Issue 1, Volume 40, March 2015

2015

1. Preface Preface Nguyen Tu Cuong PDF

   1-3
2. Tate Resolutions for Products of Projective Spaces David Eisenbud , Daniel Erman , Frank-Olaf Schreyer PDF

   ▸ Abstract

   We describe the Tate resolution of a coherent sheaf or complex of coherent sheaves on a product of projective spaces. Such a resolution makes explicit all the cohomology of all twists of the sheaf, including, for example, the multigraded module of twisted global sections, and also the Beilinson monads of all twists. Although the Tate resolution is highly infinite, any finite number of components can be computed efficiently, starting either from a Beilinson monad or from a multigraded module.

   5-36
3. The Chern Numbers and Euler Characteristics of Modules L. Ghezzi , S. Goto , J. Hong , K. Ozeki , T. T. Phuong , W. V. Vasconcelos PDF

   ▸ Abstract

   The set of the first Hilbert coefficients of parameter ideals relative to a module—its Chern coefficients—over a local Noetherian ring codes for considerable information about its structure–noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers.

   37-60
4. Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings Naoki Terai , Ken-ichi Yoshida PDF

   ▸ Abstract

   In this paper, we pose the following conjecture and give a positive answer to the case $\dim \Delta \le 2$: Let $\Delta$ be a $(d-1)$-dimensional simplicial complex on $[n]$. Fix an integer $\ell$ with $0 \le \ell \le n-d-1$. If $e(K[\Delta]) \le (\ell+1) d-\ell$ and $\beta_{\ell,\ell+d}(K[\Delta]) =0$, then $\mathrm{reg\,} K[\Delta] \le d-1$. \par Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

   61-69
5. Binomial Regular Sequences and Free Sums Winfried Bruns PDF

   ▸ Abstract

   Recently, several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note, we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequence generates a prime ideal or even normality is preserved for the residue class ring.

   71-83
6. The Tangent Cone of a Local Ring of Codimension 2 Mousumi Mandal , Maria Evelina Rossi PDF

   ▸ Abstract

   Let $(S, \mathfrak{n}) $ be a regular local ring and let $I \subseteq \mathfrak{n}^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting from information on the ideal $I, $ for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal $I^* $ which defines the tangent cone of $R$ or to give information on its graded structure. Motivated by papers of S.C. Kothari, S. Goto et al. concerning the leading ideal of a complete intersection $I=(f,g) $ in a regular local ring, we present results provided ht$(I)=2.$ If $I$ is a complete intersection, we prove that the Hilbert function of $R$ determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if $\nu(I) >2$ and a careful investigation can be provided when $\nu(I)=3. $ Several examples illustrating our results are given.

   85-100
7. On Modules of Linear Type Kosuke Fukumuro , Hirofumi Kume , Koji Nishida PDF

   ▸ Abstract

   Let $A$ be an $m \times n$ matrix with entries in a Noetherian ring $R$, where $m, n$ are positive integers such that $m \leq n$. By $M$ we denote the cokernel of the $R$-linear map $R^m \longrightarrow R^n$ defined by ${}^t{A}$. The purpose of this paper is to give an elementary proof to the result due to Avramov (J. Algebra 73, 248–263, 1981), that characterizes the condition for $M$ to be a module of linear type in terms of determinantal ideals of $A$.

   101-110
8. Segre Product, H-Polynomials, and Castelnuovo-Mumford Regularity Marcel Morales , Nguyen Thi Dung PDF

   ▸ Abstract

   The purpose of this paper is to extend the bilinearity of the Segre product that has been proved recently by Ilse Fischer and Martina Kubitzke under some restricted hypotheses. As a consequence, we get some formulas involving Eulerian polynomials and a nice formula that will be used by the first author to solve the Simon Newcomb problem. We apply these results to compute the postulation number of a series and extend partially the results about Castelnuovo-Mumford regularity of the Segre product of polynomial rings of David A. Cox and Evgeny Materov.

   111-124
9. Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth Huy Tài Hà , Mengyao Sun PDF

   ▸ Abstract

   In a recent work (Kaiser et al., J. Comb. Theory Ser. A 123, 239–251, 2014), Kaiser et al. provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs and, thus, give counterexamples to a conjecture of Francisco et al. (Discrete Math. 310, 2176–2182, 2010). The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.

   125-137
10. A General Volume = Multiplicity Formula Steven Dale Cutkosky PDF

    ▸ Abstract

    In this paper we prove a general Volume = Multiplicity formula for graded families of $m_R$-primary ideals in a Noetherian local ring $R$.

    139-147
11. Residual Complex on the Tangent Cone of a Numerical Semigroup Ring I-Chiau Huang PDF

    ▸ Abstract

    A residual complex on the tangent cone of a numerical semigroup ring is constructed in terms of modules of homomorphisms. Macaulay’s inverse system is generalized. Applications of the constructions include a criterion for the Gorensteiness of the tangent cone in terms of the symmetry of the set of basis monomials.

    149-160
12. On the Limit of Frobenius in the Grothendieck Group Kazuhiko Kurano , Kosuke Ohta PDF

    ▸ Abstract

    Considering the Grothendieck group of finitely generated modules modulo numerical equivalence, we obtain the finitely generated lattice $\overline{G_0(R)}$ for a Noetherian local ring $R$. Let $C_{CM}(R)$ be the cone in $\overline{G_0(R)}_{\Bbb R}$ spanned by cycles of maximal Cohen-Macaulay $R$-modules. We shall define the fundamental class $\overline{\mu_R}$ of $R$ in $\overline{G_0(R)}_{\Bbb R}$, which is the limit of the Frobenius direct images (divided by their rank) $[{}^e R]/p^{de}$ in the case ${ch}(R) = p > 0$. The homological conjectures are deeply related to the problems whether $\overline{\mu_R}$ is in the Cohen-Macaulay cone $C_{CM}(R)$ or the strictly nef cone $SN(R)$ defined below. In this paper, we shall prove that $\overline{\mu_R}$ is in $C_{CM}(R)$ in the case where $R$ is FFRT or F-rational.

    161-172
13. A Remark on Vanishing of Chain Complexes Yuji Yoshino PDF

    ▸ Abstract

    For chain complexes $W \in D^{\pm} _{fg} (R)$ and $X \in D(R)$ in the derived category over a commutative noetherian ring $R$, we prove that $\mathrm{RHom} _R (W, X) = 0$ if and only if $W {\, {}^{\mathrm L}\otimes}_R X =0$.

    173-177
14. Graded Annihilators and Uniformly F-Compatible Ideals Rodney Y. Sharp PDF

    ▸ Abstract

    Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is $F$-finite (that is, is finitely generated when viewed as a module over itself via the Frobenius homomorphism). Two of the afore-mentioned three sets have links to tight closure, via test ideals. Among the aims of the paper are a proof that two of the sets are equal, and a proposal for a generalization of I. M. Aberbach's and F. Enescu's splitting prime.

    179-195
15. Maximal Cohen–Macaulay Approximations and Serre’s Condition Hiroki Matsui , Ryo Takahashi PDF

    ▸ Abstract

    This paper studies the relationship between Serre's condition $(\mathsf{R}_n)$ and Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved that a Gorenstein local ring satisfies $(\mathsf{R}_n)$ if and only if every maximal Cohen--Macaulay module is a direct summand of a maximal Cohen--Macaulay approximation of a (Cohen--Macaulay) module of codimension $n+1$.

    197-203