Related papers: Characterizing Regular Local Rings via Analogues o…

It is proved that a noetherian commutative local ring A containing a field is regular if there is a complex M of free A-modules with the following properties: M_i=0 for i not in [0,dim A]; the homology of M has finite length; H_0(M)…

Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring $A$ essentially finitely presented over a valuation ring $V$. We discuss several properties of this notion of regularity for…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the classes $ \mathcal{P}_C $ and $ \mathcal{I}_C $, we extend the notions of perfect and coperfect modules introduced by D.Rees \cite{R} and…

We investigate the transfer of regularity between commutative, noetherian, local rings through a class of local homomorphisms which we call basically regular. We give numerical characterizations of these maps, investigate their behavior…

Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property ''power stable range one'' if for all a, b $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $\lambda$…

We prove a mixed-characteristic analogue of Kunz's theorem in terms of perfectoid towers: a Noetherian local ring of residue characteristic $p$ is regular if and only if it admits a flat map to a Noetherian ring that extends to a perfectoid…

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 $(A,\mathfrak{m}, k=A/\mathfrak{m})$ be a noetherian local ring. Then it is equivalent $n = \dim A = \dim_k \mathfrak{m}/\mathfrak{m}^2$ and $\mathrm{Tor}^A_i(k,k) = 0$ for all $i \gg 0$. The article gives a proof with the…

A well-known result of K\"{o}the and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to…

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length…

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg_R (M,N) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We…

We are working in the category of commutative unital rings and denote by $\mathrm U(R)$ the group of units of a nonzero ring $R$. An extension of rings $R\subseteq S$, satisfying $\mathrm U(R)=R \cap\mathrm U(S)$ is usually called local.…

Koszul modules over Noetherian local rings $R$ were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincare' series is rational. It is an interesting problem to characterize classes of…

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher $\tor$ counterparts $\i t_i=\underset{n \to \infty}{\lim} \l(\tor_i(k,{}^{f^n}…

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper show that such rings have rigid homological properties; for instance, they…

Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the…

This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and…

Let $B$ be a regular local ring and $G\subset\Aut(B)$ a finite group of local automorphisms. Assume that $G$ is cyclic of prime order $p$, where $p$ is equal to the residue characteristic of $B$. We give conditions under which the ring of…