#### DMCA

## On rings and algebras of finite Cohen-Macaulay type (2013)

Venue: | ADV. MATH |

Citations: | 12 - 0 self |

### Citations

644 | Representation Theory of Artin Algebras
- Auslander, Reiten, et al.
- 1995
(Show Context)
Citation Context ...n that case it coincides with the full subcategory of finite projective modules), see Theorem 5.2. This has some consequences for the, still open, Gorenstein Symmetry Conjecture, (GSC) for short, see =-=[29, 17, 26]-=-, which predicts that any Artin algebra with finite one-sided self-injective dimension is Gorenstein. We show that (GSC) admits the following representation-theoretic formulation: an Artin algebra Λ w... |

197 | Triangulated categories of singularities and D-branes in LandauGinzburg models, Tr
- Orlov
(Show Context)
Citation Context ... approximations, cluster-tilting objects. 1 ON ALGEBRAS OF FINITE CM-TYPE 2 instance GprojR admits a full triangulated embedding into the triangulated category of singularities of R, in Orlov’s sense =-=[60]-=-, which is an equivalence if and only if R is Gorenstein. On the other hand it was a discovery of many people that also representation-theoretic properties of the category GprojR have important conseq... |

190 |
Stable module theory
- Auslander, Bridger
- 1969
(Show Context)
Citation Context ...stein-Projective Modules 32 References 35 1. Introduction Finitely generated modules of Gorenstein dimension zero over a Noetherian ring R were introduced by Maurice Auslander in the mid-sixties, see =-=[6, 11]-=-, as a natural generalization of finite projective modules in order to provide finer homological invariants of Noetherian rings. Since then they found important applications in commutative algebra, Al... |

187 | On triangulated orbit categories,
- Keller
- 2005
(Show Context)
Citation Context ...f finite-dimensional modules over the path algebra kQ under the action of the automorphism group generated by ON ALGEBRAS OF FINITE CM-TYPE 34 X ÞÑ S´1pXrdsq, where S “ ´bLkQDpkQq. As shown by Keller =-=[48]-=-, C pdq Q is a triangulated category and the projection Dbpmod-kQq ÝÑ C pdq Q is a triangle functor. If d “ 2, we write C pdq Q “ CQ. If T is a k-linear triangulated category over a field k with finit... |

173 |
Representable functors, Serre functors, and reconstructions
- Bondal, Kapranov
- 1989
(Show Context)
Citation Context ...angle functor. If d “ 2, we write C pdq Q “ CQ. If T is a k-linear triangulated category over a field k with finite-dimensional Hom-spaces, then a Serre functor for T, in the sense of Bondal-Kapranov =-=[30]-=-, is a triangulated equivalence S : T ÝÑ T such that: DHomTpA,Bq « ÝÑ HomTpB, SAq naturally @A,B P T. If T admits a Serre functor S, then S is unique up to an isomorphism of triangulated functors, and... |

163 | Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, Available from https://tspace.library.utoronto.ca/handle/1807/16682
- Buchweitz
- 1987
(Show Context)
Citation Context ...Since then they found important applications in commutative algebra, Algebraic Geometry, singularity theory and relative homological algebra. Later on the influential works of Auslander and Buchweitz =-=[12, 33]-=- gave a more categorical approach to the study of modules of Gorenstein-dimension zero. In general the size and the homological complexity of the category GprojR of finitely generated modules of Goren... |

152 |
Application of contravariantly finite subcategories
- Auslander, Reiten
(Show Context)
Citation Context ...cal algebra based on Gorenstein-projective modules, see the book [41] for more details and Christensen’s book [36] for an hyperhomological approach. In the context of Artin algebras, Auslander-Reiten =-=[14, 15]-=- and Happel [42], see also [33, 29], studied finitely generated Gorenstein-projective modules and they established strong connections with cotilting theory and the shape of the singularity category. T... |

