On the abundance of silting modules.(English)Zbl 1422.16009
Martsinkovsky, Alex (ed.) et al., Surveys in representation theory of algebras. Maurice Auslander distinguished lectures and international conference, Woods Hole Oceanographic Institute, Woods Hole, MA, USA, April 26 – May 1, 2017, April 29 – May 4, 2015, April 18–23, 2013. Selected lectures. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 716, 1-23 (2018).
Summary: Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or of a finite dimensional algebra of finite representation type, can be parametrised by silting modules. In these notes, we give a brief introduction to the fairly recent concepts of silting and cosilting module, and we explain the classification results mentioned above.
For the entire collection see [Zbl 1419.16001].
For the entire collection see [Zbl 1419.16001].
MSC:
| 16G10 | Representations of associative Artinian rings |
| 16D90 | Module categories in associative algebras |
| 16E35 | Derived categories and associative algebras |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
Cite
References:
| [1] | Adachi, Takahide; Iyama, Osamu; Reiten, Idun, \( \tau \)-tilting theory, Compos. Math., 150, 3, 415-452 (2014) ·Zbl 1330.16004 ·doi:10.1112/S0010437X13007422 |
| [2] | Aihara, Takuma; Iyama, Osamu, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2), 85, 3, 633-668 (2012) ·Zbl 1271.18011 ·doi:10.1112/jlms/jdr055 |
| [3] | Angeleri H\`“ugel, Lidia; Archetti, Maria, Tilting modules and universal localization, Forum Math., 24, 4, 709-731 (2012) ·Zbl 1319.16009 ·doi:10.1515/form.2011.080 |
| [4] | Angeleri H\`“ugel, Lidia; Hrbek, Michal, Silting modules over commutative rings, Int. Math. Res. Not. IMRN, 13, 4131-4151 (2017) ·Zbl 1405.13018 ·doi:10.1093/imrn/rnw147 |
| [5] | Angeleri H\`“ugel, Lidia; Marks, Frederik; Vit\'”oria, Jorge, Silting modules, Int. Math. Res. Not. IMRN, 4, 1251-1284 (2016) ·Zbl 1367.16005 ·doi:10.1093/imrn/rnv191 |
| [6] | Angeleri H\`“ugel, Lidia; Marks, Frederik; Vit\'”oria, Jorge, Silting modules and ring epimorphisms, Adv. Math., 303, 1044-1076 (2016) ·Zbl 1407.16006 ·doi:10.1016/j.aim.2016.08.035 |
| [7] | L. Angeleri Hugel, F. Marks, J. Vitoria, A characterisation of \(\)-tilting finite algebras,https://arxiv.org/abs/1801.04312. ·Zbl 1423.16011 |
| [8] | L. Angeleri Hugel, F. Marks, J. Stovicek, R. Takahashi, J. Vitoria, Flat ring epimorphisms and universal localisations of commutative rings, in preparation. ·Zbl 1467.13025 |
| [9] | Angeleri H\`“ugel, Lidia; S\'”anchez, Javier, Tilting modules arising from ring epimorphisms, Algebr. Represent. Theory, 14, 2, 217-246 (2011) ·Zbl 1260.16009 ·doi:10.1007/s10468-009-9186-x |
| [10] | Angeleri H\`“ugel, Lidia; Tonolo, Alberto; Trlifaj, Jan, Tilting preenvelopes and cotilting precovers, Algebr. Represent. Theory, 4, 2, 155-170 (2001) ·Zbl 0999.16007 ·doi:10.1023/A:1011485800557 |
| [11] | Bazzoni, S., Cotilting modules are pure-injective, Proc. Amer. Math. Soc., 131, 12, 3665-3672 (2003) ·Zbl 1045.16004 ·doi:10.1090/S0002-9939-03-06938-7 |
| [12] | Bazzoni, Silvana, When are definable classes tilting and cotilting classes?, J. Algebra, 320, 12, 4281-4299 (2008) ·Zbl 1167.16004 ·doi:10.1016/j.jalgebra.2008.08.028 |
| [13] | Bazzoni, Silvana; Herbera, Dolors, One dimensional tilting modules are of finite type, Algebr. Represent. Theory, 11, 1, 43-61 (2008) ·Zbl 1187.16008 ·doi:10.1007/s10468-007-9064-3 |
| [14] | Bazzoni, Silvana; \v St’ov\'\i\v cek, Jan, Smashing localizations of rings of weak global dimension at most one, Adv. Math., 305, 351-401 (2017) ·Zbl 1386.13043 ·doi:10.1016/j.aim.2016.09.028 |
| [15] | Bergman, George M.; Dicks, Warren, Universal derivations and universal ring constructions, Pacific J. Math., 79, 2, 293-337 (1978) ·Zbl 0359.16001 |
