Related papers: Depth formula for modules of finite reducing proje…

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product…

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the…

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…

Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this paper, we…

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module,…

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 give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The…

Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective…

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…

Let $\Lambda$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate…

For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\cid_R M$ and Cohen-Macaulay dimension $\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In…

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in…

In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these…

For a Noetherian ring $R$ and a cotilting $R$-module $T$ of injective dimension at least $1$, we prove that the derived dimension of $R$ with respect to the category $\mathcal{X}_T$ is precisely the injective dimension of $T$ by applying…

In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers.…

This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized…

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of mod R with respect to a fixed semidualizing…

Let $S=K[x_1, \dots, x_m, y_1, \dots, y_n]$ be the standard bigraded polynomial ring over a field $K$. Let $M$ be a finitely generated bigraded $S$-module and $Q=(y_1, \dots, y_n)$. We say $M$ has maximal depth with respect to $Q$ if there…

This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely…

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…