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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)
##### Keywords:
Boij-Söderberg theory; Betti diagrams
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