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Closure properties of lim C

Publikace na Matematicko-fyzikální fakulta |
2022

Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".Abstrakt

Let C be a class of modules and L = limRIGHTWARDS ARROW C the class of all direct limits of modules from C. The class L is well understood when C consists of finitely presented modules: L then enjoys various closure properties. Our first goal here is to study the closure properties of L in the general case when C SUBSET OF OR EQUAL TO   Mod-R is arbitrary. Then we concentrate on two important particular cases, when C = add M and C = Add M, for an arbitrary module M.

In the first case, we prove that lim

RIGHTWARDS ARROW add M = {N ELEMENT OF Mod-R |

THERE EXISTS F ELEMENT OF FS : N TILDE OPERATOR+D91= F CIRCLED TIMES  S M} where S = End M, and

FS is the class of all flat right S-modules. In the second case, lim

RIGHTWARDS ARROW Add M = {F S M | F ELEMENT OF FS} where S is the endomorphism ring of M endowed with the finite topology,

FS is the class of all right S-contramodules that are direct limits of direct systems of projective right S-contramodules, and F S M is the contratensor product of the right Scontramodule F with the discrete left S-module M.

For various classes of modules D, we show that if M ELEMENT OF D then lim

RIGHTWARDS ARROW add M = limRIGHTWARDS ARROW Add M (e.g., when D consists of pure projective modules), but the equality for an arbitrary module

M remains open. Finally, we deal with the question of whether lim

RIGHTWARDS ARROW Add M = AddM where AddM is the class of all pure epimorphic images of direct sums of copies of a module M.

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