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On finitely generated n-SG-projective modules
... It turns out, in particular, that these Gorenstein homological dimensions are refinements of the classical dimensions of a module M , in the sense that Gpd R (M ) ≤ pd R (M ), Gid R (M ) ≤ id R (M ) and Gfd R (M ) ≤ fd R (M ) with equality each time the corresponding classical homological dimension is finite. The reader is referred to [3,6,12,14,15,19,20,[23][24][25][28][29][30] for basics and recent investigations on Gorenstein homological theory, as well as some topics related to resolutions of flat modules. ...
This paper discusses a variant theory for the Gorenstein flat dimension. Actually, since it is not yet known whether the category GF(R) of Gorenstein flat modules over a ring R is projectively resolving or not, it appears legitimate to seek alternate ways of measuring the Gorenstein flat dimension of modules which coincide with the usual one in the case where GF(R) is projectively resolving, on the one hand, and present nice behavior for an arbitrary ring R, on the other. In this paper, we introduce and study one of these candidates called the generalized Gorenstein flat dimension of a module M and denoted by GGfd(R) (M) via considering exact sequences of modules of finite flat dimension. The new entity stems naturally from the very definition of Gorenstein flat modules. It turns out that the generalized Gorenstein flat dimension enjoys nice behavior in the general setting. First, for each R-module M, we prove that GGfdR(M) - Gid(R)(Hom(Z)(M, Q/Z)) whenever GGf(R)(M) is finite. Also, we show that GF(R) is projectively resolving if and only if the Gorenstein flat dimension and the generalized Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then GGfdR(M) = Gfd(R)(M) for any R-module M. Moreover, the global dimension associated to the generalized Gorenstein flat dimension, called the generalized Gorenstein weak global dimension and denoted by GG-wgldim(R), turns out to be the best counterpart of the classical weak global dimension in Gorenstein homological algebra. In fact, it is left-right symmetric and it is related to the cohomological invariants r-sfli(R) and l-sfli(R) by the formula GG-wgldim (R) = max {r-sfli (R), l-sfli (R)}.
... It turns out, in particular, that these Gorenstein homological dimensions are refinements of the classical dimensions of a module M , in the sense that Gpd R (M ) ≤ pd R (M ) and Gid R (M ) ≤ id R (M ) with equality each time the corresponding classical homological dimension is finite. The reader is referred to [4,6,8,10,12,17,18,20,23,[30][31][32][33] for basics and recent investigations on Gorenstein homological theory as well as some related themes to our subject. Nevertheless, the finiteness of Gorenstein homological dimensions remains one of the key problems of Gorenstein homological algebra (see the survey [11] and the introduction of [12] for a further discussion of this issue). ...
We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R)=max{leftsilp(R),leftspli(R)}, recovering a recent theorem of [I. Emmanouil, J. Algebra 372 (2012), 376-396]. Moreover, this formula permits to recover the main theorem of [D. Bennis and N. Mahdou, Proc. Amer. Math. Soc. 138 (2010), 461-465]. Furthermore, we prove that, in the setting of a left and right Noetherian ring, the Gorenstein global dimension is left-right symmetric, generalizing a theorem of Enochs and Jenda. Finally, using recent work of I. Emmanouil and O. Talelli, we compute the Gorenstein global dimension for various types of rings such as commutative ℵ 0 -Noetherian rings and group rings.
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.
We investigate the structure of pure-syzygy modules in a pure-projective resolution of any right R-module over an associative ring R with an identity element. We show that a right R-module M is pure-projective if and only if there exists an integer n greater than or equal to 0 and a pure-exact sequence 0 --> M --> P-n --> ... P-0 --> M --> 0 with pure-projective modules P-n,..., P-0. As a consequence we get the following version of a result in Benson and Goodearl, 2000: at module M is projective if M admits an exact sequence 0 --> M --> F-n --> ... F-0 --> M --> 0 with projective modules F-n,..., F-0.
In this paper, we introduce a new generalization of coherent rings using the Gorenstein projective dimension. Let n be a positive integer or n = ∞. A ring R is called a left G n -coherent ring in case every finitely generated submodule of finitely generated free left R-modules whose Gorenstein projective dimension ≤ n − 1 is finitely presented. We characterize G n -coherent rings in various ways, using G n -flat, G n -injective modules and cotorsion theory.
An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.
http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=763375&tocid=100086734&page=303-309