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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 3.
УДК 512.552 DOI 10.22405/2226-8383-2020-21-3-89-128
Теорема Размыслова — Кемера — Брауна для афинных
Pi-алгебр1
Канель-Белов А. Я., Роуэн Луис Халли
Александр Яковлевич Канель-Белов —доктор физико-математических наук, федеральный профессор математики МП' Г. профессор Университета им. Бар-Илана (Израиль). e-mail: kanelster@gmail.com,
Роуэн Луис Халли — Университет им. Бар-Илана (Израиль). e-mail: kanelster@gmail.com
Аннотация
Дается замкнутое в себе альтернативное комбинаторное изложение доказательство теоремы Размыслова-Кемера-Ббрауна о нильпотентности радикала афинной PI-алгебры над нетеровым ассоциативно-коммутативным кольцом. В свое время сообщество не верило в справедливость этого результата и вопреки общественному мнению соответствующий вопрос был поставлен В.Н.Латышевым в его докторской диссертации. Начнем с определения:
1. Алгебра А является аффинной над коммутативным кольцом С, если А порождается как алгебра над С конечным числом элементов а\,... ,ai; в этом случае мы пишем А = С{а^,..., ai}.
2. Мы говорим, что алгебра А является конечной, если А порождено как С-модуль конечным числом элементов.
3. Алгебры над полем называются PI алгебрами, если они удовлетворяют (нетривиальным) полиномиальным тождествам.
4. Многочлен Капелли Capfc степени 2к определяется так:
Capfc (х!,...,хк; yi,...,yk) = ^ sgn^)^!)^! ••• х^(к)Ук
5. Jac(A) обозначат радикал Джекобсона алгебры А, который для PI-алгебр является пересечением максимальных идеалов А по теореме Каплапского.
Ключевые слова: алгебры с полиномиальными тождествами, многообразия алгебр; представимые алгебры; относительно свободные алгебры; Ряды Гильберта; Проблема Шпехта
Библиография: 34 названий. Для цитирования:
Канель-Белов А.Я., Роуэн Луис Халли Теорема Размыслова — Кемера — Брауна для афинных PI-алгебр // Чебышевский сборник, 2020, т. 21, вып. 3, с. 89-128.
1А.Я.Белов был поддержан РНФ (грант 17-11-01377), Л.Роуэн ISF 1623/16.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 3.
UDC 512.552 DOI 10.22405/2226-8383-2020-21-3-89-128
The Braun-Kemer-Razmyslov Theorem for affine Pi-algebras
Alexei Kanel Belov, Louis Rowen
Alexei Kanel Belov — doctor of physical and mathematical sciences, federal professor MIPT,
professor Bar Ilan University (Israel).
e-mail: kanelster@gmail.com,
Louis Rowen — Bar-Ilan University (Israel).
e-mail: kanelster@gmail.com
Abstract
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 С ii A is generated as an algebra over С by a finite number of elements ai,..., af, in this case we write A = С{ai,..., a,£}.
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 Capfc of degree 2k is defined as
Capfc (xi,...,xk; yi,...,yk) = ^ sgn(^)^(i)yi ••• х^(к)Ук
4. 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 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
Bibliography: 34 titles. For citation:
A. Kanel Belov, L. Rowen, 2020, "The Braun-Kemer-Razmyslov Theorem for affine PI-algebras" , Chebyshevskii sbornik, vol. 21, no. 3, pp. 89-128.
Светлой памяти Виктора Николаевича Латышева посвящается
1. The BKR Theorem 1.1. Introduction
One of the major theorems in the theory of PI algebras is the Braun-Kemer-Razmyslov Theorem (Theorem 1.1 below). We preface its statement with some basic definitions.
Definition 1. 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 ai,...,a¿; in this case we write A = C{ai,..., a¿}.
