Classifying subcategories of modules over Noetherian algebras.(English)Zbl 07842511
Summary: The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring \(R\) and a module-finite \(R\)-algebra \(\Lambda\), we study the set \(\mathsf{tors} \Lambda\) (respectively, \(\mathsf{torf} \Lambda\)) of torsion (respectively, torsionfree) classes of the category of finitely generated \(\Lambda\)-modules. We construct a bijection from \(\mathsf{torf} \Lambda\) to \(\prod_{\mathfrak{p}} \mathsf{torf}(\kappa(\mathfrak{p}) \otimes_R \Lambda)\), and an embedding \(\Phi_{\mathrm{t}}\) from \(\mathsf{tors} \Lambda\) to \(\mathbb{T}_R(\Lambda) : = \prod_{\mathfrak{p}} \mathsf{tors}(\kappa(\mathfrak{p}) \otimes_R \Lambda)\), where \(\mathfrak{p}\) runs over all prime ideals of \(R\). When \(\Lambda = R\), these give classifications of torsionfree classes, torsion classes and Serre subcategories of \(\mathsf{mod} R\) due to Takahashi, Stanley-Wang and Gabriel. To give a description of \(\operatorname{Im} \Phi_{\mathrm{t}}\), we introduce the notion of compatible elements in \(\mathbb{T}_R(\Lambda)\), and prove that all elements in \(\operatorname{Im} \Phi_{\mathrm{t}}\) are compatible. We give a sufficient condition on \((R, \Lambda)\) such that all compatible elements belong to \(\operatorname{Im} \Phi_{\mathrm{t}}\) (we call \((R, \Lambda)\) compatible in this case). For example, if \(R\) is semi-local and \(\dim R \leq 1\), then \((R, \Lambda)\) is compatible. We also give a sufficient condition in terms of silting \(\Lambda\)-modules. As an application, for a Dynkin quiver \(Q\), \((R, R Q)\) is compatible and we have a poset isomorphism \(\mathsf{tors} R Q \simeq \operatorname{Hom}_{\operatorname{poset}}(\operatorname{Spec} R, \mathfrak{C}_Q)\) for the Cambrian lattice \(\mathfrak{C}_Q\) of \(Q\).
MSC:
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 13D30 | Torsion theory for commutative rings |
| 16E35 | Derived categories and associative algebras |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 18G80 | Derived categories, triangulated categories |
Keywords:
Noetherian algebra;torsion class;torsionfree class;Serre subcategory;silting complex;silting moduleCite
References:
| [1] | Adachi, T.; Iyama, O.; Reiten, I., τ-tilting theory, Compos. Math., 150, 3, 415-452, 2014 ·Zbl 1330.16004 |
| [2] | Aihara, T., Tilting-connected symmetric algebras, Algebr. Represent. Theory, 16, 3, 873-894, 2013 ·Zbl 1348.16010 |
| [3] | Aihara, T.; Honma, T.; Miyamoto, K.; Wang, Q., Report on the finiteness of silting objects, Proc. Edinb. Math. Soc. (2), 64, 2, 217-233, 2021 ·Zbl 1505.16013 |
| [4] | Aihara, T.; Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. (2), 85, 3, 633-668, 2012 ·Zbl 1271.18011 |
| [5] | Angeleri-Hügel, L., On the abundance of silting modules, (Surveys in Representation Theory of Algebras. Surveys in Representation Theory of Algebras, Contemp. Math., vol. 716, 2018, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-23 ·Zbl 1422.16009 |
| [6] | Angeleri-Hügel, L.; Hrbek, M., Silting modules over commutative rings, Int. Math. Res. Not., 13, 4131-4151, 2017 ·Zbl 1405.13018 |
| [7] | Angeleri-Hügel, L.; Marks, F.; Vitória, J., Silting modules, Int. Math. Res. Not., 4, 1251-1284, 2016 ·Zbl 1367.16005 |
| [8] | Angeleri-Hügel, L.; Pospíšil, D.; Št́ovíček, J.; Trlifaj, J., Tilting, cotilting, and spectra of commutative Noetherian rings, Trans. Am. Math. Soc., 366, 7, 3487-3517, 2014 ·Zbl 1291.13018 |
| [9] | Antieau, B.; Stevenson, G., Derived categories of representations of small categories over commutative Noetherian rings, Pac. J. Math., 283, 1, 21-42, 2016 ·Zbl 1406.13019 |
| [10] | Aoki, T., Classifying torsion classes for algebras with radical square zero via sign decomposition, J. Algebra, 610, 167-198, 2022 ·Zbl 1514.16014 |
| [11] | Aoki, T.; Higashitani, A.; Iyama, O.; Kase, R.; Mizuno, Y., Fans and polytopes in tilting theory I: foundations |
