A V Jayanthan

Let (R,&) be a Noetherian local ring of dimension d>0 with infinite residue field. Let M be a finitely generated proper R-submodule of a free R-module F with ℓ(F∕M)<∞ and having rank r. In this article, we study the fiber multiplicity f0(M) of the module M. We prove that if (R,&) is a two-dimensional Cohen–Macaulay local ring, then f0(M)≤br1(M)-br0(M)+l(F/M)+ m (M)-r, where bri(M) denotes the ith Buchsbaum-Rim coefficient of M.

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, Hoa, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers.

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 depth (Formula Presented.), then the relation types of R(I) and FL(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.

We study the complete intersection property of monomial curves in the family. We prove that if is a complete intersection for j ≫ 0, then Γa+j+an is a complete intersection for j ≫ 0. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if is a complete intersection for j ≫ 0, then is a complete intersection. We also characterize the complete intersection property of this family when n = 3. © 2013 American Mathematical Society.

Let (Formula presented.) be a graded Gorenstein Artin algebra. (Formula presented.) for some (Formula presented.) in the divided power algebra (Formula presented.). Suppose that (Formula presented.) is a height one ideal generated by (Formula presented.) quadrics so that (Formula presented.) after a possible change of variables. Let (Formula presented.). Then (Formula presented.) and (Formula presented.) is said to be (Formula presented.) -generic if (Formula presented.). In this article we prove necessary conditions, in terms of (Formula presented.) , for an ideal to be (Formula presented.) -generic. With some extra assumptions on the exponents of terms of (Formula presented.) , we obtain a characterization for height four ideals (Formula presented.) to be (Formula presented.) -generic.

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of I(G)q in terms of certain combinatorial invariants associated with G. We also prove a weaker version of a conjecture by Alilooee, Banerjee, Beyarslan and Hà on an upper bound for the regularity of I(G)q and we prove the conjectured upper bound for the class of vertex decomposable graphs. Using these results, we explicitly compute the regularity of I(G)q for several classes of graphs.

In this article, we show that the depths of the associated graded ring and fiber cone of a lex-segment ideal in K[x,y] are equal. © 2010 Heldermann Verlag.

In this article, we study certain homological properties of the graded rings associated with stretched m-primary ideals in a Cohen-Macaulay local ring (A, m). We compute the h-polynomial of the fiber cone and using this expression we show that the fiber cone is Gorenstein is equivalent to being hypersurface under certain assumptions. We obtain some inequalities between the Hilbert coefficients of the fiber cone and obtain sufficient conditions for the equality.

Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in S = K[x1,…, xn, y1,…, yn] We compute the Castelnuovo-Mumford regularity of S/JG when JG is the binomial edge ideal of a Cohen-Macaulay bipartite graph. We achieve this by computing the regularity of certain bipartite subgraphs and some intermediate graphs, called k-fan graphs. In this process, we also obtain a class of graphs which satisfy the regularity conjecture of Saeedi Madani and Kiani.

Let G be a finite simple graph on n vertices. Let JG ⊂ K[x1,..., xn] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo-Mumford regularity of JsG for all s ≥ 1 for certain classes of graphs G.