[4] Eisenbud, D.; Schreyer, D., Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc., 22, 3, 859-888, (2009) · Zbl 1213.13032
[5] Herzog, J.; Kühl, M., On the Betti numbers of finite pure and linear resolutions, Commun. Algebra, 12, 1627-1646, (1984) · Zbl 0543.13008
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