2. We say the algebra A is finite if A is spanned as a C-module by finitely m,any elements.
3. Algebras over a field are called PI algebras if they satisfy (nontrivial) polynomial identities.
4- The Capelli polynomial Capfc of degree 2k is defined as
Capfc (xi, ...,xk; yi,...,yk) = ^ sgn^^yi ■ ■ ■ x^{k)yk
■KESk
5. Jac(A) denotes the Jacobson radical of the algebra A which, for PI-algebras is the intersection of the m,axim,al ideals of A, in view of Kaplansky's theorem.
THEOREM 1.1 (The Braun-Kemer-Razmyslov Theorem). The Jacobson radical Jac(A) of any afiine PI algebra A over a field is nilpotent.
The aim of this article is to present a readable combinatoric proof (essentially self-contained in characteristic 0).
Let us put the BKR Theorem into its broader context in PI theory. We say a ring is Jacobson if the Jacobson radical of every prime homomorphic image is 0. For Pi-rings, this means every prime ideal is the intersection of maximal ideals. Obviously any field is Jacobson, since its only prime ideal 0 is maximal. Furthermore, any commutative afiine algebra over a field is Noetherian by the Hilbert Basis Theorem and is Jacobson, in view of [28, Proposition 6.37], often called the "weak Nullstellensatz," implying the following two results:
• (cf. Proposition 1.11) If a commutative algebra A is affine over a field, then Jac(A) is nilpotent.
• (Special case of Theorem 2) If A is a finite algebra over an affine central subalgebra Z over a field, then Jac(A) is nilpotent. (Sketch of proof: Passing to homomorphic images modulo prime ideals, we may assume that A is prime PI, and Z is an affine domain over which A is torsion-free. The maximal ideals of Z № up to maximal ideals of A, in view of Nakavama's lemma, implying Z n Jac(A) C Jac(Z) = 0. If 0 = a £ Jac(A), then writing a as integral over Z, we have the nonzero constant term in Z n Jac(A) = 0, a contradiction.)
Since either of these hypotheses implies that A is a PI-algebra, it is natural to try to find an umbrella result for affine PI-algebras, which is precisely the Braun-Kemer-Razmyslov Theorem. This theorem was proved in several stages. Amitsur fl, Theorem 5], generalizing the weak Nullstellensatz, proved that if A is affine over a commutative Jacobson ring, then Jac(A) is nil. In particular, A is a Jacobson ring. (Later, Amitsur and Procesi [3, Corollary 1.3] proved that Jac(A) is locally nilpotent.) Thus, it remained to prove that every nil ideal of A is nilpotent.
It was soon proved that this does hold for an affine algebra which can be embedded into a matrix algebra, see Theorem 1 below. However, examples of Small [33] showed the existence of affine PI algebras which can not be embedded into any matrix algebra. Thus, the following theorem by Razmvslov [22] was a major breakthrough in this area.
Theorem 1.2 (Razmvslov). If an affine algebra A over a field satisfies a Capelli identity, then its Jacobson radical Jac(A) is nilpotent.
Although Razmyslov's theorem was given originally in characteristic zero, he later found a proof that works in any characteristic. As we shall see, the same ideas yield the parallel result:
Theorem 1.3. Let A be an affine algebra over a commutative Noetherian ring C. If A satisfies a Capelli identity, then any nil ideal of A is nilpotent.
Following Razmyslov's theorem, Kemer [15] then proved
Theorem 1.4. [15] In characteristic zero, any affine PI algebra satisfies some Capelli identity (see Theorem 3.3).
Thus, Kemer completed the proof of the following theorem:
Theorem 1.5 (Kemer-Razmvslov). If A is an affine PI-algebra over a field F of characteristic zero, then its Jacobson radical Jac(A) is nilpotent.
Then, using different methods relying on the structure of Azumava algebras, Br aun proved the following result, which together with the Amitsur-Procesi Theorem immediately yields Theorem 1.1:
Theorem 1.6. Any nil ideal of an affine PI-algebra over an arbitrary commutative Noetherian ring is nilpotent.
