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We show that any quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein Buchsbaum local ring with $I$-invariant $1$ admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings…

In the present paper, it is proved that any complete local domain of mixed characteristic has a weakly almost Cohen-Macaulay algebra in the sense that some system of parameters is a weakly almost regular sequence, which is a notion defined…

We investigate Cohen factorizations of local ring homomorphisms from three perspectives. First, we prove a "weak functoriality" result for Cohen factorizations: certain morphisms of local ring homomorphisms induce morphisms of Cohen…

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of "homological conjectures" in commutative algebra [H1][HH2]. Namely, for any local homomorphism $ R\to R'$ of complete local domains, there…

We provide a simplified proof of a theorem proved by Tavanfar and Shimomoto which states that a quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein local ring $(A,m,k)$ with $H^2_m(A)=k$ admits a small Cohen-Macaulay module.

In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay…

Hochster's Monomial Conjecture and Canonical Element Conjecture date back some thirty resp. twenty years. They concern all noetherian commutative local rings, and were proved by their originator right away in equal characteristic. Thanks to…

This article generalizes the correspondence between matrix factorizations and maximal Cohen-Macaulay modules over hypersurface rings due to Eisenbud and Yoshino. We consider factorizations with several factors in a purely categorical…

The aim of this article is to establish the existence of big Cohen-Macaulay algebras in mixed characteristic in some special situation. The main result follows from the so-called almost purity theorem proved by Davis and Kedlaya.

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide…

This is the author's diploma thesis. We describe a simplification in the construction of Khovanov-Rozansky's categorification of quantum sl(n) link homology using the theory of maximal Cohen-Macaulay modules over hypersurface singularities…

In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links…

Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show…

We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen-Macaulay type of a nearly Gorenstein monomial curve in $\mathbb{A}^4$ is at most $3$, answering a…

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…

We introduce quasi-Gorenstein morphisms of commutative local dg-algebras and use a Gorenstein version of the virtually small property to characterize them, a result which is new even for homomorphisms of local rings. In a different…

The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local…

We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals…