Search

Let R be a Noetherian local k-algebra whose derivation module $${\mathrm{Der}}_k(R)$$ is finitely generated. Our main goal in this paper is to investigate the impact of assuming that $${\mathrm{Der}}_k(R)$$ has finite projective dimension (or finite Gorenstein dimension), mainly in connection with freeness, under a suitable hypothesis concerning the vanishing of (co)homology or the depth of a certain tensor product. We then apply some of our results towards the critical case $${\mathrm{depth}}\,R=3$$ of the Herzog–Vasconcelos conjecture and consequently to the strong version of the Zariski–Lipman conjecture.

The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.

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 [ { X i , j } ] P\subset R=K[\{X_{i,j}\}] generated by the maximal minors of an ( n + 1 ) × n (n+1)\times n generic matrix ( X i , j ) (X_{i,j}) over an arbitrary field K K with n ≥ 2 n\geq 2 . We also characterize when the derivation module of R / P R/P is Ulrich, and we investigate this property if we replace R / P R/P by determinantal rings arising from simple degenerations of the generic case.

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., Comment: Final version, to appear in J. Pure Appl. Algebra

A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective $3$-space is a reflexive vector bundle of reduction number $1$ and Gorenstein Rees ring.

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of 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 $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular., 18 pages

We establish a prescribed upper bound for the projective dimension of a finitely generated module over a Cohen-Macaulay local ring (with canonical module) satisfying certain cohomological condition...

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

In this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

Let S be a polynomial ring over a field of characteristic zero, and let \(I\subset S\) be an ideal of intersection type assumed moreover to have no embedded primary component. Our main goal in this paper is to provide an effective sufficient condition for a given monomial prime ideal to avoid the sets of prime divisors of the powers of I, and in particular to avoid the celebrated set of asymptotic prime divisors of I, which will follow from a new and quite surprising double-colon stability property. Further, we briefly describe some other applications, e.g., on the topology of a suitable blowing-up.

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.

LetSbe a polynomial ring over a fieldKof characteristic zero and letM⊂Sbe an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case whereMis a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime idealP⊂ScontainingMbe such that the localisationMPhasnon-maximal analytic spread. Our technique describes, in fact, a concrete obstruction forPto be an asymptotic prime divisor ofMwith 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.

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.

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).

Our main goal in this note is to give a characteristic-free, general version of Seidenberg’s well-known theorem on the existence of primary decomposition in the class of differential ideals in commutative Noetherian rings containing the rational numbers. Our approach is through the study of general logarithmic derivation modules, here dubbed tangential idealizers, and we first provide a primary decomposition of the tangential idealizer of any given ideal without embedded primary component. Also, we describe a large class of ideals whose radical as well as ordinary and symbolic powers possess the same tangential idealizer, extending a (real analytic) study due to Hauser and Risler. Other results are obtained; for instance, we show that the symbolic powers and the content ideal of a differential ideal are differential as well.

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.