The Betti polynomials of powers of an ideal

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that β_i(S/I^k) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree.     

S-Noetherian properties on amalgamated algebras along an ideal

Let A and B   be commutative rings with identity, f:A→B$f:A\to B$ a ring homomorphism and J an ideal of B  . Then the subring A?^fJ:={(a,f(a)+j)|a∈A and j∈J}$A{?}^{f}J:=\{(a,f(a)+j)|a\in A\text{and}j\in J\}$ of A×B$A\times B$ is called the amalgamation of A with B along with J with respect to f. In this paper, we investigate a general concept of the Noetherian property, called the S  -Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A?^fJ$A{?}^{f}J$ for a multiplicative subset S   of A?^fJ$A{?}^{f}J$. As particular cases of the amalgamation, we also devote to study the transfers of the S  -Noetherian property to the constructions D+(_X1,…,_Xn)E[_X1,…,_Xn]$D+(X_1,\dots ,X_n)E[X_1,\dots ,X_n]$ and D+(_X1,…,_Xn)E?_X1,…,_Xn?$D+(X_1,\dots ,X_n)E?X_1,\dots ,X_n?$ and Nagata?s idealization.     

3-standardness of the maximal ideal

We study a notion called n-standardness (defined by M.E. Rossi (2000) in [10] and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is 3-standard, first proving results for when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic 0 case. As an application, we extend a result due to T. Puthenpurakal (2005) [9] and show that a certain length associated with a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.     

An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras

Both authors were partially supported by the NSF     

Enveloping algebras that are principal ideal rings

Let L be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of L is a principal ideal ring if and only if L is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.     

On the asymptotic properties of the Rees powers of a module

Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules E_n, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread.     

On the ring of global sections of the blowing-up along a two-generated ideal

The effective relations and the relation type of an ideal generated by two regular elements can be expressed in terms of the ring of global sections of the blowing-up along the ideal. As a consequence, we find the minimal overring where the ideal is syzygetic or of linear type.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

Complexity of the normalization of algebras

We introduce and develop new techniques to study the complexity of normalization processes of graded algebras. The construction of a new degree function on graded modules, with a global nature, permits a broad extension of recent bounds for the length of the chains of subalgebras that general algorithms must transverse to build the integral closure, particularly of blowup algebras. It achieves this by relating the values of the new degree with invariants of the algebra known ab initio. As a by-product, it reveals new inequalities among Hilbert coefficients. The second author was partially supported by the NSF.     

Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=?_n≥0R_n be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R₊=?_n>0R_n. Let b?b₀+R₊, where b₀ is an ideal of R₀. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:

(i)

and (R₀,m₀) is a local ring;

(ii)

dim(R₀)≤1 and R₀ is either a finite integral extension of a domain or essentially of finite type over a field;

(iii)

i≤g_b(M), where g_b(M) denotes the cohomological finite length dimension of M with respect to b.

Also, we establish some results about the Artinian property of certain submodules and quotient modules of .     

Stability results for projective modules over Rees algebras

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R-module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.     

On equations defining Veronese rings

For a standard graded noetherian algebra S that is of weakly linear type, the defining equations of the Veronesian subrings S^(d) are described explicitly, for d sufficiently large. Received: 19 January 2005     

Desingularization of regular algebras

We identify families of commutative rings that can be written as a direct limit of a directed system of noetherian regular rings and investigate the homological properties of such rings.     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

Counterexamples regarding symmetric tensors and divided powers

We investigate the similarities and differences between the module of symmetric tensors and the module of divided powers . There is a canonical map which is an isomorphism in many important cases. We give examples showing that this map need neither be surjective nor injective in general. These examples also show that the functor does not in general commute with base change.     

Homomorphisms of direct powers of algebras

We give an example of n-coconnected algebra that is not (n+1)-coconnected, for any n ≥ 2. Received June 8, 2005; accepted in final form August 28, 2005.     

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let f:A→B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A?^fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D’Anna and Fontana in 2007, and other classical constructions (such as the A+XB[X], the A+XB?X? and the D+M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.     

Stabilization of Boij–Söderberg decompositions of ideal powers

Given an ideal I we investigate the decompositions of Betti diagrams of the graded family of ideals ${\{I^k\}}_k$ formed by taking powers of I. We prove conjectures of Engström from [5] and show that there is a stabilization in the Boij–Söderberg decompositions of $I^k$ for $k>>0$ when I is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for $k>>0$, the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in k. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams.