22 research outputs found
Strict Mittag-Leffler modules and purely generated classes
We study versions of strict Mittag-Leffler modules relativized to a class
\cK (of modules), that is, \emph{strict} versions (in the technical sense of
Raynaud and Gruson) of \cK-Mittag-Leffler modules, as investigated in the
preceding paper, {\em Mittag-Leffler modules and definable subcategories}, in
this very series (as well as the arXiv)
Strict Mittag-Leffler conditions and locally split morphisms
summary:In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective
On certain homological finiteness conditions
In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions
on the ring and certain properties of the (strict) Mittag-Leffler modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimensio
Almost free modules and Mittag-Leffler conditions
AbstractDrinfeld recently suggested to replace projective modules by the flat Mittag-Leffler ones in the definition of an infinite dimensional vector bundle on a scheme X (Drinfeld, 2006 [8]). Two questions arise: (1) What is the structure of the class D of all flat Mittag-Leffler modules over a general ring? (2) Can flat Mittag-Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi-coherent sheaves on X?We answer (1) by showing that a module M is flat Mittag-Leffler, if and only if M is ℵ1-projective in the sense of Eklof and Mekler (2002) [10]. We use this to characterize the rings such that D is closed under products, and relate the classes of all Mittag-Leffler, strict Mittag-Leffler, and separable modules. Then we prove that the class D is not deconstructible for any non-right perfect ring. So unlike the classes of all projective and flat modules, the class D does not admit the homotopy theory tools developed recently by Hovey (2002) [26]. This gives a negative answer to (2)
Mittag-Leffler conditions on modules
We study Mittag-Leffler conditions on modules providing relative versions of
classical results by Raynaud and Gruson. We then apply our investigations to
several contexts. First of all, we give a new argument for solving the Baer
splitting problem. Moreover, we show that modules arising in cotorsion pairs
satisfy certain Mittag-Leffler conditions. In particular, this implies that
tilting modules satisfy a useful finiteness condition over their endomorphism
ring. In the final section, we focus on a special tilting cotorsion pair
related to the pure-semisimplicity conjecture.Comment: 45 page
Modules whose maximal submodules are supplements
We study modules whose maximal submodules are supplements (direct summands). For a locally projective module, we prove that every maximal submodule is a direct summand if and only if it is semisimple and projective. We give a complete characterization of the modules whose maximal submodules are supplements over Dedekind domains.TÃœBÄ°TAK project number 107T70