| [12] | Ariki, S.; Lyle, S.; Speyer, L., Schurian-finiteness of blocks of type A Hecke algebras, J. Lond. Math. Soc. (2), 108, 6, 2333-2376, 2023 ·Zbl 1535.20024 |
| [13] | Asai, S., Semibricks, Int. Math. Res. Not., 16, 4993-5054, 2020 ·Zbl 1467.16009 |
| [14] | Asai, S.; Iyama, O., Semistable torsion classes and canonical decompositions in Grothendieck groups ·Zbl 07945747 |
| [15] | Assem, I.; Simson, D.; Skowroński, A., Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65, 2006, Cambridge University Press: Cambridge University Press Cambridge ·Zbl 1092.16001 |
| [16] | Auslander, M.; Bridger, M., Stable Module Theory, Memoirs of the American Mathematical Society, vol. 94, 1969, American Mathematical Society: American Mathematical Society Providence, RI, 146 pp. ·Zbl 0204.36402 |
| [17] | Auslander, M.; Smalø, S. O., Preprojective modules over Artin algebras, J. Algebra, 66, 1, 61-122, 1980 ·Zbl 0477.16013 |
| [18] | Baur, K.; Laking, R., Torsion pairs and cosilting in type \(\widetilde{A} \), J. Pure Appl. Algebra, 226, 10, Article 107057 pp., 2022, 38 pp. ·Zbl 1504.16019 |
| [19] | Beilinson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers, (Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981. Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981, Asterisque, vol. 100, 1982, Soc. Math. France: Soc. Math. France Paris), 5-171, (French) ·Zbl 1390.14055 |
| [20] | Brüstle, T.; Yang, D., Ordered exchange graphs, (Advances in Representation Theory of Algebras. Advances in Representation Theory of Algebras, EMS Ser. Congr. Rep., 2013, Eur. Math. Soc.: Eur. Math. Soc. Zürich), 135-193 ·Zbl 1353.16012 |
| [21] | Buan, A.; Iyama, O.; Reiten, I.; Scott, J., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., 145, 4, 1035-1079, 2009 ·Zbl 1181.18006 |
| [22] | Chan, A.; Demonet, L., Classifying torsion classes of gentle algebras |
| [23] | Crawley-Boevey, W., Locally finitely presented additive categories, Commun. Algebra, 22, 1644-1674, 1994 ·Zbl 0798.18006 |
| [24] | Curtis, C. W.; Reiner, I., Methods of Representation Theory. Vol. I. With Applications to Finite Groups and Orders, Pure and Applied Mathematics, 1981, John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York ·Zbl 0469.20001 |
| [25] | Demonet, L.; Iyama, O.; Jasso, G., τ-tilting finite algebras, bricks and g-vectors, Int. Math. Res. Not., 3, 852-892, 2019 ·Zbl 1485.16013 |
| [26] | Demonet, L.; Iyama, O.; Reading, N.; Reiten, I.; Thomas, H., Lattice theory of torsion classes: beyond τ-tilting theory, Trans. Amer. Math. Soc. Ser. B, 10, 542-612, 2023 ·Zbl 1533.16015 |
| [27] | Eisenbud, D., Commutative Algebra. With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, 1995, Springer-Verlag: Springer-Verlag New York ·Zbl 0819.13001 |
| [28] | Enomoto, H., IE-closed subcategories of commutative rings are torsion-free classes |
| [29] | Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. Fr., 90, 323-448, 1962 ·Zbl 0201.35602 |
| [30] | Garkusha, G.; Prest, M., Torsion classes of finite type and spectra, (K-Theory and Noncommutative Geometry. K-Theory and Noncommutative Geometry, EMS Ser. Congr. Rep., 2008, Eur. Math. Soc.: Eur. Math. Soc. Zürich), 393-412 ·Zbl 1211.14007 |
| [31] | Gnedin, W., Silting theory of orders modulo a regular sequence, Oberwolfach Rep., 17, 1, 182-185, 2020, Representation theory of quivers and finite dimensional algebras |
| [32] | Gnedin, W.; Iyengar, S.; Krause, H., A class of Gorenstein algebras and their dualities ·Zbl 1546.16018 |
| [33] | Guralnick, R. M., Lifting homomorphisms of modules, Ill. J. Math., 29, 1, 153-156, 1985 ·Zbl 0558.16003 |
| [34] | Hovey, M., Classifying subcategories of modules, Trans. Am. Math. Soc., 353, 8, 3181-3191, 2001 ·Zbl 0981.13006 |
| [35] | Iima, K.; Matsui, H.; Shimada, K.; Takahashi, R., When is a subcategory Serre or torsionfree? ·Zbl 07958229 |