Note that to prove directly that Jac(A) is nilpotent it is enough to prove Theorem 1.6 and show that Jac(A) is nil, which is the case case when A is Jacobson, and is called the "weak Nullstellensatz." But the weak Nullstellensatz requires some assumption on the base ring C. It can be proved without undue difficulty that A is Jacobson when C is Jacobson, cf. [26, Theorem 4.4.5]. Thus, in this case the proper general formulation of the nilpotence of Jac(A) is:
Theorem 1.7 (Braun). If A is an affine PI-algebra over a Jacobson Noetherian base ring, then Jac(A) is nilpotent.
Small has pointed out that Theorems 1.6 and 1.7 actually are equivalent, in view of a trick of [25]. Indeed, as just pointed out, Theorem 1.6 implies Theorem 1.7. Conversely, assuming Theorem 1.7, one needs to show that Jac(A) is nil. Modding out the nilradical, and then passing to prime images, one may assume that A is prime. Then one embeds A into the polynomial algebra A[A] over the Noetherian ring C[A], and localizes at the monic polynomials over C[A], yielding a Jacobson base ring by [25, Theorem 2.8].
Braun's qualitative proof was also presented in [27, Theorem 6.3.39], and a detailed exposition, by L'vov [19], is available in Russian. A sketch of Braun's proof is also given in [5, Theorem 3.1.1].
Meanwhile, Kemer [17] proved:
Theorem 1.8. [17] If A is a PI algebra (not necessarily affine) over a field F of characteristic p > 0 then A satisfies some Capelli identity.
Together with a characteristic-free proof of Razmyslov's theorem 1.2 due to Zubrilin [34], Kemer's Theorems 1.4 and 1.8 yield another proof of the Braun-Kemer-Razmvslov Theorem 1.1. The paper [34] is given in rather general circumstances, with some non-standard terminology. Zubrilin's method was given in [7], although [7, Remark 2.50] glosses over a key point (given here as Lemma 2.13), so a complete combinatorio proof had not yet appeared in print with all the details. Furthermore, full combinatorio details were provided in [7] only in characteristic 0 because the conclusion of the proof required Kemer's difficult Theorem 1.8. We need the special case, which
we call "Kemer's Capelli Theorem," that every affine PI-algebra A over an arbitrary field satisfies some Capelli identity. This can be proved in two steps: First, that A satisfies a "sparse" identity, and then a formal argument that every sparse identity implies a Capelli identity. The version given here (Theorem 4.4) uses the representation theory of the symmetric group Sn, and provides a reasonable quartic bound ((p — 1)p{"+1), where u = 2pe(g-1) ) for the degree of the sparse identitv of A in terms of the degree d of the given PI of A.
It should be noted that every proof that we have cited of the Braun-Kemer-Razmvslov Theorem ultimately utilizes an idea of Razmvslov defining a module structure on generalized polynomials with coefficients in the base ring, but we cannot locate full details of its implementation anywhere in the literature. One of the objectives of this paper is to provide these details, in §2.5 and §2.6.1. Although the proof is rather intricate for a general expository paper, we feel that the community deserves the opportunity to see the complete argument in print.
We emphasize the combinatorio approach here. Aside from the intrinsic interest in having such a proof available of this important theorem (and characteristic-free), these methods generalize easily to nonassociative algebras, and we plan to use this approach as a framework for the nonassociative Pl-theorv, as initiated by Zubrilin. (The proofs are nearly the same, but the statements are somewhat more complicated. See [6] for a clarification of Zubrilin's work in the nonassociative case.) To keep this exposition as readable as we can, we emphasize the case where the base ring C
Theorem 1.6 and the weak Nullstellensatz, although we also treat these general cases.
§2 follows Zubrilin's short paper [34], and gives full details of Zubrilin's proof of Razmv-lov's theorem 1.2. This self-contained proof is characteristic free.
