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The semigroup of Betti diagrams. (English) Zbl 1173.13013
Let \(S\) be the polynomial ring \(k[x_1,\dots,x_n],\) where \(k\) is any field. For a finitely generated graded \(R\)-module \(M\) let \(F_i = \bigoplus_j S(-j)^{\beta_{i,j}(M)}\) denote the \(i\)-th free module in a minimal free resolution of \(M\) as \(S\)-module. That is \(\beta_{i,j}(M) = \text{Tor}_S^i(k,M)_j.\) Write \(\beta(M)\) for the Betti diagram of \(M\) considered as an element of the vector space \(\bigoplus_j \bigoplus_{i=0}^p \mathbb Q, p = \text{projdim} M,\) with coordinates \(\beta_{i,j}(M).\) The image of the set of finitely generated graded \(S\)-modules forms in \(\bigoplus_j \bigoplus_{i=0}^p \mathbb Q\) a subsemigroup. Furthermore, the restriction to any subsemigroup of \(S\)-modules is also a semigroup. A degree sequence is an integral vector \(d = (d_0,\dots,d_p) \in \mathbb N^{p+1},\) where \(d_i < d_{i+1}.\) Fix two degree sequences \(\underline{d}, \overline{d}\) of length \(p\) such that \(\underline{d}_i \leq \overline{d}_i\) for all \(i.\) The author considers the semigroup \(\mathcal Z\) of graded \(S\)-modules \(M\) such that (1) \(M\) is of projective dimension \(\leq p,\) and (2) the Betti number \(\beta_{i,j}(M)\) is nonzero only if \(i \leq p\) and \(\underline{d}_i \leq j \leq \overline{d}_i.\) The main objects of the author’s investigations are (1) \(B_{mod } = B_{mod }(\underline{d},\overline{d}) = \text{Im} \beta |_{\mathcal Z}\), (2) \(B_{\mathbb Q},\) the positive rational cone over the semigroup of Betti diagrams, and (3) the semigroup of virtual Betti diagrams \(B_{\mathbb N}\) defind as the semigroup of lattice points in \(B_{\mathbb Q}.\)
The author’s main results are summarized as follows: (1) The semigroup of Betti diagrams \(B_{mod }\) is finitely generated. (2) \(B_{\mathbb N} = B_{mod }\) for projective dimension 1 and for projective dimension 2 level modules. (3) By examples the author shows that for projective dimension grater than 2 the semigroups \(N_{\mathbb N}\) and \(B_{mod }\) diverge. In particular, \[ E_{\alpha} = \begin{pmatrix} 2 + \alpha & 3 & 2 & - \\ - & 5+6\alpha & 7+8\alpha & 3+3\alpha \end{pmatrix} \] is not the Betti diagram of a module for any \(\alpha \in \mathbb N,\) while each \(E_{\alpha}\) belongs to the cone of Betti diagrams. The author’s investigations give a deep insight into the fine structure of Betti diagrams. They are based on the recent work of M. Boij and J. Söderberg [J. Lond. Math. Soc., II. Ser. 78, No. 1, 85–106 (2008; Zbl 1189.13008) and “Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case”, arXiv:0803.1645], D. Eisenbud, G. Floystadt and J. Weyman [“Betti numbers of graded modules and cohomology of vector bundles”, arXiv:0712.1843] and D. Eisenbud and F.-O. Schreyer [“Cohomology of coherent sheaves and series of supernatural bundles”, arXiv:0902.1594].

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13D25 Complexes (MSC2000)
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