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Specht's problem for associative affine algebras over commutative Noetherian rings


Authors: Alexei Belov-Kanel, Louis Rowen and Uzi Vishne
Journal: Trans. Amer. Math. Soc. 367 (2015), 5553-5596
MSC (2010): Primary 16R10; Secondary 16G20
DOI: https://doi.org/10.1090/tran/5983
Published electronically: April 3, 2015
MathSciNet review: 3347183
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Abstract: In a series of papers by the authors we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a ``$ \bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $ \Gamma $'', i.e., a nonidentity of the representable algebra $ A$ arising from $ \Gamma $, whose ideal of evaluations in $ A$ is closed under multiplication by $ \bar q$-powers of the characteristic coefficients of matrices corresponding to the generators of $ A$, where $ \bar q$ is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring $ C$ involves localizing at finitely many elements a kind of $ C$, and reducing to the field case by a local-global principle.


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  • [1] S. A. Amitsur, On the characteristic polynomial of a sum of matrices, Linear and Multilinear Algebra 8 (1979/80), no. 3, 177-182. MR 560557 (82a:15014), https://doi.org/10.1080/03081088008817315
  • [2] A. Ya. Belov, Counterexamples to the Specht problem, Mat. Sb. 191 (2000), no. 3, 13-24 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 3-4, 329-340. MR 1773251 (2001g:16043), https://doi.org/10.1070/SM2000v191n03ABEH000460
  • [3] A. Ya. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 1, 3-134 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 1, 1-126. MR 2655238 (2011e:16039), https://doi.org/10.1070/IM2010v074n01ABEH002481
  • [4] A. Ya. Belov, On rings asymptotically close to associative rings [translation of MR2485366], Siberian Adv. Math. 17 (2007), no. 4, 227-267. MR 2643374, https://doi.org/10.3103/S1055134407040013
  • [5] Alexei Kanel-Belov and Louis Halle Rowen, Computational aspects of polynomial identities, Research Notes in Mathematics, vol. 9, A K Peters Ltd., Wellesley, MA, 2005. MR 2124127 (2006b:16001)
  • [6] Alexei Belov-Kanel, Louis Rowen, and Uzi Vishne, Structure of Zariski-closed algebras, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4695-4734. MR 2645047 (2012m:16035), https://doi.org/10.1090/S0002-9947-10-04993-7
  • [7] Alexei Belov-Kanel, Louis H. Rowen, and Uzi Vishne, Full quivers of representations of algebras, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5525-5569. MR 2931338, https://doi.org/10.1090/S0002-9947-2012-05565-6
  • [8] Alexei Belov-Kanel, Louis H. Rowen, and Uzi Vishne, PI-varieties associated to full quivers of representations of algebras, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2681-2722. MR 3020112, https://doi.org/10.1090/S0002-9947-2012-05709-6
  • [9] A. Belov, L. H. Rowen, and U. Vishne, Application of full quivers to polynomial identities, Comm. in  Alg., to appear (2012), 26 pp.
  • [10] George M. Bergman and Warren Dicks, On universal derivations, J. Algebra 36 (1975), no. 2, 193-211. MR 0387353 (52 #8196)
  • [11] Amiram Braun, The nilpotency of the radical in a finitely generated PI ring, J. Algebra 89 (1984), no. 2, 375-396. MR 751151 (85m:16007), https://doi.org/10.1016/0021-8693(84)90224-2
  • [12] Vesselin Drensky, On the Hilbert series of relatively free algebras, Comm. Algebra 12 (1984), no. 19-20, 2335-2347. MR 755919 (86f:16005), https://doi.org/10.1080/00927878408823112
  • [13] A. V. Grishin, Examples of $ T$-spaces and $ T$-ideals of characteristic 2 without the finite basis property, Fundam. Prikl. Mat. 5 (1999), no. 1, 101-118 (Russian, with English and Russian summaries). MR 1799541 (2002a:16028)
