Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture.

*(English)*Zbl 1189.13008The authors study the graded Betti numbers of Cohen-Macaulay algebras and modules. In this direction, they introduce the Betti diagrams and the pure diagrams as two essential structures in their study. From a general point of view, the main idea in the paper under review is that the Betti diagrams are linear combinations of pure diagrams by positive rational numbers, see conjecture (2.4). In addition, the authors conjecture that all the pure diagrams in such expansions can be chosen from the same chain, see conjecture (2.10). One of the main motivations for proposing the above conjectures come from the multiplicity conjecture of Herzog, Huneke and Srinivasan in [C. Huneke and M. Miller, Can. J. Math. 37, 1149–1162 (1985; Zbl 0572.13004)] and [J. Herzog and H. Srinivasan, Trans. Am. Math. Soc. 350, No. 7, 2879–2902 (1998; Zbl 0899.13026)].

Let \(R=k[x_1,x_2,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(k\) and \(A=\frac{R}{J}\) for some ideal \(J\) of \(R\), and let \(M\) be a graded Cohen-Macaulay \(R\)-module.

In Section 3, Theorem (3.4), it is shown that the above conjectures are true when \(M\) is of codimension one, generated in any degrees, or when it is of codimension two, generated in a single degree. As an important corollary of this theorem, it is shown that the multiplicity conjecture holds for Cohen-Macaulay \(R\)-modules of codimension two which is generated in a single degree.

In Section 4, the authors by using nice techniques prove that when \(A\) is complete intersection (\(J\) is generated by an \(R\)-regular sequence) or when it is Gorenstein of codimension three (\(J\) is a height-three Gorenstein ideal in \(R\)), then conjecture (2.4) holds for \(A\) as an \(R\)-module.

Let \(R=k[x_1,x_2,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(k\) and \(A=\frac{R}{J}\) for some ideal \(J\) of \(R\), and let \(M\) be a graded Cohen-Macaulay \(R\)-module.

In Section 3, Theorem (3.4), it is shown that the above conjectures are true when \(M\) is of codimension one, generated in any degrees, or when it is of codimension two, generated in a single degree. As an important corollary of this theorem, it is shown that the multiplicity conjecture holds for Cohen-Macaulay \(R\)-modules of codimension two which is generated in a single degree.

In Section 4, the authors by using nice techniques prove that when \(A\) is complete intersection (\(J\) is generated by an \(R\)-regular sequence) or when it is Gorenstein of codimension three (\(J\) is a height-three Gorenstein ideal in \(R\)), then conjecture (2.4) holds for \(A\) as an \(R\)-module.

Reviewer: Tirdad Sharif (Tehran)

##### MSC:

13C14 | Cohen-Macaulay modules |

13D02 | Syzygies, resolutions, complexes and commutative rings |

13H15 | Multiplicity theory and related topics |