Related papers: Lim Ulrich sequences and Lech's conjecture

Let $(R,m)\to (S,n)$ be a flat local extension of local rings. Lech conjectured in 1960 that there should be a general inequality $e(R)\leq e(S)$ on the Hilbert-Samuel multiplicities. This conjecture is known when the base ring $R$ has…

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for…

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by…

We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if…

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

The existence of Ulrich modules for local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether local domains always have Ulrich modules. In this paper, we answer the question of existence for…

It is proved that Ulrich modules exist for a large class of local rings of dimension two. This complements earlier work of the authors and Ziquan Zhuang that described complete intersection domains of dimension two that admit no Ulrich…

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the…

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of…

Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are…

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in the universal enveloping algebra $U(L)$. We study the conditions on $L$ under which the $Ext$-groups $Ext (k,k)$ for the trivial $L$-module $k$ are the same…

We introduce the notion of a lim Cohen-Macaulay sequence of modules. We prove the existence of such sequences in positive characteristic, and show that their existence in mixed characteristic implies the long open conjecture about…

This article is concerned with nonzero modules of finite length and finite projective dimension over a local ring. We show the Loewy length of such a module is larger than the regularity of the ring whenever the ring is strict…

Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

Let R be a Cohen-Macaulay local ring. In this paper we study the structure of Ulrich $R$-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich R-modules, and clarify when the Ulrich R-modules are…

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$…

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class…

We study certain filtrations of indecomposable injective modules over classical Lie superalgebras, applying a general approach for noetherian rings developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the consequences of our…