  • [14] A. V. Iltyakov, Finiteness of the basis of identities of a finitely generated alternative $ {\rm PI}$-algebra over a field of characteristic zero, Sibirsk. Mat. Zh. 32 (1991), no. 6, 61-76, 204 (Russian); English transl., Siberian Math. J. 32 (1991), no. 6, 948-961 (1992). MR 1156745 (93c:17056), https://doi.org/10.1007/BF00971199
  • [15] A. V. Iltyakov, Polynomial identities of Finite Dimensional Lie Algebras, monograph (2003).
  • [16] Nathan Jacobson, Basic algebra. II, W. H. Freeman and Co., San Francisco, Calif., 1980. MR 571884 (81g:00001)
  • [17] A. R. Kemer, Representability of reduced-free algebras, Algebra i Logika 27 (1988), no. 3, 274-294, 375 (Russian); English transl., Algebra and Logic 27 (1988), no. 3, 167-184 (1989). MR 997959 (90e:16027), https://doi.org/10.1007/BF01978562
  • [18] A. R. Kemer, Identities of finitely generated algebras over an infinite field, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 726-753 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 1, 69-96. MR 1073084 (91j:16027)
  • [19] Alexander R. Kemer, Identities of associative algebras, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 351-359. MR 1159223 (93d:16027)
  • [20] Alexander Kemer, On some problems in PI-theory in characteristic $ p$ connected with dividing by $ p$, Proceedings of the Third International Algebra Conference (Tainan, 2002), Kluwer Acad. Publ., Dordrecht, 2003, pp. 53-66. MR 2026093 (2004k:16062)
  • [21] Jacques Lewin, A matrix representation for associative algebras. I, II, Trans. Amer. Math. Soc. 188 (1974), 293-308; ibid. 188 (1974), 309-317. MR 0338081 (49 #2848)
  • [22] Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. MR 576061 (82a:16021)
  • [23] Louis H. Rowen, Ring theory. Vol. II, Pure and Applied Mathematics, vol. 128, Academic Press Inc., Boston, MA, 1988. MR 945718 (89h:16002)
  • [24] Louis Halle Rowen, Graduate algebra: noncommutative view, Graduate Studies in Mathematics, vol. 91, American Mathematical Society, Providence, RI, 2008. MR 2462400 (2009k:16001)
  • [25] L. M. Samoĭlov, Prime varieties of associative algebras and related nil-problems, Dr. of Sci. dissertation, Moscow, 2010. 161 pp.
  • [26] V. V. Shchigolev, Examples of infinitely basable $ T$-spaces, Mat. Sb. 191 (2000), no. 3, 143-160 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 3-4, 459-476. MR 1773258 (2001f:16044), https://doi.org/10.1070/SM2000v191n03ABEH000467
  • [27] A. Ya. Vaĭs and E. I. Zelmanov, Kemer's theorem for finitely generated Jordan algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 6 (1989), 42-51 (Russian); English transl., Soviet Math. (Iz. VUZ) 33 (1989), no. 6, 38-47. MR 1017777 (90m:17042)
  • [28] M. R. Vaughan-Lee, Varieties of Lie algebras, Quart. J. Math. Oxford Ser. (2) 21 (1970), 297-308. MR 0269710 (42 #4605)
  • [29] A. N. Zubkov, Matrix invariants over an infinite field of finite characteristic, Sibirsk. Mat. Zh. 34 (1993), no. 6, 68-74, ii, viii (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 6, 1059-1065. MR 1268158 (95c:13003), https://doi.org/10.1007/BF00973469
  • [30] K. A. Zubrilin, Algebras that satisfy the Capelli identities, Mat. Sb. 186 (1995), no. 3, 53-64 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 3, 359-370. MR 1331808 (96c:16032), https://doi.org/10.1070/SM1995v186n03ABEH000021

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Additional Information

Alexei Belov-Kanel
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: belova@math.biu.ac.il

Louis Rowen
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: rowen@math.biu.ac.il

Uzi Vishne
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: vishne@math.biu.ac.il

DOI: https://doi.org/10.1090/tran/5983
Received by editor(s): September 2, 2012
Received by editor(s) in revised form: June 13, 2013
Published electronically: April 3, 2015
Additional Notes: This work was supported by the Israel Science Foundation (grant no. 1207/12).
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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