| [36] | Iyama, O.; Jorgensen, P.; Yang, D., Intermediate co-t-structures, two-term silting objects, τ-tilting modules, and torsion classes, Algebra Number Theory, 8, 10, 2413-2431, 2014 ·Zbl 1305.18048 |
| [37] | Iyama, O.; Reiten, I., Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Am. J. Math., 130, 4, 1087-1149, 2008 ·Zbl 1162.16007 |
| [38] | Iyama, O.; Reiten, I.; Thomas, H.; Todorov, G., Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc., 47, 4, 639-650, 2015 ·Zbl 1397.16011 |
| [39] | Iyama, O.; Wemyss, M., Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math., 197, 3, 521-586, 2014 ·Zbl 1308.14007 |
| [40] | Iyama, O.; Wemyss, M., Singular derived categories of \(\mathbb{Q} \)-factorial terminalizations and maximal modification algebras, Adv. Math., 261, 85-121, 2014 ·Zbl 1326.14033 |
| [41] | Iyama, O.; Yang, D., Silting reduction and Calabi-Yau reduction of triangulated categories, Trans. Am. Math. Soc., 370, 11, 7861-7898, 2018 ·Zbl 1443.18006 |
| [42] | Ingalls, C.; Thomas, H., Noncrossing partitions and representations of quivers, Compos. Math., 145, 6, 1533-1562, 2009 ·Zbl 1182.16012 |
| [43] | Jasso, G., Reduction of τ-tilting modules and torsion pairs, Int. Math. Res. Not., 16, 7190-7237, 2015 ·Zbl 1357.16028 |
| [44] | Kanda, R., Classifying Serre subcategories via atom spectrum, Adv. Math., 231, 3-4, 1572-1588, 2012 ·Zbl 1255.18011 |
| [45] | Kanda, R., Extension groups between atoms and objects in locally Noetherian Grothendieck category, J. Algebra, 422, 53-77, 2015 ·Zbl 1314.18012 |
| [46] | Kaplansky, I., Commutative Rings, 1974, University of Chicago Press: University of Chicago Press Chicago, Ill.-London ·Zbl 0296.13001 |
| [47] | Kimura, Y., Tilting and silting theory of Noetherian algebras, Int. Math. Res. Not., 2, 1685-1732, 2024 ·Zbl 1548.13032 |
| [48] | Koshio, R.; Kozakai, Y., Induced modules of support τ-tilting modules and extending modules of semibricks over blocks of finite groups, J. Algebra, 628, 524-544, 2023 ·Zbl 1512.20038 |
| [49] | Krause, H., Thick subcategories of modules over commutative Noetherian rings (with an appendix by Srikanth Iyengar), Math. Ann., 340, 4, 733-747, 2008 ·Zbl 1143.13012 |
| [50] | Krause, H., Krull-Schmidt categories and projective covers, Expo. Math., 33, 4, 535-549, 2015 ·Zbl 1353.18011 |
| [51] | Li, J., Piecewise hereditary algebras under field extensions, Czechoslov. Math. J., 71(146), 4, 1025-1034, 2021 ·Zbl 1524.16013 |
| [52] | Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8, 1989, Cambridge University Press: Cambridge University Press Cambridge ·Zbl 0666.13002 |
| [53] | Mousavand, K.; Paquette, C., Minimal (τ-)tilting infinite algebras, Nagoya Math. J., 249, 221-238, 2023 ·Zbl 1518.16014 |
| [54] | Neeman, A., Triangulated Categories, Annals of Mathematics Studies, vol. 148, 2001, Princeton University Press: Princeton University Press Princeton, NJ ·Zbl 0974.18008 |
| [55] | Reading, N., Sortable elements and Cambrian lattices, Algebra Univers., 56, 411-437, 2007 ·Zbl 1184.20038 |
| [56] | Saito, S., Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories ·Zbl 07975197 |
| [57] | Schroll, S.; Treffinger, H.; Valdivieso, Y., On band modules and τ-tilting finiteness, Math. Z., 299, 3-4, 2405-2417, 2021 ·Zbl 1493.16012 |
| [58] | Sentieri, F., A brick version of a theorem of Auslander, Nagoya Math. J., 249, 88-106, 2023 ·Zbl 1523.16019 |
| [59] | Stanley, D.; Wang, B., Classifying subcategories of finitely generated modules over a Noetherian ring, J. Pure Appl. Algebra, 215, 11, 2684-2693, 2011 ·Zbl 1267.13020 |
| [60] | Takahashi, R., Classifying subcategories of modules over a commutative Noetherian ring, J. Lond. Math. Soc. (2), 78, 3, 767-782, 2008 ·Zbl 1155.13008 |
| [61] | Yurikusa, T., Wide subcategories are semistable, Doc. Math., 23, 35-47, 2018 ·Zbl 1410.16019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