145 | Cluster-tilted algebras are Gorenstein and stably Calabi-Yau
- Keller, Reiten
(Show Context)
Citation Context ...s devoted to the study of finitely generated Gorenstein-projective rigid modules, i.e. modules without self-extensions, or cluster tilting Gorenstein-projective modules, in the sense of Keller-Reiten =-=[49]-=- or Iyama [44], over an Artin algebra Λ in connection with the property that Λ is of finite CM-type. We show that Cohen-Macaulay finiteness of Λ is intimately related to the representation finiteness ... |

144 | Gorenstein dimensions,
- Christensen
- 2000
(Show Context)
Citation Context ...orth denoted by GProjR. Enochs and his collaborators studied extensively relative homological algebra based on Gorenstein-projective modules, see the book [41] for more details and Christensen’s book =-=[36]-=- for an hyperhomological approach. In the context of Artin algebras, Auslander-Reiten [14, 15] and Happel [42], see also [33, 29], studied finitely generated Gorenstein-projective modules and they est... |

115 |
The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences,
- Ringel
- 1991
(Show Context)
Citation Context ... consider modules having good filtrations which arise in the study of highest weight categories. Let Λ be a quasi-hereditary algebra. We denote by ∆ the Weyl modules and by ∇ the induced modules, see =-=[61]-=- for more details. Let Filt Add∆, resp. Filt∆, be the full subcategory of Mod-Λ, resp. mod-Λ, consisting of the modules with Weyl filtration. Dually let Filt Add∇, resp. Filt∇, the full subcategory of... |

112 |
Gorenstein injective and projective modules,
- Enochs, Jenda
- 1995
(Show Context)
Citation Context ...f Gorenstein dimension zero, then R is Gorenstein and an isolated singularity, which is a simple hypersurface singularity if in addition R is complete. In the non-commutative setting, Enochs-Jenda in =-=[40]-=- generalized the notion of finitely generated modules of Gorenstein dimension zero over a Noetherian ring in order to cover, not necessarily finitely generated, modules over any ring R, under the name... |

101 |
Relative homological algebra, de Gruyter Expositions
- Enochs, Jenda
- 2000
(Show Context)
Citation Context ... the corresponding full subcategory is henceforth denoted by GProjR. Enochs and his collaborators studied extensively relative homological algebra based on Gorenstein-projective modules, see the book =-=[41]-=- for more details and Christensen’s book [36] for an hyperhomological approach. In the context of Artin algebras, Auslander-Reiten [14, 15] and Happel [42], see also [33, 29], studied finitely generat... |

98 |
Representation dimension of Artin algebras
- Auslander
- 1971
(Show Context)
Citation Context ...the infinitely generated versions of the above results rarely hold in the commutative setting. In section 6, motivated by Auslander’s homological theory of representation-finite Artin algebras Λ, see =-=[7]-=-, we consider relative Auslander algebras ApXq, defined as endomorphism rings of representation generators, i.e. modules containing as direct summands all types of indecomposable modules, of contravar... |

96 | The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel - Auslander, Buchweitz - 1987 |

95 | The stable derived category of a Noetherian scheme,
- Krause
- 2005
(Show Context)
Citation Context ...te length as an R-module. It follows that the R-algebra Λ is Noetherian and therefore, by [34], Λ is self-injective Artinian, so the Frobenius category Mod-Λ is a locally finite. By results of Krause =-=[54]-=-, the stable category GInjR is compactly generated and its compact objects are given, up to direct factors, by the Verdier quotientDbpmod-Rq{perfR, where perfR is the full subcategory ofDbpmod-Rq form... |

88 |
Anneaux de Gorenstein, et torsion en algèbre commutative, in: Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67, Secrétariat mathématique
- Auslander
- 1967
(Show Context)
Citation Context ...stein-Projective Modules 32 References 35 1. Introduction Finitely generated modules of Gorenstein dimension zero over a Noetherian ring R were introduced by Maurice Auslander in the mid-sixties, see =-=[6, 11]-=-, as a natural generalization of finite projective modules in order to provide finer homological invariants of Noetherian rings. Since then they found important applications in commutative algebra, Al... |