| [16] | Breaz, Simion; Pop, Flaviu, Cosilting modules, Algebr. Represent. Theory, 20, 5, 1305-1321 (2017) ·Zbl 1376.16005 ·doi:10.1007/s10468-017-9688-x |
| [17] | S. Breaz, J. Zemlicka, Torsion classes generated by silting modules,http://arxiv.org/abs/1601.06655. ·Zbl 1414.16026 |
| [18] | Buan, Aslak Bakke; Zhou, Yu, A silting theorem, J. Pure Appl. Algebra, 220, 7, 2748-2770 (2016) ·Zbl 1362.16013 ·doi:10.1016/j.jpaa.2015.12.009 |
| [19] | Buan, Aslak Bakke; Zhou, Yu, Silted algebras, Adv. Math., 303, 859-887 (2016) ·Zbl 1383.16010 ·doi:10.1016/j.aim.2016.07.004 |
| [20] | Buan, Aslak Bakke; Zhou, Yu, Endomorphism algebras of 2-term silting complexes, Algebr. Represent. Theory, 21, 1, 181-194 (2018) ·Zbl 1416.16012 ·doi:10.1007/s10468-017-9709-9 |
| [21] | Cohn, P. M., Free rings and their relations, London Mathematical Society Monographs 19, xxii+588 pp. (1985), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London ·Zbl 0659.16001 |
| [22] | Colpi, Riccardo; Tonolo, Alberto; Trlifaj, Jan, Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes, J. Pure Appl. Algebra, 211, 1, 223-234 (2007) ·Zbl 1137.16009 ·doi:10.1016/j.jpaa.2007.01.004 |
| [23] | Crawley-Boevey, W. W., Regular modules for tame hereditary algebras, Proc. London Math. Soc. (3), 62, 3, 490-508 (1991) ·Zbl 0768.16003 ·doi:10.1112/plms/s3-62.3.490 |
| [24] | Crawley-Boevey, William, Locally finitely presented additive categories, Comm. Algebra, 22, 5, 1641-1674 (1994) ·Zbl 0798.18006 ·doi:10.1080/00927879408824927 |
| [25] | Crawley-Boevey, William, Infinite-dimensional modules in the representation theory of finite-dimensional algebras. Algebras and modules, I, Trondheim, 1996, CMS Conf. Proc. 23, 29-54 (1998), Amer. Math. Soc., Providence, RI ·Zbl 0920.16007 |
| [26] | L. Demonet, O. Iyama, G. Jasso, \(\)-tilting finite algebras, bricks, and g-vectors, to appear in Int. Math. Res. Notices IMRNhttps://doi.org/10.1093/imrn/rnx135. ·Zbl 1485.16013 |
| [27] | Gabriel, P.; de la Pe\~na, J. A., Quotients of representation-finite algebras, Comm. Algebra, 15, 1-2, 279-307 (1987) ·Zbl 0609.16013 ·doi:10.1080/00927878708823421 |
| [28] | Geigle, Werner; Lenzing, Helmut, Perpendicular categories with applications to representations and sheaves, J. Algebra, 144, 2, 273-343 (1991) ·Zbl 0748.18007 ·doi:10.1016/0021-8693(91)90107-J |
| [29] | Hoshino, Mitsuo; Kato, Yoshiaki; Miyachi, Jun-Ichi, On \(t\)-structures and torsion theories induced by compact objects, J. Pure Appl. Algebra, 167, 1, 15-35 (2002) ·Zbl 1006.18011 ·doi:10.1016/S0022-4049(01)00012-3 |
| [30] | Hrbek, Michal, One-tilting classes and modules over commutative rings, J. Algebra, 462, 1-22 (2016) ·Zbl 1348.13015 ·doi:10.1016/j.jalgebra.2016.05.014 |
| [31] | Igusa, Kiyoshi; Schiffler, Ralf, Exceptional sequences and clusters, J. Algebra, 323, 8, 2183-2202 (2010) ·Zbl 1239.16019 ·doi:10.1016/j.jalgebra.2010.02.003 |
| [32] | Ingalls, Colin; Thomas, Hugh, Noncrossing partitions and representations of quivers, Compos. Math., 145, 6, 1533-1562 (2009) ·Zbl 1182.16012 ·doi:10.1112/S0010437X09004023 |
| [33] | O. Iyama, Rejective subcategories of artin algebras and orders,http://arxiv:math/0311281. ·Zbl 1124.16017 |
| [34] | Herbera, Dolors; P\v r\'\i hoda, Pavel, Reconstructing projective modules from its trace ideal, J. Algebra, 416, 25-57 (2014) ·Zbl 1333.16005 ·doi:10.1016/j.jalgebra.2014.06.010 |
| [35] | Keller, Bernhard; Nicol\'as, Pedro, Weight structures and simple dg modules for positive dg algebras, Int. Math. Res. Not. IMRN, 5, 1028-1078 (2013) ·Zbl 1312.18007 ·doi:10.1093/imrn/rns009 |
| [36] | Keller, B.; Vossieck, D., Aisles in derived categories, Bull. Soc. Math. Belg. S\'”er. A, 40, 2, 239-253 (1988) ·Zbl 0671.18003 |
