arXiv subjects
Publications and source records attributed to Paolo Mantero.
We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the Cohen-Macaulay ones.
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.
We introduce the concept of $j$-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched $\mathfrak{m}$-primary ideals of Sally and Rossi-Valla, as well as the concept of ideals of minimal and almost minimal $j$-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a $j$-stretched ideal, the associated graded ring is Cohen-Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem, presenting numerical conditions which ensure the almost Cohen-Macaulayness of the associated graded ring of a $j$-stretched ideal, provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the Cohen-Macaulayness or almost Cohen-Macaulayness of the associated graded module by several authors, including Sally, Rossi-Valla, Wang, Elias, Corso-Polini-Vaz Pinto, Huckaba, Marley and Polini-Xie.
We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.
Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $¶^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes Serre's $(S_1)$ property holds. Specifically, we prove that for any positive integers $h, e \ge 2$ with $(h,e) \neq (2,2)$ and $p \ge 5$ there is a homogeneous ideal $I$ in a polynomial ring over $K$ such that (1) the height of $I$ is $h$, (2) the Hilbert-Samuel multiplicity of $R/I$ is $e$, (3) the projective dimension of $R/I$ is at least $p$ and (4) the ideal $I$ is primary to a linear prime $(x_1,..., x_h)$. This result is in stark contrast to Manolache's characterization of Cohen-Macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2 and multiplicity 2.
Given a finite module $M$ over a Noetherian local ring $(R, \m)$, we introduce the concept of $j$-stretched ideals on $M$. Thanks to a crucial specialization lemma, we show that this notion greatly generalizes (to arbitrary ideals, and with respect to modules) the classical definition of stretched $\m$-primary ideals of Sally and Rossi-Valla, as well as the notion of minimal and almost minimal $j$-multiplicity given recently by Polini-Xie. For $j$-stretched ideals $I$ on a Cohen-Macaulay module $M$, we show that ${\rm gr}_I(M)$ is Cohen-Macaulay if and only if two classical invariants of $I$, the reduction number and the index of nilpotency, are equal. Moreover, for the same class of ideals, we provide a generalized version of Sally's conjecture (proving the almost Cohen-Macaulayness of associated graded rings). Our work unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the (almost) Cohen-Macaulayness %and almost Cohen-Macaulayness of associated graded modules by several authors, including Sally, Rossi-Valla, Wang, Elias, Rossi, Corso-Polini-Vaz Pinto, Huckaba and Polini-Xie.
In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically a complete intersection perfect ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered this question and gave a positive answer for special classes of ideals. We give a positive answer for some classes of ideals, however, we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings showing that in general the answer is negative. The homogeneous examples have been found thanks to the help of J. C. Migliore. Furthermore, the counterexamples show the sharpness of our main result. As a by-product, we exhibit several classes of Cohen-Macaulay ideals whose square is not Cohen-Macaulay. Our methods work both in the homogeneous and in the local settings.