To complete the proof of the BKR Theorem, it remains to prove Kemer's Capelli Theorem. In §3 we provide the proof in characteristic 0, by means of Young diagrams, and §4 contains the characteristic p analog (for affine algebras). An alternative proof could be had by taking the second author's "pumping procedure" which he developed to answer Specht's question in characteristic p, and applying it to the "identity of algebraicitv" [7, Proposition 1.59]. We chose the representation-theoretic approach since it might be more familiar to a wider audience. The proof of Theorem 1.6, over arbitrary commutative Noetherian rings, is given in §5.
Remark 1.9. An early version of Theorem 3.16 was written by Amitai Regev, to whom we are indebted for suggesting this project and providing helpful suggestions all along the way. Belov belongs lemma 3.8
1.2. Structure of the proof
We assume that A is an affine C-algebra and satisfies the n + l Capelli identity Capra+1 (but not necessarily the n Capelli identity Capra), and we induct on n: if such A satisfies Capra then we assume that Jac(A) is nilpotent, and we prove this for Capra+1. For the purposes of this sketch, in Steps 1 through 3 and Step 7 we assume that C is a field F.
The same argument shows that any nil ideal N of an affine algebra A over a Noetherian ring is nilpotent, yielding Theorem 1.3. For this result we would replace Jac(A) by N throughout our sketch.
We write C{x,y,t} for the free (associative) algebra over the base ring C, with inde-terminates Xi,yj ,tk, z, containing one extra indeterminate z for further use. This is a free module over C, whose basis is the set of words, i.e., formal strings of the letters Xi,yj,tk,2. The x and y indeterminates play a special role and need to be treated separately. We write C{i} for the free subalgebra generated by the tk and z, omitting the x and y indeterminates.
1. The induction starts with n = l. Then n + l = 2, and any algebra satisfying Cap2 is
commutative. We therefore need to show that if A is commutative affine over a field F, then Jac(A) is nilpotent. This classical case is reviewed in §1.3.1.
2. Next is the finite case: If A is affine over a field F and a finite module over an affine central subalgebra, then Jac(A) is nilpotent. This case was known well before Razmvslov's Theorem, and is reviewed in §1.3.2. Theorem 1.3 follows whenever A is a subring of a finite dimensional algebra over a field.
3. Let CAVn = T(Capra) be the T-ideal generated by Capra, and let CAVn(A) C A be the ideal generated in A by the evaluations of Capn on A, so A/CAVn(A) satisfies Capra. Therefore, by induction on n, Jac(A/CAVn(A)) is nilpotent. Hence there exists q such that
Jac(A)9 C CAPn(A), so Jac(A)2« C (CAPn(A))2.
4. In §2.2.4 we work over an arbitrary base ring C (which need not even be Noetherian), and for any algebra A introduce the ideal In,A C fa commuting indeterminates (n,A, which provides "generic" integrality relations for elements of A Let C{x,y,t} := C{x,y,t}/CAPn+1, the relatively free algebra of Capn+1. Taking the "doubly alternating" polynomial
f = ti Capn(xi,.. .,xn)t2 Capn(yi,.. .,yn)t3,
we construct, in Section 2.2.1, the key C{i}-module M C C{x, y, t}, which contains the polynomial /. A combinatoric argument given in Proposition 2.17 applied to C{x,y,t} (together with substitutions) shows that I —,, ■ M = 0.
5. We introduce the obstruction to integrality Obstra(A) = A n In,A C A and show that A/Obstra(A) ^rn be embedded into a finite algebra over an affine central F-subalgebra; hence Jac(A/Obstra(A)) is nilpotent. This implies that there exists m such that
Jac(A)m C Obst„(A).
The proof of this step applies Shirshov's Height Theorem [32], [7, Theorem 2.3].
6. We prove that Obst„(A) ■ (CAVn(A))2 = 0 over an arbitrary ring C. This is obtained from Step 4 via a sophisticated specialization argument involving free products.
