Approximations and Mittag-Leffler conditions – the applications. (English) Zbl 1432.16009
Summary: A classic result by Bass says that the class of all projective modules is covering if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules \(\mathcal{C}\), which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [the second author, Isr. J. Math. 226, No. 2, 737–756 (2018; Zbl 1432.16010)] and give a positive answer when \(\mathcal{C} = \mathcal{A}\), or \(\mathcal{C}\) is the class of all locally \(\mathcal{A}^{\leq\omega}\)-free modules, where \(\mathcal{A}\) is any class of modules fitting in a cotorsion pair \((\mathcal{A}, \mathcal{B})\) such that \(\mathcal{B}\) is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and Artin algebras of infinite representation type.
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 |
Citations:
Zbl 1432.16010References:
| [1] | Angeleri Hügel, L., Covers and envelopes via endoproperties of modules, Proceedings of the London Mathematical Society, 86, 649-665, (2003) · Zbl 1041.16003 · doi:10.1112/S002461150201393X |
| [2] | Angeleri Hügel, L., A key module over pure-semisimple hereditary rings, Journal of Algebra, 307, 361-376, (2007) · Zbl 1151.16016 · doi:10.1016/j.jalgebra.2006.06.025 |
| [3] | Angeleri Hügel, L.; Herbera, D., Mittag-Leffler conditions on modules, Indiana university Mathematics Journal, 57, 2459-2517, (2008) · Zbl 1206.16002 · doi:10.1512/iumj.2008.57.3325 |
| [4] | Angeleri Hügel, L.; Herbera, D., Auslander-Reiten-components over puresemisimple hereditary rings, Journal of Algebra, 331, 285-303, (2011) · Zbl 1247.16016 · doi:10.1016/j.jalgebra.2010.10.039 |
| [5] | Angeleri Hügel, L.; Herbera, D.; Trlifaj, J., Tilting modules and Gorenstein rings, Forum Mathematicum, 18, 211-229, (2006) · Zbl 1107.16010 |
| [6] | Angeleri Hügel, L.; Herbera, D.; Trlifaj, J., Baer modules and Mittag-Leffler modules over tame hereditary algebras, Mathematische Zeitschrift, 265, 1-19, (2010) · Zbl 1235.16012 · doi:10.1007/s00209-009-0499-6 |
| [7] | Angeleri Hügel, L.; Kerner, O.; Trlifaj, J., Large tilting modules and representation type, Manuscripta Mathematica, 132, 483-499, (2010) · Zbl 1233.16011 · doi:10.1007/s00229-010-0356-2 |
| [8] | Angeleri Hügel, L.; Saorín, M., Modules with perfect decompositions, Mathematica Scandinavica, 98, 19-43, (2006) · Zbl 1143.16006 · doi:10.7146/math.scand.a-14981 |
| [9] | Angeleri Hügel, L.; Šaroch, J.; Trlifaj, J., On the telescope conjecture for module categories, Journal of Pure and Applied Algebra, 212, 297-310, (2008) · Zbl 1141.16010 · doi:10.1016/j.jpaa.2007.05.027 |
| [10] | Baer, D., Homological properties of wild hereditary Artin algebras, Representation Theory. I (Ottawa, 1177, 1-12, (1986) · Zbl 0648.16024 |
| [11] | D˜ung, N. V.; García, J. L., Indecomposable modules over pure semisimple hereditary rings, Journal of Algebra, 371, 577-595, (2012) · Zbl 1285.16007 · doi:10.1016/j.jalgebra.2012.09.004 |
| [12] | R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics, Vol. 41, Walter de Gruyter, Berlin, 2012. · Zbl 1292.16001 |
| [13] | A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents, Institut de Hautes Études Scientifiques. Publications Mathématiques 11 (1961). · Zbl 0122.16102 |
| [14] | Herbera, D., Definable classes and Mittag-Leffler conditions, Ring Theory and its Applications, 609, 137-166, (2014) · Zbl 1296.16011 · doi:10.1090/conm/609/12123 |
| [15] | Herbera, D.; Trlifaj, J., Almost free modules and Mittag-Leffler conditions, Advances in Mathematics, 229, 3436-3467, (2012) · Zbl 1279.16001 · doi:10.1016/j.aim.2012.02.013 |
| [16] | Kerner, O.; Takane, M., Mono orbits, epi orbits, and elementary vertices of representation infinite quivers, Communications in Algebra, 25, 51-77, (1997) · Zbl 0877.16005 · doi:10.1080/00927879708825839 |
| [17] | M. Prest, Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, Vol. 121, Cambridge University Press, Cambridge, 2009. · Zbl 1205.16002 |
| [18] | Šaroch, J., Approximations and Mittag-Leffler conditions. the tools, Israel Journal of mathematics, 226, 737-756, (2018) · Zbl 1432.16010 · doi:10.1007/s11856-018-1710-4 |
| [19] | Šaroch, J.; Št’ovíček, J., The countable telescope conjecture for module categories, Advances in Mathematics, 219, 1002-1036, (2008) · Zbl 1210.16007 · doi:10.1016/j.aim.2008.05.012 |
| [20] | Slávik, A.; Trlifaj, J., Approximations and locally free modules, Bulletin of the London Mathematical Society, 46, 76-90, (2014) · Zbl 1309.16004 · doi:10.1112/blms/bdt069 |
| [21] | J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, Vol. 1634, Springer, Berlin, 1996. · Zbl 0860.16002 |
| [22] | Zimmermann, W., Existenz von AR-folgen, Archiv der Mathematik, 40, 40-49, (1983) · Zbl 0513.16019 · doi:10.1007/BF01192750 |
| [23] | Zimmermann, W., (σ-)algebraic compactness of rings, Journal of Pure and Applied Algebra, 23, 319-328, (1982) · Zbl 0474.16016 · doi:10.1016/0022-4049(82)90104-9 |
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