On superstability in the class of flat modules and perfect rings

Mazari-Armida, Marcos

doi:10.1090/proc/15359

On superstability in the class of flat modules and perfect rings

Author: Marcos Mazari-Armida
Journal: Proc. Amer. Math. Soc. 149 (2021), 2639-2654
MSC (2020): Primary 03C48, 16B70; Secondary 03C45, 03C60, 13L05, 16L30, 16D10
DOI: https://doi.org/10.1090/proc/15359
Published electronically: March 22, 2021
MathSciNet review: 4246813
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Abstract:

We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings.

\begingroup\itshapeTheorem 0.1. For a ring $R$ the following are equivalent.

$R$ is left perfect.
The class of flat left $R$-modules with pure embeddings is superstable.
There exists a $\lambda \geq (|R| + \aleph _0)^+$ such that the class of flat left $R$-modules with pure embeddings has uniqueness of limit models of cardinality $\lambda$.
Every limit model in the class of flat left $R$-modules with pure embeddings is $\Sigma$-cotorsion.

\endgroup

A key step in our argument is the study of limit models in the class of flat modules. We show that limit models with chains of long cofinality are cotorsion and that limit models are elementarily equivalent.

We obtain a new characterization via limit models of the rings characterized in Rothmaler [Comm. Algebra 30 (2002), pp. 3077–3085]. We show that in these rings the equivalence between left perfect rings and superstability can be refined. We show that the results for these rings can be used to extend Shelah result [1.2, Notre Dame J. Form. Log. 58 (2017), pp. 159–177] to classes of flat modules not axiomatizable in first-order logic.

References

John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009. MR 2532039
John T. Baldwin and Ralph N. McKenzie, Counting models in universal Horn classes, Algebra Universalis 15 (1982), no. 3, 359–384. MR 689770, DOI https://doi.org/10.1007/BF02483731
Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI https://doi.org/10.1090/S0002-9947-1960-0157984-8
L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. MR 1832549, DOI https://doi.org/10.1017/S0024609301008104
Will Boney, Tameness and extending frames, J. Math. Log. 14 (2014), no. 2, 1450007, 27. MR 3304119, DOI https://doi.org/10.1142/S021906131450007X

Will Boney and Monica VanDieren, Limit Models in Strictly Stable Abstract Elementary Classes, Preprint. URL: arXiv:1508.04717.

R. T. Bumby, Modules which are isomorphic to submodules of each other, Arch. Math. (Basel) 16 (1965), 184–185. MR 184973, DOI https://doi.org/10.1007/BF01220018

Gabriel Sabbagh and Paul Eklof, Definability problems for modules and rings, J. Symbolic Logic 36 (1971), 623–649. MR 313050, DOI https://doi.org/10.2307/2272466

Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI https://doi.org/10.1007/BF02760849

Steven Garavaglia, Decomposition of totally transcendental modules, J. Symbolic Logic 45 (1980), no. 1, 155–164. MR 560233, DOI https://doi.org/10.2307/2273362

Rami Grossberg, A Course in Model Theory, in Preparation, 201X.

Rami Grossberg, Classification theory for abstract elementary classes, Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, 2002, pp. 165–204. MR 1928390, DOI https://doi.org/10.1090/conm/302/05080

Rami Grossberg and Saharon Shelah, A nonstructure theorem for an infinitary theory which has the unsuperstability property, Illinois J. Math. 30 (1986), no. 2, 364–390. MR 840135

Rami Grossberg and Monica VanDieren, Galois-stability for tame abstract elementary classes, J. Math. Log. 6 (2006), no. 1, 25–48. MR 2250952, DOI https://doi.org/10.1142/S0219061306000487

Rami Grossberg, Monica VanDieren, and Andrés Villaveces, Uniqueness of limit models in classes with amalgamation, MLQ Math. Log. Q. 62 (2016), no. 4-5, 367–382. MR 3549555, DOI https://doi.org/10.1002/malq.201500033

Rami Grossberg and Sebastien Vasey, Equivalent definitions of superstability in tame abstract elementary classes, J. Symb. Log. 82 (2017), no. 4, 1387–1408. MR 3743614, DOI https://doi.org/10.1017/jsl.2017.21

Pedro A. Guil Asensio and Ivo Herzog, Sigma-cotorsion rings, Adv. Math. 191 (2005), no. 1, 11–28. MR 2102841, DOI https://doi.org/10.1016/j.aim.2004.01.006

Pedro A. Guil Asensio and Ivo Herzog, Pure-injectivity in the category of flat modules, Algebra and its applications, Contemp. Math., vol. 419, Amer. Math. Soc., Providence, RI, 2006, pp. 155–166. MR 2279115, DOI https://doi.org/10.1090/conm/419/08002

Pedro A. Guil Asensio and Ivo Herzog, Model-theoretic aspects of $\Sigma $-cotorsion modules, Ann. Pure Appl. Logic 146 (2007), no. 1, 1–12. MR 2311160, DOI https://doi.org/10.1016/j.apal.2006.11.001

Pedro A. Guil Asensio, Berke Kalebog̃az, and Ashish K. Srivastava, The Schröder-Bernstein problem for modules, J. Algebra 498 (2018), 153–164. MR 3754408, DOI https://doi.org/10.1016/j.jalgebra.2017.11.029

