Pullback of the Normal Module of Ideals with Low Codimension

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.

ANALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

SYMBOLIC ANALYTIC SPREAD: UPPER BOUNDS AND APPLICATIONS

The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$ . Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.

Bounds related to projective equivalence classes of good filtrations

Let [Formula: see text] be a regular ideal in noetherian ring [Formula: see text]. Mc Adam and Ratliff showed the existence of the unique minimal reduction number of [Formula: see text], noted [Formula: see text], such that for every minimal reduction [Formula: see text] of [Formula: see text], [Formula: see text] and [Formula: see text]. They showed that the set of integers [Formula: see text] is bounded in terms of the analytic spread of [Formula: see text]. Here, we extend these results to good filtrations. Let [Formula: see text] be a good filtration on [Formula: see text], we show that the set of integers [Formula: see text] is bounded.

On coefficient ideals

Let (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals I ⊆ I[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < s − k. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

Equimultiple coefficient ideals

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.