Hossein Faridian
This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring $R$, we present a characterization of indecomposable projective $R$-modules and describe a one-to-one correspondence between the projective indecomposable $R$-modules and the simple $R$-modules.
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In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of Gorenstein projective, injective, and flat modules and the notion of the usual projective, injective, and flat modules.
The main aim of this paper is to investigate rings over which all (finitely generated strongly) Gorenstein projective modules are projective. We consider this propriety under change of rings, and give various examples of rings with and without this propriety.
In \cite{Ouarghi}, the authors discuss the rings over which all modules are strongly Gorenstein projective. In this paper, we are interesting to an extension of this idea. Thus, we discuss the rings over which every Gorenstein projective (resp. flat) module is strongly Gorenstein projective (resp, flat). Our aim is to give examples of rings with different Gorenstein global dimension satisfied this condition.
Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N \in \mathcal{P}_{w}^{\dagger_\infty}$ (see, \cite{FLQ}). In this paper, we introduce and study some properties of weak $w$-projective modules. And we use these modules to characterize some classical rings, for example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective, $R$ is a Von Neumann regular ring if and only if every FP-projective is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projecti…
This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call, respectively, strongly n-Gorenstein projective, injective and flat modules. These three classes of modules give us a new characterization of the first modules, and they are a generalization of the notions of strongly Gorenstein projective, injective, and flat modules respectively.
One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.
Let $R$ be a ring. An $R$-module $M$ is said to be an absolutely $w$-pure module if and only if $\Ext^1_R(F,M)$ is a GV-torsion module for any finitely presented module $F$. In this paper, we introduce and study the concept of $w$-FP-projective module which is in some way a generalization of the notion of $\rm FP$-projective module. An $R$-module $M$ is said to be $w$-$\rm FP$-projective if $\Ext^1_R(M,N)=0$ for any absolutely $w$-pure module $N$. This new class of modules will be used to characterize (Noetherian) $DW$ rings. Hence, we introduce the $w$-$\rm FP$-projective dimensions of module…
On the projective dimension of tensor products of modules
该文研究交换Noetherian环上有限生成模张量积的投射维数问题，构造了无限投射维数模的张量积为有限投射维数的例子，并给出了此类现象无法出现的非平凡条件。
Let $R$ be a commutative ring. An $R$-module $M$ is said to be super finitely presented if there is an exact sequence of $R$-modules $\cdots\rightarrow P_n\rightarrow\cdots \rightarrow P_1\rightarrow P_0\rightarrow M\rightarrow 0$ where each $P_i$ is finitely generated projective. In this paper it is shown that if $R$ has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if…
Let $R$ be a commutative ring. A quasi-Gorenstein $R$-module is an $R$-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Noetherian factorial domain $R$ extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative Noetherian factorial domain.
This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437--445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.
For a semidualizing module $C$ over a ring $R$, we study the following classes modulo exact zero divisors: $\g_C$--projectives, $\mathcal G_C$; the Auslander class $\mathcal A_C$; the Bass class $\mathcal B_C$; $\mathcal{P}_C$--projective; $ {\mathcal F}_C$--projective; and ${\mathcal I}_C$--injective dimensions.
We characterize Ding modules and complexes over Ding-Chen rings. We show that over a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Ding flat) R-modules coincide with the Gorenstein projective (resp. Gorenstein injective, resp. Gorenstein flat) modules, which in turn are nothing more than modules appearing as a cycle of an exact complex of projective (resp. injective, resp. flat) modules. We prove a similar characterization for chain complexes of R-modules: A complex is Ding projective (resp. Ding injective, resp. Ding flat) if and only if each component is Ding projective …
Gorenstein projective precovers and finitely presented modules
针对任意环上 Gorenstein 投射预覆盖的存在性问题，本文证明可归约到有限表现模：若其 Gorenstein 投射维数一致有界，则该模类特殊预覆盖。
In this paper, first we obtain some new and interesting results on projective modules and on the upper topology of an ordinal number. Then it is shown that the rank map of a locally of finite type projective module is continuous with respect to the upper topology (by contract, it is well known this map is not necessarily continuous with respect to the discrete topology). It is also proved that a finitely generated flat module is projective if and only if its rank map is continuous with respect to the upper topology.
