Abstract
Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. 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 [16], and it does not provide for approximations when R has cardinality ≤ ℵ0, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class 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 [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. 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 [22].
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References
L. Angeleri Hügel, S. Bazzoni and D. Herbera, A solution to the Baer splitting problem, Transactions of the American Mathematical Society 360 (2008), 2409–2421.
L. Angeleri Hügel and D. Herbera, Mittag-Leffler conditions on modules, Indiana University Mathematics Journal 57 (2008), 2459–2517.
L. Angeleri Hügel, J. Šaroch and J. Trlifaj, On the telescope conjecture for module categories, Journal of Pure and Applied Algebra 212 (2008), 297–310.
L. Angeleri Hügel, J. Šaroch and J. Trlifaj, Approximations and Mittag-Leffler conditions. The applications, Israel Journal of Mathematics 226 (2018), 757–780.
G. Azumaya and A. Facchini, Rings of pure global dimension zero and Mittag-Leffler modules, Journal of Pure and Applied Algebra 62 (1989), 109–122.
H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Transactions of the American Mathematical Society 95 (1960), 466–488.
S. Bazzoni and D. Herbera, One dimensional tilting modules are of finite type, Algebras and Representation Theory 11 (2008), 43–61.
S. Bazzoni and J. Št’ovíček, Flat Mittag-Leffler modules over countable rings, Proceedings of the American Mathematical Society 140 (2012), 1527–1533.
K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer, New York–Berlin, 1982.
V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry: an introduction, in The Unity of Mathematics, Progress in Mathematics, Vol. 244, Birkhäuser, Boston, MA, 2006, pp. 263–304.
P. C. Eklof and A. H. Mekler, Almost Free Modules, North-Holland Mathematical Library, Vol. 65, North–Holland, Amsterdam, 2002.
S. Estrada, P. Guil Asensio, M. Prest and J. Trlifaj, Model category structures arising from Drinfeld vector bundles, Advances in Mathematics 231 (2012), 1417–1438.
R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics, Vol. 41, Walter de Gruyter, Berlin, 2012.
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents, Institut des Hautes Études Scientifiques. Publications Mathématiques 11 (1961).
D. Herbera, Definable classes and Mittag-Leffler conditions, in Ring Theory and its Applications, Contemporary Mathematics, Vol. 609, American Mathematical Society, Providence, RI, 2014, pp. 137–166.
D. Herbera and J. Trlifaj, Almost free modules and Mittag-Leffler conditions, Advances in Mathematics 229 (2012), 3436–3467.
S. Kabbaj and N. Mahdou, Trivial extensions of local rings and a conjecture of Costa, in Commutative Ring Theory and Applications (Fez, 2001), Lecture Notes in Pure and Applied Mathematics, Vol. 231, Dekker, New York, 2003, pp. 301–311.
M. Prest, Model Theory and Modules, London Mathematical Society Lecture Note Series, Vol. 130, Cambridge University Press, Cambridge, 1988.
M. Prest, Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, Vol. 121, Cambridge University Press, Cambridge, 2009.
M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Inventiones Mathematicae 13 (1971), 1–89.
P. Rothmaler, Mittag-Leffler modules and positive atomicity, Habilitationsschrift, Christian–Albrechts–Universität zu Kiel, 1994.
J. Šaroch, On the non-existence of right almost split maps, Inventiones Mathematicae 209 (2017), 463–479.
J. Šaroch and J. Št’ovíček, The countable Telescope Conjecture for module categories, Advances in Mathematics 219 (2008), 1002–1036.
J. Šaroch and J. Trlifaj, Kaplansky classes, finite character, and 1-projectivity Forum Mathematicum 24 (2012), 1091–1109.
A. Slávik and J. Trlifaj, Approximations and locally free modules, Bulletin of the London Mathematical Society 46 (2014), 76–90.
A. Slávik and J. Trlifaj, Very flat, locally very flat, and contraadjusted modules, Journal of Pure and Applied Algebra 220 (2016), 3910–3926.
W. Zimmermann, Modules with chain conditions for finite matrix subgroups, Journal of Algebra 190 (1997), 68–87.
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Research supported by grant GAČR 14-15479S.
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Šaroch, J. Approximations and Mittag-Leffler conditions the tools. Isr. J. Math. 226, 737–756 (2018). https://doi.org/10.1007/s11856-018-1710-4
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DOI: https://doi.org/10.1007/s11856-018-1710-4