84 |
Homological and homotopical aspects of torsion theories,
- Beligiannis, Reiten
- 2002
(Show Context)
Citation Context ...is a direct sum of finitely generated modules. We show that the class of algebras Λ enjoying p:q coincides with the class of virtually Gorenstein algebras of finite Cohen-Macaulay type, introduced in =-=[29, 26]-=-. Thus we solve the problem stated in [35]. This is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under fil... |

83 |
Almost split sequences in subcategories,
- Auslander, Smalø
- 1981
(Show Context)
Citation Context ...berg [57], the category X is also covariantly finite. It follows that X is a functorially finite extension closed subcategory ofmod-Λ. Therefore X is a dualizing R-subvariety of mod-Λ in the sense of =-=[18]-=-. In particular X has left and right almost split maps, in the sense of [18], and there exists an equivalence pmod-Xopqop « mod-X, see [13]. (i) ñ (ii) Assuming (i), let T be a representation generato... |

79 |
Coherent functors
- Auslander
- 1965
(Show Context)
Citation Context ...iant additive functors C op ÝÑ Ab is locally finitely presented and fpMod-C “ mod-C is the full subcategory of coherent functors over C . Recall that an additive functor F : C Ñ Ab is called coherent =-=[5]-=-, if there exists an exact sequence C p´, Xq ÝÑ C p´, Y q ÝÑ F ÝÑ 0. The categories C -Mod and C -mod are defined by C -Mod “ Mod-C op, and C -mod “ mod-C op respectively. Note thatmod-C is abelian if... |

67 | Functors and morphisms determined by objects, in: Representation theory of algebras (Proc. Conf., - Auslander - 1976 |

50 | Relative homological algebra and purity in triangulated category
- Beligiannis
(Show Context)
Citation Context ...q “ 0 if and only if A “ 0. In general for the representation functor H : T ÝÑ FlatpTcptq we have, in general non-reversible, implications: H is faithful ñ H is full ñ H is surjective on objects, see =-=[22]-=- for details. 2.3.2. Purity and the Ziegler Spectrum. A triangle pT q : A ÝÑ B ÝÑ C ÝÑ Ar1s in T is called pure if the induced sequence 0 ÝÑ HpAq ÝÑ HpBq ÝÑ HpCq ÝÑ 0 is exact in LpTq. An object E in ... |

44 |
Model-theoretic algebra with particular emphasis on fields, rings, modules
- Jensen
- 1989
(Show Context)
Citation Context ...tive, i.e. the representation category Mod-GprojΛ is perfect. It is well-known that in a perfect functor category any projective functor is a direct sum of finitely generated projective functors, see =-=[46]-=-. It follows that HpAq “ ‘jPJHpX 1 jq, where the X 1 j lie in GprojΛ. Since H is fully faithful we conclude that A is a direct sum of finitely generated modules and (i) holds. Finally we show that (v)... |

42 |
den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc
- Bondal, Van
- 1996
(Show Context)
Citation Context ...q “ dom. dimΛ ě 2, since Λ is an Auslander algebra. As a consequence we have the following bound on the dimension dimDbpmod-Λq of the bounded derived category of mod-Λ in the sense of Rouquier, see =-=[67, 31]-=-. Note that if X “ projΛ, then dimDbpmod-Λq ď gl. dimΛ “ res. dimXmod-Λ, see [67]. Corollary 6.7. Let X representation-finite resolving subcategory of mod-Λ. Then: dimDbpmod-Λq ď max 2, res. dimX mod-... |

38 | Acyclicity versus total acyclicity for complexes over Noetherian rings,
- Iyengar, Krause
- 2006
(Show Context)
Citation Context ...he complex HomRpP ‚, P q is acyclic, @P P ProjR. Hence any acyclic complex of projective R-modules is totally acyclic. Since R is complete, it is well-known that R admits a dualizing complex. Then by =-=[55]-=- it follows that R is Gorenstein. Then R is Artinian by Proposition 5.14. (ii) ñ (i) Clearly R is complete, and KR “ GProjR by Proposition 5.14. If d “ idR, then any R-module admits a finite resolutio... |

