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A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module... more
A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was proved by Kaplansky's technique of d\'evissage already in \cite[II.\S3]{RG}. Since vector bundles coincide with $\aleph_0$-restricted Drinfeld vector bundles, a question arose in \cite{EGPT} of whether locality holds for $\kappa$-restricted Drinfeld vector bundles for each infinite cardinal $\kappa$. We give a positive answer here by replacing the d\' evissage with its recent refinement involving $\mathcal C$-filtrations and the Hill Lemma.
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We generalize two major ways of obtaining derived equivalences, the tilting process by Happel, Reiten and Smal and Happel's Tilting The- orem, to the setting of nitely presented modules over right coherent rings. Moreover, we extend... more
We generalize two major ways of obtaining derived equivalences, the tilting process by Happel, Reiten and Smal and Happel's Tilting The- orem, to the setting of nitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi{tilted artin algebras as the almost hereditary ones to all right noetherian rings. We also give a streamlined and general presentation of how to obtain derived equivalences without tilting objects, using torsion pairs instead.
Let R be a right noetherian ring and let P<1 be the class of all nitely presented modules of nite projective dimension. We prove that ndimR = n< 1 i there is an (innitely generated) tilting module T such that pdT = n and T? =(... more
Let R be a right noetherian ring and let P<1 be the class of all nitely presented modules of nite projective dimension. We prove that ndimR = n< 1 i there is an (innitely generated) tilting module T such that pdT = n and T? =( P<1)? .I fR is an artin algebra, then T can be taken to be
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ABSTRACT We generalize Hill’s lemma,in order to obtain a large family of C- filtered submodules,from a single C-filtration of a module. We use this to prove the following generalization of Kaplansky’s structure theorem for projective... more
ABSTRACT We generalize Hill’s lemma,in order to obtain a large family of C- filtered submodules,from a single C-filtration of a module. We use this to prove the following generalization of Kaplansky’s structure theorem for projective modules: for any ring R, a cotorsion pair (A,B) in Mod-R is of countable type if and only if every module M 2 A is A,!-filtered. We also prove rank versions of these results for torsion-free modules over commutative domains. As an application, we solve a problem of Bazzoni and Salce [3] by showing that strongly flat modules over any valuation domain coincide with the exten- sions of free modules,by divisible torsion-free modules. Another application yields a short proof of the structure of Matlis localizations of commutative rings.
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We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special... more
We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.