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Let S be a Cohen-Macaulay ring which is local or standard graded over a field, and let I be an unmixed ideal that is generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal J ⊆ I (m) equals the m -th symbolic power I (m) of I. Second, we provide a saturation-type formula to compute I (m) and employ it to deduce a theoretical criterion for when I (m) = I m. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of I (m). Along the way, we prove a generalized version of a conjecture raised by Eisenbud and Mazur about ann S (I (m) / I m) , and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals. [ABSTRACT FROM AUTHOR]

AbstractWe describe an Ulrich module on the coordinate ring of the projective hypersurface X defined by the determinant of an n×n linear matrix ϕ , and show that the module defines a vector bundle on X if and only if the ideal generated by the n−1 order minors of ϕ is zero-dimensional; this property does not force X to be smooth, while the existing literature focuses on the smooth case. If X is any smooth hypersurface, we characterize when its derivation module is Ulrich. We raise the same question in higher codimension and show the Segre threefold has this property. [ABSTRACT FROM AUTHOR]

We contribute to the theory of G-dimension relative to a semidualizing module C , in connection to the properties of a module being totally C -reflexive, C - k -torsionless, and C - k -syzygy, where k ≥ 0 is an integer. We extend several known results, and for an integer q ≥ 0 we introduce the class of C - q -Gorenstein rings. We also consider C -duals and initiate the study of C -valued derivation modules over a local ring R of low depth; if C = R and R is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when C is general. [ABSTRACT FROM AUTHOR]

We establish new characterizations of complete intersection ideals in Cohen-Macaulay local rings as a byproduct of our investigation on a certain André-Quillen homology vector space in connection to the vanishing of (finitely many) Ext modules over a suitable quasi-deformation. Along the way, by means of (co)homology vanishing, we also detect criteria for quasi-complete intersection ideals. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

In this paper, we contribute to the theory of reduced G-perfection and horizontal linkage of modules over a commutative, Noetherian (typically, local) ring R, in the general setting where properties and operations are considered relatively to a semidualizing module C. We investigate when reduced GC-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules. Moreover, we show how to produce reduced G C -perfect modules that are also C-k-torsionless ( k ≥ 0 is an integer) but fail to be G C -perfect, and we illustrate that, unlike the usual grade, the relative reduced grade depends on the choice of C. [ABSTRACT FROM AUTHOR]

An important problem in algebra and related fields (such as algebraic and complex analytic geometry) is to find an explicit, well-structured, minimal set of generators for the module of logarithmic derivations of classes of homogeneous ideals in polynomial rings. In this note we settle the case of the ideal P ⊂ R = K[{Xi,j}] generated by the maximal minors of an (n + 1) × n generic matrix (Xi,j) over an arbitrary field K with n ≥ 2. We also characterize when the derivation module of R/P is Ulrich, and we investigate this property if we replace R/P by determinantal rings arising from simple degenerations of the generic case. [ABSTRACT FROM AUTHOR]

Abstract We establish a criterion for the strong F -regularity of a (non-Gorenstein) Cohen–Macaulay reduced complete local ring of dimension at least 2 and prime characteristic p. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an n × (n − 1) matrix X of indeterminates. For a perfect field k of characteristic p ≥ 5 , these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring over k given by the maximal minors of X is strongly F -regular. [ABSTRACT FROM AUTHOR]

Let S be a polynomial ring over a field K of characteristic zero and let M ⊂ S be an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case where M is a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime ideal P ⊂ S containing M be such that the localisation MP has non-maximal analytic spread. Our technique describes, in fact, a concrete obstruction for P to be an asymptotic prime divisor of M with respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes. [ABSTRACT FROM PUBLISHER]

This paper deals with blowup algebras of certain classical modules related to a quasi-homogeneous hypersurface in characteristic zero. Motivated by Aluffi's problem, which asks for an appropriate adaptation of his quasi-symmetric algebra into the context of modules (or sheaves), we propose and study our general candidate – which is shown to actually retrieve Aluffi's definition in the situation of ideals – and further we compute it explicitly in the case of the module of tangent vector fields on the (possibly singular) given divisor. Along the way, we work out the Rees ring of the corresponding logarithmic derivation module and we apply a structural result of Corso–Polini–Ulrich in order to determine its core (intersection of all reductions) under certain conditions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to furnish new criteria for freeness of (algebraic, homogeneous) divisors, especially by means of the minimal number of generators of certain graded derivation modules. Our approach is based on the description of the graded syzygies of the derivation module in the hypersurface case, which allows us to derive several other applications. We investigate, under certain conditions, the Castelnuovo–Mumford regularity and the Hilbert function of such module, as well as an Eisenbud matrix factorization of the given polynomial. We also obtain the defining ideals of the blowup algebras of the derivation module, as a dual version, in the hypersurface case, of the so-called tangent algebras   introduced by Simis, Ulrich and Vasconcelos. Finally, we give an explicit Ulrich ideal and the Hilbert polynomial in the distinguished case of linear free divisors (in the sense of Buchweitz and Mond). [ABSTRACT FROM AUTHOR]

We establish characterizations of smoothness for a complex affine algebraic variety at a given Cohen-Macaulay point. Our main result treats the case of surfaces with (at most) rational singularities, by a technique that requires the vanishing of suitable Ext modules rather than of sheaf cohomology groups. We also prove results in arbitrary dimension which, in addition to smoothness, detect curves and surfaces as a global feature. [ABSTRACT FROM AUTHOR]

Abstract: This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered. [Copyright &y& Elsevier]

For decades, differential ideals have played an important role in algebra. In this paper, if A is a Noetherian local ring with positive residual characteristic, we characterize when a maximally differential ideal P ⊂ A is an integrally closed ideal of finite projective dimension. Our main argument yields, in a characteristic-free setting, that if P has finite projective dimension then P must be a complete intersection. This generalizes a well-known result which assumes A to be regular. We derive further homological criteria and, along the way, we conjecture (in positive residual characteristic) that if P is maximally differential with respect to the entire derivation module, then A must be a complete intersection ring if P is integrally closed and has finite complete intersection dimension. [ABSTRACT FROM AUTHOR]

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module. [ABSTRACT FROM AUTHOR]