We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-known but nontrivial fact that class groups of rings of integers are torsion. We provide an independent proof using elementary techniques.

Let ${\mathbb{𝔸}}_{n,m}$ be the polynomial ring $\text{Sym}({ℂ}^n\otimes {ℂ}^m)$ with the natural action of ${\text{GL}}_m(ℂ)$. We consider a family of ${\text{GL}}_m(ℂ)$-stable ideals $J_{n,m}$ in ${\mathbb{𝔸}}_{n,m}$, each equivariantly generated by one homogeneous polynomial of degree $2$ and show that the regularity of this family is unbounded. Using this, we negatively answer a question raised by Erman, Sam and Snowden on a generalization of Stillman’s conjecture.

Given a collection of $t$ subspaces in an $n$-dimensional vector space $W$ we can associate to them $t$ linear ideals in the symmetric algebra $\mathcal{𝒮}(W^{\ast })$. Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of $t$ linear ideals is at most $t$. We show that analogous results hold when we work over the exterior algebra $\bigwedge (W^{\ast })$ (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor $\mathrm{\Omega }$ from the category of polynomial functors to itself. The functor $\mathrm{\Omega }$ transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.

Our aim is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings of algebraic curves at singular points. We characterize canonical ideals by means of a symmetry relation between lengths of certain quotients of associated ideals to a pair of dual ideals. In particular, we extend the symmetry among absolute and relative maximals in the sets of values of pairs of dual fractional ideals to other kinds of maximal points. Our results generalize and complement previous ones by other authors.

We study certain properties of Noetherian rings with weak normality and seminormality in mixed characteristic. It is known that the two concepts can differ in the equal prime characteristic case, while they coincide by definition in the equal characteristic zero case. We exhibit some examples in the mixed characteristic case. We also establish the local Bertini theorem for weak normality in mixed characteristic under a certain condition.

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras $A$, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra $A$ or finite $A$-module $M$ that we consider here is the set of Jordan types of elements of the maximal ideal $\mathfrak{𝔪}$ of $A$, acting on $M$. Here, the Jordan type of $\ell \in {\mathfrak{𝔪}}_A$ is the partition giving the Jordan blocks of the multiplication map $m_{\ell }:M\to M$. In particular, we consider the Jordan type of a generic linear element $\ell$ in $A_1$, or in the case of a local ring $\mathcal{𝒜}$, that of a generic element $\ell \in {\mathfrak{𝔪}}_{\mathcal{𝒜}}$, the maximum ideal.

We often take $M=A$, the graded algebra, or $M=\mathcal{𝒜}$ a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of $M$. However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra $A$ or $A$-module $M$, except, importantly, in modular invariant theory, or in the study of commuting Jordan types.

We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of $\ell \in {\mathfrak{𝔪}}_{\mathcal{𝒜}}$ and a certain local Hilbert function. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We also propose open problems.

The Waring rank of the generic $d\times d$ determinant is bounded above by $d\cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum decomposition. We describe some of the symmetries of the decomposition and set-theoretic defining equations for the terms of the decomposition.

A new class of abelian $p$-groups is introduced, the countably totally projective groups, that contains the well-known class of totally projective groups. A countably totally projective group is shown to have the property that every fully inert subgroup is commensurable with a fully invariant subgroup. This generalizes results of Goldsmith, Salce and Zanardo (2014), who proved that a direct sum of cyclic $p$-groups has this property. It also answers affirmatively two questions recently posed in the literature.

We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Drawing inspiration from the representation theory of orders in separable algebras, we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.