35 |
The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co)stabilization,
- Beligiannis
- 2000
(Show Context)
Citation Context ...R is Artinian. Proof. Since R is Gorenstein, there exists a cotorsion pair pGProjR,Projă8Rq in Mod-R, where Projă8R denotes the full subcategory of all R-modules with finite projective dimension, see =-=[21, 40]-=-. If A lies in K ProjR, let p:q : 0 ÝÑ YA ÝÑ XA ÝÑ A ÝÑ 0 be an exact sequence, where XA is Gorenstein-projective and YA has finite projective dimension. Then clearly YA lies in K ProjR, and since YA ... |

35 | The spectrum of a module category
- Krause
(Show Context)
Citation Context ...f ZgpA q, i.e. a subset U of ZgpA q is Ziegler-closed if U “ UΦ for some collection of maps Φ in fppA q. A subset C of ZgpA q is Ziegler-open if its complement ZgpA qzC is Ziegler-closed. We refer to =-=[53]-=- for more details. 2.2. Contravariantly Finite Subcategories, Filtrations, and Cotorsion Pairs. Let A be an additive category. If V is a full subcategory of A , then a map f : A ÝÑ B in A is called V-... |

34 |
Reiten: Stable equivalence of dualizing R-varieties
- Auslander, I
- 1974
(Show Context)
Citation Context ...efore X is a dualizing R-subvariety of mod-Λ in the sense of [18]. In particular X has left and right almost split maps, in the sense of [18], and there exists an equivalence pmod-Xopqop « mod-X, see =-=[13]-=-. (i) ñ (ii) Assuming (i), let T be a representation generator of X, i.e. X “ addT . If Γ :“ EndΛpT q op, then LpA q “ Mod-X « Mod-Γ. Since Γ is an Artin algebra we have A “ limÝÑX « FlatpprojΓq « Pro... |

31 | From triangulated categories to abelian categories: cluster tilting in a general framework
- Koenig, Zhu
(Show Context)
Citation Context ... and therefore Ω´1A is a direct summand of Ω´1T 1, or equivalently A is a direct summand of T 1, i.e. A P addT . Hence addT “ tA P GprojΛ | HompT ,Ω´1Aq “ 0u. Since addT is contravariantly finite, by =-=[51]-=- it follows that T is a cluster tilting object in GprojΛ. Assume now that T is a cluster tilting object in GprojΛ. Then part (a) follows from (i). By a result of Keller-Reiten [49], the endomorphism a... |

30 |
On the structure of triangulated categories with finitely many indecomposables,
- Amiot
- 2007
(Show Context)
Citation Context ...y by using that the stable triangulated category GprojΛ of finitely generated Gorenstein projective modules over a virtually Gorenstein algebra Λ admits Serre duality [26] and recent results of Amiot =-=[1]-=- and Keller-Reiten [49, 50], we give descriptions of GprojΛ, in case the latter is 2-Calabi-Yau in the sense of [49] and EndΛpT q has finite global dimension, in terms of the cluster category CQ assoc... |

24 |
Large modules over artin algebras. In: Algebra, Topology and Category Theory.
- Auslander
- 1976
(Show Context)
Citation Context ...acterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits, thus generalizing a classical result of Auslander =-=[9]-=- and Ringel-Tachikawa [64]. In the commutative case, if R admits a non-free finitely generated Gorenstein-projective module, then we show that R is of finite Cohen-Macaulay type iff R is Gorenstein an... |

24 |
Application of cotorsion pairs
- Krause, Solberg
- 2003
(Show Context)
Citation Context ...ojective modules over an Artin algebra. Proof. Consider the representation functor H : A ÝÑ LpA q “ Mod-X, HpAq “ HomΛp´, Aq|X, which induces an equivalence A « FlatpXq. By a result of Krause-Solberg =-=[57]-=-, the category X is also covariantly finite. It follows that X is a functorially finite extension closed subcategory ofmod-Λ. Therefore X is a dualizing R-subvariety of mod-Λ in the sense of [18]. In ... |

