Geodesic
The Braun--Kemer--Razmyslov theorem for affine $PI$-algebras
Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 89-128.

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A self-contained, combinatoric exposition is given for the Braun–Kemer–Razmyslov Theorem over an arbitrary commutative Noetherian ring.At one time, the community did not believe in the validity of this result, and contrary to public opinion, the corresponding question was posed by V.N. Latyshev in his doctoral dissertation. One of the major theorems in the theory of PI algebras is the Braun-Kemer-Razmyslov Theorem. We preface its statement with some basic definitions. 1. An algebra $A$ is affine over a commutative ring $C$ if $A$ is generated as an algebra over $C$ by a finite number of elements $a_1, \dots, a_\ell;$ in this case we write $A = C \{ a_1, \dots, a_\ell \}.$ We say the algebra $A$ is finite if $A$ is spanned as a $C$-module by finitely many elements. 2. Algebras over a field are called PI algebras if they satisfy (nontrivial) polynomial identities. 3. The Capelli polynomial $\mathrm{Cap}_k$ of degree $2k$ is defined as $$\mathrm{Cap}_k(x_1, \dots, x_k; y_1, \dots, y_k)= \sum_{\pi \in S_k}\mathrm{sgn} (\pi) x_{\pi (1)}y_1 \cdots x_{\pi (k)} y_k.$$ 4. $\operatorname{Jac}(A)$ denotes the Jacobson radical of the algebra $A$ which, for PI-algebras is the intersection of the maximal ideals of $A$, in view of Kaplansky's theorem. The aim of this article is to present a readable combinatoric proof of the theorem: The Braun-Kemer-Razmyslov Theorem The Jacobson radical $\operatorname{Jac}(A)$ of any affine PI algebra $A$ over a field is nilpotent.
Keywords: algebras with polynomial identity, varieties of algebras, representable algebras, relatively free algebras, Hilbert series, Specht problem. 
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Alexei Kanel Belov; Louis Rowen. The Braun--Kemer--Razmyslov theorem for affine $PI$-algebras. Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 89-128. http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a10/

[1] S.A. Amitsur, “A generalization of Hilbert Nullstellensatz”, Proc. Amer. Math. Soc., 8 (1957), 649–656 | MR | Zbl

[2] S.A. Amitsur, “A note on P.I. rings”, Israel J. Math., 10 (1971), 210–211 | DOI | MR | Zbl

[3] S.A. Amitsur, C. Procesi, “Jacobson rings and Hilbert algebras with polynomial identities”, Ann. Mat. Pura Appl., 71 (1966), 67–72 | DOI | MR

[4] S.A. Amitsur, A. Regev, “P.I. algebras and their cocharacters”, J. of Algebra, 78 (1982), 248–254 | DOI | MR | Zbl

[5] S.A. Amitsur, L. Small, “Affine algebras with polynomial identities”, Supplemento ai Rendiconti del Circolo Matematico di Palermo, 31 (1993) | MR | Zbl

[6] A. Belov, L. Bokut, L.H. Rowen, J. T. Yu, “The Jacobian Conjecture, together with Specht and Burnside-type problems”, Proc. Groups of Automorphisms in Birational and Affine Geometry, eds. M. Zaidenberg, M. Rich, M. Reizakis, Springer (to appear) | MR

[7] A.K. Belov, L. H. Rowen, Computational aspects of Polynomial Identities, A. K. Peters, 2005 | MR | Zbl

[8] A. Braun, “The nilpotency of the radical in a finitely generated PI-ring”, J. Algebra, 89 (1984), 375–396 | DOI | MR | Zbl

[9] M. Fayers, “Irreducible Specht modules for Hecke algebras of type A”, Advances in Math., 193 (2005), 438–452 | DOI | MR | Zbl

[10] M. Fayers, S. Lyle, S. Martin, “p-restriction of partitions and homomorphisms between Specht modules”, J. Algebra, 306 (2006), 175–190 | DOI | MR | Zbl

[11] N. Jacobson, Basic Algebra, v. II, second edition, Freeman and company, 1989 | MR | Zbl

[12] G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., 682, Springer-Verlag, New York, NY, 1978 | DOI | MR | Zbl

[13] G. James, A. Mathas, “The irreducible Specht modules in characteristic $2$”, Bull. London Math. Soc., 31 (1999), 457–62 | DOI | MR

[14] G. D. James, A. Kerber, The Representation Theory of the Symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison–Wesley, Reading, MA, 1981 | MR | Zbl

[15] Soviet Math. Dokl., 22 (1980), 750–753 | MR | Zbl

[16] A.R. Kemer, Ideals of identities of associative algebras, Translations of monographs, 87, Amer. Math. Soc., 1991 | DOI | MR | Zbl

[17] A.R. Kemer, “Multilinear identities of the algebras over a field of characteristic $p$”, Internat. J. Algebra Comput., 5:2 (1995), 189–197 | DOI | MR | Zbl

[18] J. Lewin, “A matrix representation for associative algebras I and II”, Trans. Amer. Math. Soc., 188:2 (1974), 293–317 | DOI | MR

[19] Unpublished, L'vov (Russian)

[20] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press, Oxford, 1995 | MR | Zbl

[21] A. Mathas, Iwahori–Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, American Mathematical Society, Providence, RI, 1999 | DOI | MR | Zbl

[22] Algebra and Logika, 13 (1974), 192–204 | DOI | MR | MR

[23] A. Regev, “Existence of identities in $A\otimes B$”, Israel J. Math., 11 (1972), 131–152 | DOI | MR | Zbl

[24] A. Regev, “The representations of $S_n$ and explicit identities for P. I. algebras”, J. Algebra, 51 (1978), 25–40 | DOI | MR | Zbl

[25] Richard Resco, Lance W. Small, J. T. Stafford, “Krull and Global Dimensions of Semiprime Noetherian $PI$-Rings”, Transactions of the American Mathematical Society, 274:1 (1982), 285–295 | DOI | MR | Zbl

[26] L.H. Rowen, Polynomial Identities in Ring Theory, Pure and Applied Mathematics, 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1980 | MR | Zbl

[27] L.H. Rowen, Ring Theory, v. II, Pure and Applied Mathematics, 128, Academic Press, Inc., Boston, MA, 1988 | MR | Zbl

[28] L.H. Rowen, Graduate algebra: Commutative View, AMS Graduate Studies in Mathematics, 73, 2006 | DOI | MR | Zbl

[29] L.H. Rowen, Graduate algebra: Noncommutative View, AMS Graduate Studies in Mathematics, 91, 2008 | DOI | MR | Zbl

[30] B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, 203, 2nd edition, Springer-Verlag, 2000 | MR

[31] A.I. Shirshov, “On certain non associative nil rings and algebraic algebras”, Mat. Sb., 41 (1957), 381–394 (Russian) | MR | Zbl

[32] A.I. Shirshov, “On rings with identity relations”, Mat. Sb., 43 (1957), 277–283 (Russian) | MR | Zbl

[33] Small L.W., “An example in PI rings”, J. Algebra, 17 (1971), 434–436 | DOI | MR | Zbl

[34] K.A. Zubrilin, “Algebras satisfying Capelli identities”, Sbornic Math., 186:3 (1995), 359–370 | DOI | MR | Zbl