Ananthnarayan, H.; Kumar, Rajiv Modules with pure resolutions. (English) Zbl 1407.13001 Commun. Algebra 46, No. 7, 3155-3163 (2018). 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. Reviewer: S. H. Hassanzadeh (Rio de Janeiro) Cited in 2 Documents MSC: 13A02 Graded rings 13C14 Cohen-Macaulay modules 13D02 Syzygies, resolutions, complexes and commutative rings Keywords:Cohen-Macaulay defect; graded Betti numbers; Herzog-Kühl equations; pure modules; Boij-Söderberg Citations:Zbl 1189.13008 Software:BoijSoederberg PDFBibTeX XMLCite \textit{H. Ananthnarayan} and \textit{R. Kumar}, Commun. Algebra 46, No. 7, 3155--3163 (2018; Zbl 1407.13001) 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.