Mittag-Leffler conditions on modules. (English) Zbl 1206.16002

Angeleri Hügel, Lidia; Herbera, Dolors

Indiana Univ. Math. J. 57, No. 5, 2459-2517 (2008).

Summary: 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

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. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.

MSC:

16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16E30	Homological functors on modules (Tor, Ext, etc.) in associative algebras
18E15	Grothendieck categories (MSC2010)

Keywords:

Mittag-Leffler conditions; Baer splitting problem; cotorsion pairs; tilting modules; endomorphism ring; pure-semisimplicity conjecture