In this paper, we investigate the notions of $\mathcal{X}^\bot$-projective, $\mathcal{X}$-injective and $\mathcal{X}$-flat modules and give some characterizations of these modules, where $\mathcal{X}$ is a class of left $R$-modules. We prove that the class of all $\mathcal{X}^\bot$-projective modules is Kaplansky. Further, if the class of all $\mathcal{X}$-projective $R$-modules is closed under direct limits, we show the existence of $\mathcal{X}^\bot$-projective covers and $\mathcal{X}$-injective envelopes over a $\mathcal{X}^\bot$-hereditary ring $R.$ Moreover, we decompose a $\mathcal{X}^\b…
Let A be an affine algebra over the field of real numbers of dimension d. Let f \in A be an element not belonging to any real maximal ideal of A. Let P be a projective A-module of rank \geq d-1. Let (a,p) \in A_f \oplus P_f be a unimodular element. Then the projective A_f module Q=A_f \oplus P_f/(a,p)A_f is extended from A.
Two remarks on Almost Projective Modules
本文研究交换环上的几乎投射模，证明了其均为投射模当且仅当环半局部，并给出每个模有几乎投射包络的充要条件为环凝聚且局部阿尔廷，且条件不可省略。
Inspired by a recent work of Buchweitz and Flenner, we show that, for a semidualizing bimodule $C$, $C$--perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of $C$--projective modules.
We develop in this paper a stable theory for projective complexes, by which we mean to consider a chain complex of finitely generated projective modules as an object of the factor category of the homotopy category modulo split complexes. As a result of the stable theory we are able to prove that a complex of finitely generated projective modules over a generically Gorenstein ring is exact if and only if its dual complex is exact. This shows the dependence of total reflexivity conditions for modules over a generically Gorenstein ring.
Let $M$ be a finitely generated module over a local complete intersection $R$ of characteristic $p>0$. The property that $M$ has finite projective dimension can be characterized by the vanishing of $\ext_R^i({}^{f^n} R,M)$ for some $i>0$ and for some $n>0$.
Let $(A,\mathfrak{m})$ be a local Gorenstein local ring and let $M$ be an $A$ module of finite length and finite projective dimension. We prove that the Lowey length of $M$ is greater than or equal to order of $A$. This generalizes a result of Avramov, Buchweitz, Iyengar and Miller
Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid of this monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K group of R (ignoring the derivation) as an image and contains the reduced K group of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential …
Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.
We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.
Counterexamples to Peskine-Szpiro's conjecture on modules of finite projective dimension
该研究针对诺特局部环上有限投射维数模的Peskine-Szpiro同调猜想，给出反例并证明其中的维数不等式、强相交猜想和等级猜想在一般情况下不成立。
On w-copure projective modules
本文通过引入w-余纯投射模与强w-余纯投射模，推广了余纯投射模的概念，并以此研究平坦模的w-分裂维数。
Depth formula for modules of finite reducing projective dimension
本文证明 Cohen-Macaulay 局部环上两个有限生成 Tor-独立模在其一具有有限约化投射维数时满足深度公式，推广了 Bergh-Jorgensen 关于可约复杂度的定理。
Splitting criteria for projective modules over polynomial algebras
本文研究了投射模在多项式代数上的分裂问题，并证明了monic逆原则在有限生成环上的成立，同时探讨了相关理想的应用。
Splitting of projective modules over some affine algebras
本文针对特定仿射代数上的光滑仿射代数 A，定义稳定自由模的欧拉类群，分析对应射影模中 unimodular 元素的存在条件，是对 Banerjee 和 Das 相关工作的改进。
We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In section 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Goren…
Uniformly S-projective relative to a module and its dual
本文研究了相对于模及其对偶的均匀S-投射和注入概念，探讨了它们的性质及在模理论中的应用。
In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext ho…
A note on projective dimension over twisted commutative algebras
研究了有限生成模块在扭曲交换代数上的投射维数，证明其随n线性增长，验证了Le等人的猜想。
Let R be a ring of dimension d and A be one of R[Y] or R[Y,Y^{-1}]. If P is a projective A-module of rank \geq d+1 satisfying some condition, then we show that E(A\oplus P) acts transitively on Um(A\oplus P). When P is free, this result is due to Yengui (when A=R[Y]) and Abedelfatah (when A=R[Y,Y^{-1}]).
