Dimensions cohomologiques relieés aux foncteurs

Gruson, Laurent; Jensen, Christian U.

doi:10.1007/bfb0090389

Cited by 105 publications

(58 citation statements)

References 12 publications

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“…there are only finitely many indecomposable R-modules up to isomorphism). The Auslander algebra of R is Aus(R) = End [38] and independently to Auslander [6], between fun-A and fun d -A is described explicitly as follows. If…”

Section: Preadditive Categories and Their Ind-completionsmentioning

confidence: 99%

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Definable additive categories: purity and model theory

Prest

2011

Memoirs of the AMS

102

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Section: Preadditive Categories and Their Ind-completionsmentioning

confidence: 99%

“…The construction, which on objects takes M to (M ⊗ R −) R-mod, works just as well if we start with a functor category Mod-A rather than a module category over a ring. The original reference has few details but more are in [38], also see [46] and [84, §12.1.1]. pinjtoinj Theorem 5.12.…”

Section: Gruson and Jensen [36] Defined A Full Embedding Of Any Modulmentioning

confidence: 99%

Definable additive categories: purity and model theory

Prest

2011

Memoirs of the AMS

102

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“…, n, are pure-injective modules and the associated short exact sequences are pure exact. The supremum of the pure-injective dimensions of the right R-modules is called the right pure global dimension of R [7,6], and is denoted by r. pgldim(R). Thus the rings R such that r. pgldim(R) ≤ 1 provide a natural source of completely pure-injective modules.…”

Section: Rings Of Pure Global Dimension Less Than or Equal To Onementioning

confidence: 99%

“…As a consequence of Theorem 2.1 we see that, in Corollary 1.9, it is enough to assume that every proper pure quotient of E/JE is pure-injective, instead of requiring that E/JE be completely pure-injective. An interesting class of rings of right pure global dimension ≤ 1 is the class of countable rings [6,7]. For instance, it follows from the preceding results that every countable ring R such that every finitely generated submodule of E(R R ) embeds in a finitely presented module of projective dimesion ≤ 1 is finite-dimensional.…”

Section: Rings Of Pure Global Dimension Less Than or Equal To Onementioning

confidence: 99%

Endomorphism rings of completely pure-injective modules

Pardo

Asensio

1996

Proc. Amer. Math. Soc.

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Abstract. Let R be a ring, E = E(R R ) its injective envelope, S = End(E R ) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension ≤ 1, then every finitley generated right S/J-module X is canonically isomorphic to Hom R (E, X ⊗ S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is ≤ 1 and hence when R is a countable ring), then S is semiperfect and R R is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.

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“…Thus Theorem 1 remains true if we merely assume that every quotient of Af by a pure subobject is pure Af-injective. There is a good supply of modules satisfying this condition; for example, all the pure injective modules over a ring of right pure global dimension < 1 (and, in particular, over any countable ring [7,Theoreme 7.10]). Observe also that, in fact, the same arguments show that for a finitely presented object M satisfying the above condition, every flat cyclic right S-module is pure S-injective (without assuming that S is regular).…”

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confidence: 99%

Complete pure injectivity and endomorphism rings

Pardo¹,

Dũng²,

Wisbauer³

1993

Proc. Amer. Math. Soc.

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Abstract. It is shown that if M is a finitely presented completely pure injective object in a locally finitely generated Grothendieck category C such that S = Endc M is von Neumann regular, then 5 is semisimple. This is a generalized version of a well-known theorem of Osofsky, which includes also a result of Damiano on PCI-rings. As an application, we obtain a characterization of right hereditary rings with finitely presented injective hull.In [11,12] Osofsky showed that a ring, all of whose cyclic right modules are injective, is semisimple (Artinian). Faith [6] studied the structure of right PCI-rings, i.e., rings whose proper right cyclic modules are injective, and he left open the question of whether right PCI-rings must be right Noetherian. In In this note we prove a general version for Grothendieck categories of Osofsky's theorem [11,12], which includes also the above-mentioned result of Damiano. Furthermore, our arguments provide a simple proof of this result. We show that if Af is a finitely presented completely (pure) A7-injective object in a locally finitely generated Grothendieck category C such that S = Endc M is von Neumann regular, then S is semisimple. Consequently, if Af is a projective completely injective object in C, then Af = ©/6/ A,, where Endc Aj are division rings and the subobjects of Ai are linearly ordered. As an application to rings, we obtain a characterization of the right hereditary rings 7? such that the injective hull E(Rr) is finitely presented. This extends a result of Colby and Rutter [2]. Note that even in the module case, Damiano's arguments cannot be applied for proving our main theorem.

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Dimensions cohomologiques relieés aux foncteurs

Cited by 105 publications

References 12 publications

Definable additive categories: purity and model theory

Definable additive categories: purity and model theory

Endomorphism rings of completely pure-injective modules

Complete pure injectivity and endomorphism rings

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