D. K. Harrison, Infinite abelian groups and homological methods, Ann. of Math. (2) 69 (1959), 366–391. MR 104728, DOI https://doi.org/10.2307/1970188

Ivo Herzog and Philipp Rothmaler, When cotorsion modules are pure injective, J. Math. Log. 9 (2009), no. 1, 63–102. MR 2665782, DOI https://doi.org/10.1142/S0219061309000835

Saharon Shelah and Oren Kolman, Categoricity of theories in $L_{\kappa \omega }$, when $\kappa $ is a measurable cardinal. I, Fund. Math. 151 (1996), no. 3, 209–240. MR 1424575

Thomas G. Kucera and Marcos Mazari-Armida, On universal modules with pure embeddings, Math. Log. Quart. 66 (2020), 395–408, DOI 10.1002/malq.201900058.

T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 1991. MR 1125071

Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Adv. Math. 346 (2019), 719–772. MR 3914179, DOI https://doi.org/10.1016/j.aim.2019.02.018

Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Weak factorization systems and stable independence, 45 pages. Preprint. arXiv:1904.05691v2

Marcos Mazari-Armida, Algebraic description of limit models in classes of abelian groups, Ann. Pure Appl. Logic 171 (2020), no. 1, 102723, 17. MR 4024675, DOI https://doi.org/10.1016/j.apal.2019.102723

Marcos Mazari-Armida, Superstability, noetherian rings and pure-semisimple rings, Ann. Pure Appl. Logic 172 (2021), no. 3, 102917, 24. MR 4172773, DOI https://doi.org/10.1016/j.apal.2020.102917

Marcos Mazari-Armida, A model theoretic solution to a problem of László Fuchs, J. Algebra 567 (2021), 196–209. MR 4158728, DOI https://doi.org/10.1016/j.jalgebra.2020.09.029

Marcos Mazari-Armida and Sebastien Vasey, Universal classes near $\aleph _1$, J. Symb. Log. 83 (2018), no. 4, 1633–1643. MR 3893293, DOI https://doi.org/10.1017/jsl.2018.37

M. Scott Osborne, Basic homological algebra, Graduate Texts in Mathematics, vol. 196, Springer-Verlag, New York, 2000. MR 1757274

Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092

Philipp Rothmaler, When are pure-injective envelopes of flat modules flat?, Comm. Algebra 30 (2002), no. 6, 3077–3085. MR 1908259, DOI https://doi.org/10.1081/AGB-120004010

Jan Šaroch and Jan Šťovíček, Singular compactness and definability for $\Sigma $-cotorsion and Gorenstein modules, Selecta Math. (N.S.) 26 (2020), no. 2, Paper No. 23, 40. MR 4076700, DOI https://doi.org/10.1007/s00029-020-0543-2

S. Shelah, Stable theories, Israel J. Math. 7 (1969), 187–202. MR 253889, DOI https://doi.org/10.1007/BF02787611

Saharon Shelah, The lazy model-theoretician’s guide to stability, Logique et Anal. (N.S.) 18 (1975), no. 71-72, 241–308. MR 539969

Saharon Shelah, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 419–497. MR 1033034, DOI https://doi.org/10.1007/BFb0082243

Saharon Shelah, Universal classes, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 264–418. MR 1033033, DOI https://doi.org/10.1007/BFb0082242

Saharon Shelah, Categoricity for abstract classes with amalgamation, Ann. Pure Appl. Logic 98 (1999), no. 1-3, 261–294. MR 1696853, DOI https://doi.org/10.1016/S0168-0072%2898%2900016-5

Saharon Shelah, Universal structures, Notre Dame J. Form. Log. 58 (2017), no. 2, 159–177. MR 3634974, DOI https://doi.org/10.1215/00294527-3800985

Saharon Shelah, Classification theory for abstract elementary classes. Vol. 2, Studies in Logic (London), vol. 20, College Publications, London, 2009. Mathematical Logic and Foundations. MR 2649290

Saharon Shelah and Andrés Villaveces, Toward categoricity for classes with no maximal models, Ann. Pure Appl. Logic 97 (1999), no. 1-3, 1–25. MR 1682066, DOI https://doi.org/10.1016/S0168-0072%2898%2900015-3

Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. MR 0389953

Monica VanDieren, Categoricity in abstract elementary classes with no maximal models, Ann. Pure Appl. Logic 141 (2006), no. 1-2, 108–147. MR 2229933, DOI https://doi.org/10.1016/j.apal.2005.10.006

Monica M. VanDieren, Superstability and symmetry, Ann. Pure Appl. Logic 167 (2016), no. 12, 1171–1183. MR 3550319, DOI https://doi.org/10.1016/j.apal.2016.05.004

Sebastien Vasey, Toward a stability theory of tame abstract elementary classes, J. Math. Log. 18 (2018), no. 2, 1850009, 36. MR 3878471, DOI https://doi.org/10.1142/S0219061318500095

Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 65, 51. MR 4021851, DOI https://doi.org/10.1007/s00029-019-0511-x

Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789

Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213. MR 739577, DOI https://doi.org/10.1016/0168-0072%2884%2990014-9