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If I is an ideal in a Gorenstein ring S, and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal L_{n} of minors of a generic 2 \times n matrix when n>3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, suppose that K is the residual intersection of L_{n} by 2n-4 general quadratic forms in L_{n}. In this situation we analyze S/K and show that I^{n-3}(S/K) is a self-dual maximal Cohen-Macaulay S/K-module with linear free resolution over S. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented. [ABSTRACT FROM AUTHOR]

Let (R , m , k) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field k of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials Ω R is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–510, 1990) and Cortiñas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of Hom R (m , m) and constructing enough torsion in Ω S , enabling us to pull back a nontrivial torsion to Ω R . [ABSTRACT FROM AUTHOR]

Let (R , m , k) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field k of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials Ω R is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–510, 1990) and Cortiñas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of Hom R (m , m) and constructing enough torsion in Ω S , enabling us to pull back a nontrivial torsion to Ω R . [ABSTRACT FROM AUTHOR]

We give explicit criteria that imply the resurgence of a self‐radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. One criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves. [ABSTRACT FROM AUTHOR]

In this paper we prove that, under mild conditions, an equicharacteristic integrally closed domain which is a finite abelian extension of a regular domain has the uniform symbolic topology property. [ABSTRACT FROM AUTHOR]

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity and the number of generators. We prove the conjecture in dimension three and establish a weaker result in full generality. [ABSTRACT FROM AUTHOR]

Given a radical ideal I in a regular ring R⁠, the Containment Problem of symbolic and ordinary powers of I consists of determining when the containment I(a) ⊆ Ib holds. By work of Ein–Lazersfeld–Smith, Hochster–Huneke and Ma–Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when R/I is F-pure, and prove tighter containments in the case when R/I is strongly F-regular. [ABSTRACT FROM AUTHOR]

Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I :R ms for positive integers s. We are interested in iterated socles in connection with the notion of integral dependence of ideals. In this article we show that iterated socles are integral over I, with reduction number at most one, provided s ≤ o(I1(φd)) − 1, where o(I1(φd)) is the order of the ideal of entries of the last map in a minimal free R-resolution of R/I. In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o(I1(φd)). This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R-resolution of R/I. Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. The generators of these socles are suitable determinants obtained from the Hilbert-Burch matrix. [ABSTRACT FROM AUTHOR]

This paper answers in the affirmative a question raised by Karl Schwede concerning an upper bound on the multiplicity of F-pure rings. [ABSTRACT FROM AUTHOR]

We prove that for all n, simultaneously, we can choose prime filtrations of R/In such that the set of primes appearing in these filtrations is finite. [ABSTRACT FROM AUTHOR]

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 ∉ 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. [ABSTRACT FROM AUTHOR]

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra. [ABSTRACT FROM AUTHOR]

This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds and give examples to show that at least in some cases it is sharp. The techniques come largely from work of Elias, Rossi, Valla, and Vasconcelos. [ABSTRACT FROM AUTHOR]

Let S = K[xl, … xn], let A, B be finitely generated graded S-modules, and let m = (xl, … xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first [(n - 1)/2] steps of the resolution of I are linear, then I² is a power of m. [ABSTRACT FROM AUTHOR]

Throughout we let ($T,{\frak m},k$) denote a commutative Noetherian local ring with maximal ideal ${\frak m}$ and residue field $k$. We let $I\,{\subseteq}\,T$ be an ideal generated by a regular sequence of length $c$ and set $R\,{\colonequal}\,T/I$. In the important paper [1], Avramov addresses the following question. Given a finitely generated $R$-module $M$, when does $M$ have finite projective dimension over a ring of the form $T/J$, where $J$ is generated by part (or all) of a set of minimal generators for $I$? He gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over $R$. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about $M$ (orits syzygies) coming from $T$, in particular, information gleaned from various Fitting ideals defined over $T$. The following theorem from Section 3 is one of our main results. We use $\Fitt_T(M)$ to denote the Fitting ideal of $M$. [ABSTRACT FROM PUBLISHER]

We prove new height inequalities for determinantal ideals in a regular local ring, or more generally in a local ring of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns. [ABSTRACT FROM AUTHOR]

This paper contains two theorems concerning the theory of maximal Cohen–Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen–Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen–Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen–Macaulay lo cal ring of finite Cohen–Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen–Macaulay local ring of finite Cohen–Macaulay type is again of finite Cohen–Macaulay type . The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result relates to work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties. [ABSTRACT FROM AUTHOR]

It is proved that an excellent ring which is the homomorphic image of a regular ring of finite Krull dimension and having a resolution of singularities satisfies the uniform Artin-Rees property. Extensions of some results of Lipman, Sathaye and Teissier are also obtained. [ABSTRACT FROM AUTHOR]

Let I be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical of I, for localizing I with respect to another ideal, and thus for finding the primary decomposition of I. Our methods rest on modern ideas from commutative algebra, and are direct in the sense that they avoid the generic projections used by Hermann (1926) and all others until now. Some of our methods are practical for certain classes of interesting problems, and have been implemented in the computer algebra system Macaulay of Bayer and Stillman (1982-1992). [ABSTRACT FROM AUTHOR]

In this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal. Our techniques come from residual intersections, integral closures of ideals, and differentials. [ABSTRACT FROM AUTHOR]

Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of I are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case , is still helpful in finding some fresh new elements in . Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals. [ABSTRACT FROM AUTHOR]

Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure Ī has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of I are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case I ≠ Ī, ✓ I is still helpful in finding some fresh new elements in Ī/ I. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals. [ABSTRACT FROM AUTHOR]

We prove that given a Noetherian ring R and a finitely generated R-inodule M, there exists a finite set of prime ideals Λ in R such that the depth of an arbitrary ideal I on M is determined by the height of I modulo each of the primes in Λ. As an application we answer a question raised by the second author and V. Srinivas concerning m-adic approximations of regular sequences in a local ring. [ABSTRACT FROM AUTHOR]

Discusses the friendship theorem. Translation of the friendship theorem into a graph theorem; Proofs of the friendship theorem; Relationship between the proofs.