Related papers: Representation-theoretic properties of balanced bi…

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

For a finitely generated, non-free module $M$ over a CM local ring $(R,\fm,k)$, it is proved that for $n\gg 0$ the length of $\tor 1RM{R/\fm^{n+1}}$ is given by a polynomial of degree $\dim R-1$. The vanishing of $\tor iRM{N/\fm^{n+1}N}$ is…

Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a…

The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(\omega)$ with $\omega^p\in…

One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some…

Let $(R,\fm)$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or…

In this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]/(xy, z^2)$ we give an…

In this paper we completely classify all the special Cohen-Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit…

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is…

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R).…

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if $A$ is a hypersurface ring with dimension $d > 0$ then the Hilbert function of $M$ \wrt $\m$ is non-decreasing. If $A = Q/(f)$ for some…

We utilize recent results of Andr\'e and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed…

Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega_R$. Let $I$ be an $\m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $n\longmapsto \ell (\Hom_R(M,{\omega_R}/{I^{n+1}…

Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension one. In an effort to address the Berger's conjecture, a crucial invariant reduced type $s(R)$ was introduced by Huneke et. al. In this article, we study this…

This work introduces a notion of complexes of maximal depth, and maximal Cohen-Macaulay complexes, over a commutative noetherian local ring. The existence of such complexes is closely tied to the Hochster's ``homological conjectures", most…

In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen--Macaulay modules…

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the…

Let $A=Q/(f)$ where $(Q,\mathfrak{n})$ be a complete regular local ring of dimension $d+1$, $f\in \mathfrak{n}^i\setminus\mathfrak{n}^{i+1}$ for some $i\geq 2$ and $M$ an MCM $A-$module with $e(M)=\mu(M)i(M)+1$ then we prove that depth…

We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $G$. By definition, these are $G$-actions on Laurent polynomial algebras that stabilize the multiplicative group consisting of…

The main result of this article is to prove that any Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay algebra. The proof of this result is based on the construction of almost Cohen-Macaulay…