| [37] | Koenig, Steffen; Yang, Dong, Silting objects, simple-minded collections, \(t\)-structures and co-\(t\)-structures for finite-dimensional algebras, Doc. Math., 19, 403-438 (2014) ·Zbl 1350.16010 |
| [38] | H. Krause, M. Saorin, On minimal approximations of modules, In: Trends in the representation theory of finite dimensional algebras (ed. by E. L. Green and B. Huisgen-Zimmermann), Contemp. Math. 229 (1998) 227-236. ·Zbl 0959.16003 |
| [39] | Krause, Henning; \v S\v tov\'\i\v cek, Jan, The telescope conjecture for hereditary rings via Ext-orthogonal pairs, Adv. Math., 225, 5, 2341-2364 (2010) ·Zbl 1242.16007 ·doi:10.1016/j.aim.2010.04.027 |
| [40] | Lam, T. Y., Lectures on modules and rings, Graduate Texts in Mathematics 189, xxiv+557 pp. (1999), Springer-Verlag, New York ·Zbl 0911.16001 ·doi:10.1007/978-1-4612-0525-8 |
| [41] | Marks, Frederik, Universal localisations and tilting modules for finite dimensional algebras, J. Pure Appl. Algebra, 219, 7, 3053-3088 (2015) ·Zbl 1350.16022 ·doi:10.1016/j.jpaa.2014.10.003 |
| [42] | Marks, Frederik, Homological embeddings for preprojective algebras, Math. Z., 285, 3-4, 1091-1106 (2017) ·Zbl 1405.16008 ·doi:10.1007/s00209-016-1739-1 |
| [43] | Marks, Frederik; \v S\v tov\'\i\v cek, Jan, Torsion classes, wide subcategories and localisations, Bull. Lond. Math. Soc., 49, 3, 405-416 (2017) ·Zbl 1407.16012 ·doi:10.1112/blms.12033 |
| [44] | F. Marks, J. Stovicek, Universal localisations via silting, to appear in Proc. Roy. Soc. Edinburgh Sect. A.http://arxiv.org/abs/1605.04222 |
| [45] | Mendoza Hern\'andez, Octavio; S\'aenz Valadez, Edith Corina; Santiago Vargas, Valente; Souto Salorio, Mar\'\i a. Jos\'e, Auslander-Buchweitz context and co-\(t\)-structures, Appl. Categ. Structures, 21, 5, 417-440 (2013) ·Zbl 1291.18017 ·doi:10.1007/s10485-011-9271-2 |
| [46] | Michler, Gerhard, Idempotent ideals in perfect rings, Canad. J. Math., 21, 301-309 (1969) ·Zbl 0201.04101 ·doi:10.4153/CJM-1969-031-0 |
| [47] | Mizuno, Yuya, Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type, Math. Z., 277, 3-4, 665-690 (2014) ·Zbl 1355.16008 ·doi:10.1007/s00209-013-1271-5 |
| [48] | P. Nicolas, M. Saorin, A. Zvonareva, Silting theory in triangulated categories with coproducts.http://arxiv.org/abs/1512.04700 ·Zbl 1436.18013 |
| [49] | Prest, Mike, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications 121, xxviii+769 pp. (2009), Cambridge University Press, Cambridge ·Zbl 1205.16002 ·doi:10.1017/CBO9781139644242 |
| [50] | Psaroudakis, Chrysostomos; Vit\'oria, Jorge, Realisation functors in tilting theory, Math. Z., 288, 3-4, 965-1028 (2018) ·Zbl 1407.18014 ·doi:10.1007/s00209-017-1923-y |
| [51] | Ringel, Claus Michael, The Catalan combinatorics of the hereditary Artin algebras. Recent developments in representation theory, Contemp. Math. 673, 51-177 (2016), Amer. Math. Soc., Providence, RI ·Zbl 1360.05018 ·doi:10.1090/conm/673/13490 |
| [52] | Schofield, A. H., Representation of rings over skew fields, London Mathematical Society Lecture Note Series 92, xii+223 pp. (1985), Cambridge University Press, Cambridge ·Zbl 0571.16001 ·doi:10.1017/CBO9780511661914 |
| [53] | A. H. Schofield, Universal localisations of hereditary rings,http://arxiv.org/0708.0257. ·Zbl 0592.16021 |
| [54] | Stenstr\`“om, Bo, Rings of quotients, viii+309 pp. (1975), Springer-Verlag, New York-Heidelberg ·Zbl 0296.16001 |
| [55] | Wei, Jiaqun, Semi-tilting complexes, Israel J. Math., 194, 2, 871-893 (2013) ·Zbl 1286.16011 ·doi:10.1007/s11856-012-0093-1 |
| [56] | Whitehead, James M., Projective modules and their trace ideals, Comm. Algebra, 8, 19, 1873-1901 (1980) ·Zbl 0447.16018 ·doi:10.1080/00927878008822551 |
| [57] | Zhang, Peiyu; Wei, Jiaqun, Cosilting complexes and AIR-cotilting modules, J. Algebra, 491, 1-31 (2017) ·Zbl 1406.16004 ·doi:10.1016/j.jalgebra.2017.07.022 |
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