Closure properties of $\varinjlim\mathcal C$

Abstract

Let C be a class of modules and L = lim − → 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 ⊆ 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 limand F S is the class of all flat right Smodules. In the second case, limwhere S is the endomorphism ring of M endowed with the finite topology, F S 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 S-contramodule F with the discrete left S-module M .For various classes of modules D, we show that if M ∈ D then limAdd M , but the equality for an arbitrary module M remains open. Finally, we deal with the case when M is an (infinitely generated) tilting module, and consider the problem of whether lim − → Add M = Add M where Add M is the class of all pure-epimorphic images of direct sums of copies of M . We prove that the equality holds, e.g., for all tilting modules over Dedekind domains.

Contramodules

“…The aim of this section is to describe a certain class of adjusted contramodules over topological rings which plays a crucial role in the arguments in [89, Section 5] and [88,Section 7], and will probably prove to be important and useful in other contexts as well (cf. [87,91,9,10,93]). To begin with, let us briefly return to the discussion of contramodules over the adic completions of Noetherian rings by centrally generated ideals from Section 2.…”

“…An even more general setting of contramodules over a complete, separated topological ring with a (not necessarily countable) base of neighborhoods of zero consisting of open right ideals is discussed in the papers [87,91,10,93]. In this context, the definition of a flat contramodule is the same as in the previous one.…”

“…In this context, the definition of a flat contramodule is the same as in the previous one. The class of flat contramodules is still closed under filtered inductive limits, but it no longer needs to be closed under the kernels of surjective morphisms [93,Example 12.4]. An example of a flat contramodule which cannot be obtained as a filtered inductive limit of projective ones can be found in [93,Example 9.2].…”

“…The class of flat contramodules is still closed under filtered inductive limits, but it no longer needs to be closed under the kernels of surjective morphisms [93,Example 12.4]. An example of a flat contramodule which cannot be obtained as a filtered inductive limit of projective ones can be found in [93,Example 9.2]. Topological rings over which the classes of projective and flat left contramodules coincide are characterized, under the name of topologically left perfect topological rings, in the paper [91, Section 14].…”

“…Mostly, these are applications to commutative algebra [7,89,88], among which the most important one, in our view, is the proof of the Very Flat Conjecture (which was formulated in the February 2014 version of the long preprint [71] and proved in the August 2017 preprint [89]). There is also an application to noncommutative ring theory [82] and an application to direct limits of classes of modules over noncommutative rings [93].…”