Related papers: Multiplicity bounds in prime characteristic
We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we…
We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation…
This paper resolves a question of Huneke and Watanabe by proving a sharp upper bound for the multiplicity of Du Bois singularities: at a point of a $d$-dimensional variety with Du Bois singularities and embedding dimension $e$, the…
For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of…
We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings,…
Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…
We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…
New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of…
The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many…
Criteria are given in terms of certain Hilbert coefficients for the fiber cone F(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay or has depth at least dim(R)-1. A version of Huneke's fundamental…
Let R = k[[x_0,...,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x_0,...,x_d]]. We investigate the question of which rings of this form have bounded Cohen--Macaulay type, that is, have a bound…
Motivated by the Bass conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond the Cohen--Macaulay property, the existence of such modules forces…
We give lower and upper bounds on the Buchsbaum-Rim multiplicity of finitely generated torsion-free modules over two-dimensional regular local rings, and conditions for them to attain the bounds. As consequences, we have formulae on the…
It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the…
We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…
We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC).…
The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra $A$ in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of $A$. All…
Blickle, Musta\c{t}\u{a} and Smith proposed two conjectures on the limits of $F$-pure thresholds. One conjecture asks whether or not the limit of a sequence of $F$-pure thresholds of principal ideals on regular local rings of fixed…
Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…
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…