Modules of logarithmic derivations in weighted projective spaces and applications to free divisors
本文提出加权对数导数模的概念，推广Saito的自由性判据，并通过加权多重特征方案构造自由除子家族，提供了多种显式例子。
The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex.
We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and Jørgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noethe…
Motivated by the Pontryagin-Hill criteria of freeness for abelian groups, we investigate conditions under which unions of ascending chains of projective modules are again projective. Several extensions of these criteria are proved for modules over arbitrary rings and domains, including a genuine generalization of Hill's theorem for projective modules over Prüfer domains with a countable number of maximal ideals. More precisely, we prove that, over such domains, modules which are unions of countable ascending chains of projective, pure submodules are likewise projective.
Let A be a Noetherian commutative ring. Assume that projective modules of rank r over polynomial extensions of A are extended from A. Then projective modules of rank r over discrete Hodge A-algebras are also extended from A. This result extends a result of T. Vorst.
We show that all the projective modules over the coordinate ring of the real algebraic sphere of dimension 3 are free
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
Construction of Gorenstein projective modules over tensor rings
本文给出张量环上完全投射分解和Gorenstein投射模的充要条件，并由此提供从基环构造张量环上Gorenstein投射模的方法，应用于平凡环扩张、Morita上下文环和三角矩阵环。
In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it.
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 a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.
Let $A$ be a ring of dimension $d$. Assume that for every finite extension ring $R$ of $A$, E_{d+1}(R) acts transitively on Um_{d+1}(R). Then we prove that E(A\oplus P) acts transitively on Um(A\oplus P), for any projective A-module P of rank d. As a consequence of this, we generalise some results of Gubeladze.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\rightarrow \mathrm{Hom}_R(P,A)\rightarrow \mathrm{Hom}_R(P,B)\rightarrow \mathrm{Hom}_R(P,C)\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S…
We provide a class of commutative Noetherian domains $R$ of dimension $d$ such that every finitely generated projective $R$-module $P$ of rank $d$ splits off a free summand of rank one. On this class, we also show that $P$ is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.
We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C=R, this recovers the standard projective dimension. We show that three natural definitions of finite C-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite C-projective dimension. In parallel, we develop the dual theory for injective dimension and C-injec…
Descending finite projective modules from a Novikov ring
本文研究了从Novikov环下降有限投射模的有效性，并证明了完美F_p-代数上向量包的下降性质。
Let A be a ring of dimension d and let P be a projective A-module of rank d. We prove that if for every finite extension R of A, R^d is cancellative, then P is cancellative. This gives an alternate proof of Bhatwadekar's result: every projective module of rank d over an affine algebra of dimension d over a C_1-field of characteristic 0 is cancellative. Let P be a projective module of rank d-1 over an affine agebra of dimension d over an algebraically closed field. Then, it is not known if P is cancellative. We prove some results in this direction also.
Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted $\KPC$, is always well generated and is compactly generated provided $\KPR$ is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension…
The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.
Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, we show: if a short local algebra $A$ has an indecomposable non-projective Gorenstein-projective module $M$, then either $A$ is self-injective (so that all modules are Gorenstein-projective) and then $|J^2| \le 1$, or else $|J^2| = |J/J^2| - 1$ and $|JM| = |J^2||M/JM|$. More generally, we focus the attention to semi-Gorenstein-projective and $\infty$-torsionfree modules…
I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex families of weight polyhedra for the general linear group.
Let A be an artin algebra. An A-module M is semi-Gorenstein-projective provided that Ext^i(M,A) = 0 for all i > 0. If M is Gorenstein-projective, then both M and its A-dual M* are semi-Gorenstein projective. As we have shown recently, the converse is not true, thus answering a question raised by Avramov and Martsinkovsky. The aim of the present note is to analyse in detail the modules M such that both M and M* are semi-Gorenstein-projective.
We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.
We extend a result of Napp Avelli, van der Put, and Rocha with a system-theoretic interpretation to the noncommutative case: Let P be a f.g. projective module over a two-sided Noetherian domain. If P admits a subdirect product structure of the form P = M x_T L over a factor module T of grade at least 2 then the torsion-free factor of M (resp. L) is projective.
We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.
We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modul…
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