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Register a new userTotal Betti numbers of modules of finite projective dimension
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Authors
Mark E. Walker
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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.
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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.
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.
Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case when $ i geq d+1 $ and $curv(I^n) > 1$ for all $n geq 1.$ We prove a similar result for the numerical function [ n mapsto ell(Tor_i^A(k, A/I^{n+1}))] under the assumption that $A$ is a CM ~ local ring. oindent We note that there are many examples of ideals satisfying the condition $curv(I^n) > 1,$ for all $ n geq 1.$ We also consider more general functions $n mapsto ell(Tor_i^A(M, A/I_n)$ for a filtration ${I_n }$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal CM ~ $A$-module and ${I_n=overline{I^n} }$ is the integral closure filtration, $I$ an $m$-primary ideal in $A.$
In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.
Let $K$ be a field and $S = K[x_1,dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.
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