# Lech's conjecture in dimension three

@article{Ma2016LechsCI,
title={Lech's conjecture in dimension three},
author={Linquan Ma},
journal={arXiv: Commutative Algebra},
year={2016}
}
• Linquan Ma
• Published 1 September 2016
• Mathematics
• arXiv: Commutative Algebra
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 dimension less than or equal to two, and remains open in higher dimensions. In this paper, we prove Lech's conjecture in dimension three when $R$ has equal characteristic. In higher dimension, our method yields substantial partial estimate: $e(R)\leq (d!/2^d… Expand Lim Ulrich sequence: proof of Lech's conjecture for graded base rings The long standing Lech's conjecture in commutative algebra states that for a flat local extension$(R,\mathfrak{m})\to (S,\mathfrak{n})$of Noetherian local rings, we have an inequality on theExpand Asymptotic Lech's inequality • Mathematics • 2019 We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an$\mathfrak{m}$-primary ideal in a Noetherian local ring$(R,\mathfrak{m})$. We prove optimalExpand Lech's inequality, the Stückrad–Vogel conjecture, and uniform behavior of Koszul homology • Mathematics • Advances in Mathematics • 2019 Abstract Let ( R , m ) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set { l ( M / I M ) e ( I , M ) } I = m is bounded below by 1 / d ! e (Expand Cohomologically Full Rings • Mathematics • International Mathematics Research Notices • 2019 Inspired by a question raised by Eisenbud–Mustaţă–Stillman regarding the injectivity of maps from${\operatorname{Ext}}$modules to local cohomology modules and the work by the third author withExpand Multiplicities and Betti numbers in local algebra via lim Ulrich points • Mathematics • 2021 This work concerns finite free complexes with finite length homology over a commutative noetherian local ring R. The focus is on complexes that have length dimR, which is the smallest possible value,Expand Annihilators of Koszul Homologies and Almost Complete Intersections In this article, we propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection$R$. The question can be stated in terms of theExpand Homological Conjectures and Lim Cohen-Macaulay Sequences We discuss the new notion of a lim Cohen-Macaulay sequence of modules over a local ring, and also a somewhat weaker notion, as well as the theory of content for local cohomology modules. We relateExpand A Trilogy, Given By Complete Tensor Product of Complete Rings Over The Coefficient Ring The purpose of this paper is triple. First off, for an equi-characteristic zero, or a p-torsion free (0, p)-mixed characteristic local ring, we settle, positively, the conjecture on the closedness ofExpand Counterexamples in the Theory of Ulrich Modules The theory of Ulrich modules has many powerful and broad applications ranging from the original purpose of giving a criterion for when a local Cohen-Macaulay ring is Gorenstein to new methods ofExpand Asymptotic Phenomena in Local Algebra and Singularity Theory • Physics • 2017 The goal of this workshop was to highlight, and further, the interactions between local algebra and singularity theory. The timing was serendipitous for both subjects have witnessed tremendousExpand #### References SHOWING 1-10 OF 38 REFERENCES Intersection multiplicities over Gorenstein rings • Mathematics • 2000 Abstract. Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed thatExpand F-signature exists Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let$R^{1/p^{e}}\$ be the ring of pe-th roots of elements of R for e∈ℕ, andExpand
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