23 | Acyclic Calabi-Yau categories, with an appendix by Michel Van den
- Keller, Reiten
- 2008
(Show Context)
Citation Context ...enstein k-algebra Λ of finite CM-type over an algebraically closed field k, under the presence of a cluster tilted object. These descriptions follow from recent results of Amiot [1] and Keller-Reiten =-=[50]-=-. First we recall that if Q is a quiver without oriented cycles, and d ě 2 or d “ 1 and Q is a Dynkin quiver, then the d-cluster category of Q over a field k, is the orbit category C pdq Q :“ D bpmod-... |

20 |
Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras,
- Beligiannis
- 2005
(Show Context)
Citation Context ...is a direct sum of finitely generated modules. We show that the class of algebras Λ enjoying p:q coincides with the class of virtually Gorenstein algebras of finite Cohen-Macaulay type, introduced in =-=[29, 26]-=-. Thus we solve the problem stated in [35]. This is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under fil... |

19 |
Direct limits of modules of finite projective dimension, Proc. Algebra Conf.
- L, Trlifaj
- 2002
(Show Context)
Citation Context ...s a direct sum of finitely generated modules. (iii) X is contravariantly finite and any X-filtered module is a direct sum of finitely generated modules. Proof. (i) ô (ii) follows from Theorem 3.1. By =-=[2]-=-, the subcategory X cogenerates a cotorsion pair of finite type pA ,Bq in Mod-Λ, where B “ XK and A “ KB, and moreover we have A fin “ X and A consists of all direct summands of X-filtered modules. Th... |

19 | All tilting modules are of finite type
- Bazzoni, Stovicek
- 2007
(Show Context)
Citation Context ...om Corollary 3.4. Example 3.6. Let T be a (possibly infinitely generated) tilting module over Λ, see for instance [3]. Then pX, TKq is a cotorsion pair of finite type in Mod-Λ, where X “ KpTKq, see =-=[20]-=-. Hence X fin is of finite representation type if and only if X fin is contravariantly finite and any module in X is a direct sum of finitely generated modules. If T is finitely generated, then any mo... |

18 | Independence of the total reflexivity conditions for modules
- Jorgensen, S¸ega
(Show Context)
Citation Context ...o their importance in the homological or representation-theoretic structure of an Artin algebra, it is useful to have convenient descriptions of the category GprojΛ. In this connection Jorgensen-Sega =-=[47]-=- and Yoshino [71] studied the problem of when a finitely generated module without non-zero extensions with the ring, i.e. a stable module, is Gorenstein-projective. By the results of [47] this is not ... |

17 | Finite Gorenstein representation type implies simple singularity, Adv
- Christensen, Piepmeyer, et al.
(Show Context)
Citation Context ... of its indecomposable objects is finite. In this case R is said to be of finite Cohen-Macaulay type, finite CM-type for short. In many cases also the converse holds. For instance in the recent paper =-=[37]-=-, Christensen, Piepmeyer, Striuli and Takahashi proved that if a commutative Noetherian local ring R is of finite CM-type and admits a non-free finitely generated module of Gorenstein dimension zero, ... |

16 | Cotilting modules over tame hereditary algebras,
- Buan, Krause
- 2003
(Show Context)
Citation Context ...sum of indecomposable modules. (iii) Any indecomposable module in A is finitely generated. If piq holds, then A “ KT , where T is a finitely generated 1-cotilting module and A is definable. Proof. By =-=[32]-=- it follows that A “ limÝÑA fin and clearly A fin is resolving. Then the assertions follow from Theorem 3.1 and Proposition 3.12. 4. Artin Algebras and Local Rings of Finite CM-Type In this section ... |

16 | Covers induced by Ext - Eklof, Trlifaj |

14 | Endomorphism rings of finite global dimension
- Leuschke
- 2003
(Show Context)
Citation Context ...7 and 6.8. ON ALGEBRAS OF FINITE CM-TYPE 31 In the commutative case we have the following consequence. Note that if R is Cohen-Macaulay, then the second inequality below was first shown by Leuschke =-=[58]-=-. Corollary 6.10. Let R be a commutative Noetherian complete local ring of finite CM-type. Then: dimDbpmod-Rq ď gl. dimApGprojRq ď max 2, dimR ( ă 8 and the second inequality is an equality if dimR ě ... |

