Approximations and Mittag-Leffler conditions – the tools. (English) Zbl 1432.16010
Summary: Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [M. Raynaud and L. Gruson, Invent. Math. 13, 1–89 (1971; Zbl 0227.14010); A. Grothendieck, Publ. Math., Inst. Hautes Étud. Sci. 17, 137–223 (1963; Zbl 0122.16102); M. Prest, Purity, spectra and localisation. Cambridge: Cambridge University Press (2009; Zbl 1205.16002)]. If \(R\) is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [D. Herbera and J. Trlifaj, Adv. Math. 229, No. 6, 3436–3467 (2012; Zbl 1279.16001)], and it does not provide for approximations when \(R\) has cardinality \(\leq \aleph_{0}\), [S. Bazzoni and J. Šťovíček, Proc. Am. Math. Soc. 140, No. 5, 1527–1533 (2012; Zbl 1256.16003)]. We remove the cardinality restriction on \(R\) in the latter result. We also prove an extension of the countable telescope conjecture [the author and J. Šťovíček, Adv. Math. 219, No. 3, 1002–1036 (2008; Zbl 1210.16007)]: a cotorsion pair \((\mathcal A, \mathcal B)\) is of countable type whenever the class \(\mathcal B\) is closed under direct limits.
In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [L. Angeleri Hügel et al., Isr. J. Math. 226, No. 2, 757–780 (2018; Zbl 1432.16009)], and enable investigation of new classes of flat modules occurring in algebraic geometry [A. Slávik and J. Trlifaj, J. Pure Appl. Algebra 220, No. 12, 3910–3926 (2016; Zbl 1348.13018)]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [the author, Invent. Math. 209, No. 2, 463–479 (2017; Zbl 1414.16015)].
In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [L. Angeleri Hügel et al., Isr. J. Math. 226, No. 2, 757–780 (2018; Zbl 1432.16009)], and enable investigation of new classes of flat modules occurring in algebraic geometry [A. Slávik and J. Trlifaj, J. Pure Appl. Algebra 220, No. 12, 3910–3926 (2016; Zbl 1348.13018)]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [the author, Invent. Math. 209, No. 2, 463–479 (2017; Zbl 1414.16015)].
MSC:
| 16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 03E10 | Ordinal and cardinal numbers |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
Citations:
Zbl 0227.14010; Zbl 0122.16102; Zbl 1205.16002; Zbl 1279.16001; Zbl 1256.16003; Zbl 1210.16007; Zbl 1348.13018; Zbl 1414.16015; Zbl 1432.16009References:
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