Closure properties of lim⟶⁡C

0557710 - MÚ 2023 RIV US eng J - Journal Article
Positselski, Leonid - Příhoda, P. - Trlifaj, J.
Closure properties of lim⟶⁡C.
Journal of Algebra. Roč. 606, September 15 (2022), s. 30-103. ISSN 0021-8693. E-ISSN 1090-266X
R&D Projects: GA ČR(CZ) GA20-13778S
Institutional support: RVO:67985840
Keywords : direct limits in module categories * pure projective modules * flat contramodules over topological rings
OECD category: Pure mathematics
Impact factor: 0.9, year: 2022
Method of publishing: Limited access
https://doi.org/10.1016/j.jalgebra.2022.04.029

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. We study the closure properties of L in the general case when C is arbitrary class of modules. 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 L is the class of all tensor products of L with flat modules over the endomorphism ring of M. In the second case, we show that L is the class of all contratensor products of M, over the endomorphism ring of M endowed with the finite topology, with contramodules that can be obtained as direct limits of projective contramodules. For modules M from various classes of modules (e.g., for pure projective modules), we prove that lim add M = lim Add M, but the general case remains open.
Permanent Link: http://hdl.handle.net/11104/0331624