Related papers: On Modules of Finite Projective Dimension

It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent.…

It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent.…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…

If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective…

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal…

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…

In this paper, we characterize several properties of commutative notherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application we prove that…

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each…

Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct…

Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$…

We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that…

A local ring $R$ is regular if and only if every finitely generated $R$-module has finite projective dimension. Moreover, the residue field $k$ is a test module: $R$ is regular if and only if $k$ has finite projective dimension. This…

This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension, or locally finite injective dimension. We extend these results by providing…

Let $R$ be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated $R$-module $C$ is determined by the existence of a Cohen--Macaulay module $M$ that satisfies an inequality…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

In this paper, we consider finitely generated modules over commutative Noetherian rings whose tensor products have finite projective dimension. We construct examples of modules of infinite projective dimension (and also of infinite…