13 |
Left triangulated categories arising from contravariantly finite subcategories
- Beligiannis, Marmaridis
- 1994
(Show Context)
Citation Context ... a resolving subcategory of A . We view the stable category A modulo projectives as a left triangulated category; then clearly the stable category X of X is a left triangulated subcategory of A , see =-=[27]-=-, so X has weak kernels and therefore the category of coherent functors mod-X over X is abelian with enough projectives. Moreover the Yoneda embedding X ÝÑ mod-X, X ÞÝÑ Homp´, Xq has image in Projmod-... |

13 |
The category of good modules over a quasi-hereditary algebra
- Ringel
- 1992
(Show Context)
Citation Context ...tly finite subcategories of mod-Λ. We leave its formulation to the reader. (ii) Let Λ be connected and let X be a contravariantly finite resolving subcategory of mod-Λ. Using Harada-Sai’s Lemma as in =-=[62]-=- it follows that X is of finite representation type if and only if the (relative) Auslander-Reiten quiver ΓpXq, defined by using irreducible maps in X, contains a component whose indecomposable module... |

11 | Right triangulated categories with right semiequivalences - Assem, Beligiannis, et al. - 1998 |

11 | DTr-periodic modules and functors, Representation theory of algebras (Cocoyoc - Auslander, Reiten - 1994 |

11 | On the Freyd categories of an additive category, - Beligiannis - 2000 |

9 | Ø.: Gorenstein algebras and algebras with dominant dimension at least 2
- AUSLANDER, SOLBERG
(Show Context)
Citation Context ...ojΛq has finite global dimension. (vi) ApGprojΛq is an Auslander algebra if and only if G-dimΛ ď 2 ď dom. dimΛ. Note that the Artin algebras Λ satisfying G-dimΛ “ 2 “ dom. dimΛ have been described in =-=[19]-=-. The following consequence gives an upper bound for the dimension of the bounded derived category of an Artin algebra of finite CM-type. Corollary 6.9. Let Λ be a virtually Gorenstein Artin algebra o... |

9 |
Gorenstein algebras of finite Cohen-Macaulay type
- Li, Zhang
- 2010
(Show Context)
Citation Context ... infinity can occur. As a consequence we characterize Gorenstein abelian categories of finite Cohen-Macaulay type, thus generalizing and, at the same time, giving a simple proof to the main result of =-=[59]-=-, see Corollary 6.13. Then we concentrate on relative, resp. stable, Auslander algebras ApGprojΛq, resp. ApGprojΛq, of Artin algebras Λ of finite CM-type which we call (stable) Cohen-Macaulay Auslande... |

8 | An Auslander-type result for Gorenstein projective modules - Chen - 2008 |

7 |
Finite representation type and periodic Hochschild (co-)homology. Trends in the representation theory of finite-dimensional algebras
- Buchweitz
- 1997
(Show Context)
Citation Context ...he dimension of Dbpmod-Rq. Finally we give an application to the periodicity of Hochschild (co)homology of the stable Cohen-Macaulay Auslander algebra ApGprojΛq which is based on results of Buchweitz =-=[34]-=-. The final section 7 of the paper is devoted to the study of finitely generated Gorenstein-projective rigid modules, i.e. modules without self-extensions, or cluster tilting Gorenstein-projective mod... |

6 |
On the telescope conjecture for module categories
- Angeleri-Hügel, Saroch, et al.
- 2006
(Show Context)
Citation Context ...A consists of all direct summands of X-filtered modules. Then (i) ô (iii) follows from Corollary 3.4. Example 3.6. Let T be a (possibly infinitely generated) tilting module over Λ, see for instance =-=[3]-=-. Then pX, TKq is a cotorsion pair of finite type in Mod-Λ, where X “ KpTKq, see [20]. Hence X fin is of finite representation type if and only if X fin is contravariantly finite and any module in X i... |

6 | Auslander–Reiten triangles, Ziegler spectra and Gorenstein rings’, K-Theory 32
- Beligiannis
- 2004
(Show Context)
Citation Context ...le pure-injective objects of T form a small set, called the Ziegler spectrum, and admits a natural topology, the Ziegler topology, as in the case of locally finitely presented categories. We refer to =-=[25]-=- for more details. Conventions and Notations. If Λ is a ring, we denote by Mod-Λ the category of right Λmodules. Left Λ-modules are treated as right Λop-modules. The category of finitely presented rig... |

