Issue 3, Volume 40, September 2015

2015

1. Absolutely Koszul Algebras and the Backelin-Roos Property Aldo Conca , Srikanth B. Iyengar , Hop D. Nguyen , Tim Römer PDF

   ▸ Abstract

   We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property.

   353-374
2. Arithmetical Rank of a Squarefree Monomial Ideal whose Alexander Dual is of Deviation Two Kyouko Kimura , Naoki Terai , Ken-ichi Yoshida PDF

   ▸ Abstract

   In this paper, we prove that the arithmetical rank of a squarefree monomial ideal $I$ of a polynomial ring $S$ is equal to the projective dimension of $S/I$ when $\mathrm{arithdeg\,} I - \mathrm{indeg\,} I = 2$ and $I$ has a linear resolution.

   375-391
3. Serre's Condition $R_{k}$ for Sums of Geometric Links Mark Johnson , Bernd Ulrich PDF

   ▸ Abstract

   We investigate what it means that the intersection of a variety with a residual intersection has a low-dimensional singular locus. For schemes having Cohen-Macaulay residual intersections, we prove, for instance, that if the intersection of the scheme with one of its geometric residual intersections has a ‘small’ singular locus, then the scheme can be defined by ‘few’ equations locally.

   393-401
4. Syzygies of Hibi Rings Viviana Ene PDF

   ▸ Abstract

   We survey recent results on resolutions of Hibi rings.

   403-446
5. Chain Polytopes and Algebras with Straightening Laws Takayuki Hibi , Nan Li PDF

   ▸ Abstract

   It will be shown that the toric ring of the chain polytope of a finite partially ordered set is an algebra with straightening laws on a finite distributive lattice. Thus, in particular, every chain polytope possesses a regular unimodular triangulation arising from a flag complex.

   447-452
6. Initial Ideals of Borel Type Fabrizio Brienza , Anna Guerrieri PDF

   ▸ Abstract

   This paper provides a survey of results on regularity, Betti numbers and reduction numbers, showing their stability in passing from an ideal to its initial ideal if the latter is of Borel type. Similar results happen when one passes from an ideal to its generic initial ideal. We also present analogous results on extremal Betti numbers.

   4530462
7. On Adjusted Hilbert-Samuel Functions Nguyen Tuan Long PDF

   ▸ Abstract

   Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$.  Let $\mathfrak{q}$ be a parameter ideal of $M$. Consider an adjusted Hilbert-Samuel function in $n$ defined by 
   $$ f_{\mathfrak {q},M}(n)=\ell (M/\mathfrak {q}^{n+1}M)-\sum \limits _{i=0}^{d}\text {adeg}_{i}(\mathfrak {q};M) \left (\begin {array}{c} n+i \\ i \end {array}\right ), $$
    where ${\rm adeg}_i(\mathfrak{q};M)$ is the $i$-th arithmetic degree of $M$ with respect to $\mathfrak{q}$.  In this paper, we prove that if $\mathfrak{q}$ is a distinguished parameter ideal  then there exists an integer $n_0$ such that $f_{\mathfrak{q}, M}(n)\geq 0$ for all $n\geq n_0.$ Moreover, if $M$ is sequentially generalized Cohen-Macaulay then $n_0$ exists independently of the choice of $\frak q$. 

   463-477
8. Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals Shreedevi K. Masuti , Tony J. Puthenpurakal , J. K. Verma PDF

   ▸ Abstract

   Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and ${\bf I}=(I_1,\ldots,I_d)$ be $\mathfrak{m}-$primary ideals in $R$. We prove that $\lambda_R([H^d_{(x_{ii}t_i:1\leq i \leq d)}(\mathcal R^\prime(\mathcal F)]_{\bf n})$ $< \infty$, for all ${\bf n} \in \mathbb N^d$, where $\mathcal F=\{\mathcal F({\bf n}):{\bf n}\in \mathbb Z^d\}$ is an ${\bf I}-$admissible filtration and $(x_{ij})$ is a strict complete reduction of $\mathcal F$ and $\mathcal R^\prime(\mathcal F)$ is the extended multi-Rees algebra of $\mathcal F.$ As a consequence we prove that the normal joint reduction number of $I,J,K$ is zero in an analytically unramified Cohen-Macaulay local ring of dimension $3$ if and only if $\overline{e}_3(IJK)-[\overline{e}_3(IJ)+\overline{e}_3(IK)+\overline{e}_3(JK)]+\overline{e}_3(I)+\overline{e}_3(J)+\overline{e}_3(K)=0.$ This generalizes a theorem of Rees on joint reduction number zero in dimension $2$. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension $3.$

   479-510
9. Combinatorial Characterizations of the Saturation and the Associated Primes of the Fourth Power of Edge Ideals Ha Thi Thu Hien , Ha Minh Lam PDF

   ▸ Abstract

   To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in Hien et al. (J. Algebra 439, 225–244, 2015) and Terai and Trung (J. Pure Appl. Algebra 218, 1117–1129, 2014). In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.

   511-526
10. Classification of the Linearly Reductive Finite Subgroup Schemes of $SL_2$ Mitsuyasu Hashimoto PDF

    ▸ Abstract

    We classify the linearly reductive finite subgroup schemes $G$ of $SL_2=SL(V)$ over an algebraically closed field $k$ of positive characteristic, up to conjugation. As a corollary, we prove that such $G$ is in one-to-one correspondence with an isomorphism class of two-dimensional $F$-rational Gorenstein complete local rings with the coefficient field $k$ by the correspondence $G\mapsto \left ((\mathrm {Sym\,}V)^{G}\right )\widehat {~}$.

    527-534
11. On the Relation Type of Fiber Cone A. V. Jayanthan , Ramakrishna Nanduri PDF

    ▸ Abstract

    In this article we study the relation type of the fiber cone of certain special classes of ideals in Noetherian local rings. We show that in any Noetherian local ring, if deviation of $I$ is $1$, and $\mathrm{depth}(G(\mathcal F_L)) \geq \ell-1$, then the relation types of $\mathcal R(I)$ and $F_L(I)$ are equal. We also prove that for lexsegment ideals in $K[x,y]$, where $K$ is a field, the relation types of the fiber cone and the Rees algebra are equal.

    535-544