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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.

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 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.

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.

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.