6 | Thick subcategories and virtually Gorenstein algebras
- Beligiannis, Krause
(Show Context)
Citation Context ...and global structural properties of the ring R, especially when R is Gorenstein. In this connection the class of virtually Gorenstein Artin algebras was introduced in [29], and studied extensively in =-=[26, 28]-=-, as a convenient common generalization of Gorenstein algebras and algebras of finite representation type. Note that Λ is virtually Gorenstein if and only if any Gorenstein-projective module is a filt... |

6 | Functors on locally finitely presented additive categories
- Krause
- 1998
(Show Context)
Citation Context ...mÝÑSubpmod-Λq Ď SubpMod-Λq. On the other hand it is well known that any monomorphism f : A B in Mod-Λ can be written as a filtered colimit of monomorphisms between finitely generated Λ-modules, see =-=[52]-=-. Therefore SubpMod-Λq “ lim ÝÑ Subpmod-Λq Since Subpmod-Λq is contravariantly finite in mod-T2pΛq, see [65], the first three conditions are equivalent by Theorem 3.1. (i) ô (iv) If Subpmod-Λq is of f... |

4 | Homotopy theory of modules and Gorenstein - Beligiannis |

4 | Rings without a Gorenstein analogue of the Govorov-Lazard theorem - Holm, Jørgensen |

3 | Higher Auslander-Reiten Theory on maximal orthogonal subcategories
- Iyama
(Show Context)
Citation Context ...he study of finitely generated Gorenstein-projective rigid modules, i.e. modules without self-extensions, or cluster tilting Gorenstein-projective modules, in the sense of Keller-Reiten [49] or Iyama =-=[44]-=-, over an Artin algebra Λ in connection with the property that Λ is of finite CM-type. We show that Cohen-Macaulay finiteness of Λ is intimately related to the representation finiteness of the stable ... |

2 |
On Gorenstein algebras, Progr
- Happel
- 1991
(Show Context)
Citation Context ... Gorenstein-projective modules, see the book [41] for more details and Christensen’s book [36] for an hyperhomological approach. In the context of Artin algebras, Auslander-Reiten [14, 15] and Happel =-=[42]-=-, see also [33, 29], studied finitely generated Gorenstein-projective modules and they established strong connections with cotilting theory and the shape of the singularity category. The results in th... |

1 |
Boevey, Locally Finitely Presented Additive Categories
- Crawley
- 1994
(Show Context)
Citation Context ...kernel of X2 ÝÑ X3. Weak cokernels are defined dually. Note that a locally finitely presented category A has products iff fpA has weak cokernels and A is abelian iff A is a Grothendieck category, see =-=[38]-=-. In this case A is called locally finite if any finitely presented object has finite Jordan-Hölder length, equivalently fpA is Artinian and Noetherian. 2.1.1. Representation Categories, Purity and t... |

1 |
On the vanishing of lim ÐÝ piq
- Jensen
- 1970
(Show Context)
Citation Context ...icular any flat R-module is Gorenstein-projective. However since R is local, it follows that the big finitistic dimension, which is equal to the Krull dimension of R, is finite. By a result of Jensen =-=[45]-=- this implies that any flat R-module has finite projective dimension. Since clearly any Gorenstein-projective module with finite projective dimension is projective, we infer that any flat R-module is ... |

1 |
On minimal approximations of modules, Contemp
- Krause, Saorin
- 1998
(Show Context)
Citation Context ...tinian. (ii)ñ (i) Since R is Artinian, it follows that R is complete and moreover we have KR “ K ProjR, since in this case any projective R-module is a direct summand of a product of copies of R, see =-=[56]-=-. Hence GProjR “ KR. Corollary 5.15. The following statements are equivalent. (i) R is complete and Ω8pMod-Rq Ď KR Ď GProjR. (ii) R is Artinian Gorenstein. If piiq holds, then the inclusions in piq ... |