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Let $R=Bbbk[x_1,...,x_m]$ be the polynomial ring over a field $Bbbk$ with the standard $mathbb Z^m$-grading (multigrading), let $L$ be a Noetherian multigraded $R$-module, let $beta_{i,alpha}(L)$ the $i$th (multigraded) Betti number of $L$ of multidegree $a$. We introduce the notion of a generic (relative to $L$) multidegree, and the notion of multigraded module of generic type. When the multidegree $a$ is generic (relative to $L$) we provide a Hochster-type formula for $beta_{i,alpha}(L)$ as the dimension of the reduced homology of a certain simplicial complex associated with $L$. This allows us to show that there is precisely one homological degree $ige 1$ in which $beta_{i,alpha}(L)$ is non-zero and in this homological degree the Betti number is the $beta$-invariant of a certain minor of a matroid associated to $L$. In particular, this provides a precise combinatorial description of all multigraded Betti numbers of $L$ when it is a multigraded module of generic type.
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Let $R = mathbb{K}[x_1, ldots, x_n]$ and $I subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by homogeneous complete intersection, then subadditivity holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on subadditivity, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain several classes of graphs whose edge ideals are strand connected. Finally, we compute upper bounds for multigraded Betti numbers of several classes of edge ideals.
The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that if M is a non-zero module of finite length and finite projective dimension over a local ring R of dimension d, then the i-th Betti number of M is at least d choose i. This conjecture implies that the sum of all the Betti numbers of such a module must be at least 2^d. We prove the latter holds in a large number of cases.
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
Let $R=bigoplus_{underline{n} in mathbb{N}^t}R_{underline{n}}$ be a commutative Noetherian $mathbb{N}^t$-graded ring, and $L = bigoplus_{underline{n}inmathbb{N}^t}L_{underline{n}}$ be a finitely generated $mathbb{N}^t$-graded $R$-module. We prove that there exists a positive integer $k$ such that for any $underline{n} in mathbb{N}^t$ with $L_{underline{n}} eq 0$, there exists a primary decomposition of the zero submodule $O_{underline{n}}$ of $L_{underline{n}}$ such that for any $P in {rm Ass}_{R_0}(L_{underline{n}})$, the $P$-primary component $Q$ in that primary decomposition contains $P^k L_{underline{n}}$. We also give an example which shows that not all primary decompositions of $O_{underline{n}}$ in $L_{underline{n}}$ have this property. As an application of our result, we prove that there exists a fixed positive integer $l$ such that the $0^{rm th}$ local cohomology $H_I^0(L_{underline{n}}) = big(0 :_{L_{underline{n}}} I^lbig)$ for all ideals $I$ of $R_0$ and for all $underline{n} in mathbb{N}^t$.
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
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