7. We put the pieces together. WThen C is a field, Step 3 shows that Jac(A)9 C CAPn(A) for some q, and Step 5 shows that Jac(A)m C Obstra(A) for some m. Hence
Jac(A)2q+m C Obstra(A) ■ (CAPn(A))2 = 0,
which completes the proof of Theorem 1.2. WThen C is Noetherian, any nil ideal N of C is nilpotent, so the analogous argument shows that Nm C Obstra(A) for some m. Hence
N2q+m C Obstra(A) ■ (CAPn(A))2 = 0,
proving Theorem 1.3.
1.3. Special cases
We need some classical special cases.
1.3.1. The commutative case
Our main objective is to prove that the Jacobson radical Jac(A) of an affine PI-algebra A (over a field) is nilpotent. We start with the classical case for which A is commutative.
Remark 1.10. Any commutative affine algebra A over a Noetherian base ring C is Noetherian, by Hilbert's Basis Theorem, and hence the intersection of its prime ideals is nilpotent, cf. [29, Theorem 16.24].
But for any ideal I < A, the algebra A/I is also Noetherian, so the intersection of the prime ideals of A containing I is nilpotent modulo I.
Proposition 1.11. If H is a commutative affine algebra over a field, then Jac(#) is nilpotent.
proof. The "weak Nullstellensatz" [28, Proposition 6.37] says that H is Jacobson, and thus the Jacobson radical Jac(#) is contained in the intersection of the prime ideals of H. But any commutative affine algebra is Noetherian, so we conclude with Remark 1.10. □
1.3.2. The finite case
To extend this to noncommutative algebras, we start with some other classical results:
1. [29, Theorem 15.23] (Wedderburn) Any nil subring of an n x n matrix algebra over a field is nilpotent, of nilpotence index ^ n (in view of [29, Lemma 15.22]).
2. [29, Theorem 15.18] (Jacobson) The Jacobson radical of an n-dimensional algebra over a field is nilpotent, and thus has nilpotence index ^ n, by (1).
3. Any algebra finite over a Noetherian central subring C, is Noetherian (This follows at once from induction applied to [28, Proposition 7.5].
Theorem 1. Suppose A = C{a\,..., ai} is an affine algebra over a commutative Noetherian ring C, with A C Mn(K) for a suitable commutative C-algebra K. Then
1. Any nil subalgebra N of A is nilpotent,, of bounded nilpotenee index ^ mn, where m is given in the proof. When K is reduced, i.e., without nonzero nilpotent elements, then m = 1, so Nn = 0.
2. If C is a field, then Jac(A) is nilpotent.
(k)\ (k)
Proof. For each 1 ^ k ^ l, write each a^an n x n matrix (a\j ), for G K, and let H be
(k)
the commutative C-subalgebra of K generated by these finitely many a^ , then H is C-affine. We can view each au in Mn(H), so A C Mn(H).
(1) Let N C A be a nil subalgebra. Now A C Mn(H), so N C Mn(H) and is nil. Let P C H be prime. The homomorphism H ^ H/P extends to
Mn(H) ^ Mn(H/P) (^ Mn(H)/Mn(P)).
Let NV be the image of N, so NV = (N + Mn(P))/Mn(P^o NV C Mn(H)/Mn(P) = Mn(H/P) C Mn(L) where L is the field of fractions of the domain H/P. By Wedderburn's theorem Nn = 0 which implies that Nn C Mn(P) (since P = 0 in H/P and in L). letting U denote the prime
radical of H, we have Nn C Mn(U). in view of Remark 1.10, we have Um = 0 for some m. (If K is reduced then U = 0 implying m = 1.) We conclude that
Nmn = (Nn)m C (Mn(U))m = Mn(Um) = 0.
(2) We need here the well-known fact [29, Exercise 15.28] that when J<A, with J nilpotent, then Jac(A/J) = Jac(A)/J. It follows at once that if Jac(A/J) is nilpotent, then Jac(A) is nilpotent.
By hypothesis H is affine OTer the field C, so Jac(^) is nilpotent, and thus Mn(Jac(H)) = Jac(Mn(H)) is nilpotent. Let A = A/(A n Mn(Jac(H))) and H = H/ Jac(ff). Then
A C Mn(H/ Jac(^)) = Mn(H),
and Jac(Ii) = 0 Thus we may assume that Jac(^) = 0, and we shall prove that Jn = 0, where J = Jac(A).
For any maximal ideal P of we see that H/P is an affine field extension of C, and thus is finite dimensional over C, by [28, Theorem 5.11]. But then the image of A in Mn(H/P) is finite dimensional over C, so the image J of J is nilpotent, implying ,Jn = 0. Hence Jn is contained in nMn(P) = Mn(nP) = 0, where P runs over the maximal ideals of H. □
Theorem 2. Suppose A is an algebra that is finite over C, itself an affine algebra over a field. Then Jac(A) is nilpotent.
proof. Since A is Noetherian, its nilradical N is nilpotent by [29, Remark 16.30(H)], so modding out N we may assume that A is semiprime, and thus the subdirect product of prime algebras {Ai = A/Pi : i e I} ^^^^e over their centers. If Jac(A)ra c Pi for each i e ^^en Jac(A)ra c nPi = 0. So we may assume that A is prime. But localizing over the center, we may assume that C is a
field. Let n = dim c A Then A is embedded via the regular representation into n x n matrices over
□
Since not every affine Pi-algebra might satisfy the hypotheses of Theorem 1, cf. [33] and [18], we must proceed further.
2. Proof of Razmyslov's Theorem
In this section we give full details for Zubrilin's proof of Theorem 1.2. 2.1. Zubrilin's approach
2.1.1. The operator
Let us fix notation for the next few sections. C is an arbitrary commutative ring. We start with a polynomial f := f (x1,... ,xn) e C{x,y,t} in x = {x1,... ,xn} (which we always notate), as well as possibly y = {y1,..., yn} (which we sometimes notate), and t = {t1,... }, all of which are noncommutative indeterminates.
Definition 2.1. Let f (x, y, t) ^e multilinear in the Xi (and perhaps involving additional indeterminates y and ^^ Take 0 ^ k ^ n, and expand
f * = f ((z + 1)X1,..., (z + l)xn,y,t),
where z is a new noncommutative indeterminate. Then we write
*&n)(f) := Si!:f)(f)(x1,...,xn,z)
for the homogeneous component of f * of degree k in the noncommutative indeterminate z. (We have suppressed y, t in the notation, as indicated above.) For example let n = ^d f = x1x2. Then
(z + 1)x1(z + 1)x2 = zx1zx2 + zx1x2 + x1zx2 + x1x2.
Hence 50f'z\x1x2) = x-\_x2, b^f2 (x1x2) = zx1x2 + x1zx2, and d^2 (x1x2) = zx1zx2.
More generally, for any h e C{t} we write ) := ^^(f)(x\_,...,xn,h), i.e., the
specialization of (f) under z ^ h.
Remark 1.
1. In calculating (f) the substitu tion Xi ^ (z+1)xi is applied to the first n positions in f but not to the other positions. For example, the last (i.e. n + 1 st) variable in f (x1,... ,xn-1,xn+1,xn) is xn, not xn+1. Hence, to calculate (x-\_,..., xn-1, xn+1,xn)) we apply Xi ^ (z + 1)xi to all
Xi s GXCGpt X'fy*
2. We can also write
6kf)(f ^ ...,Xn,t))= f ^ ...,X^,t) \Xi. ^zxi. =
— ^ ^ f (%1, . . . , ZXii, . . . ZXik , . . . , X^, t).
3. In case f = f (x-\_,..., xn, y1,..., yn) also involves indeterminates y1,..., yn, we still have
) (f) f \xijij , K i 1<"< ik
indicating that the other indeterminates y1,...,yn remain fixed. Analogously,
^k'Z )(/) = ^ f \vij Vij ,
and the indeterminates x1,... ,xn are fixed.
Definition 2.2. A polynomial f (x]_,..., xn, t) is alternating in x1,... ,xnii / is multilinear in the Xi and
f (x-]_,... ,Xi,... ,Xj,..., xn, t) + f (x-]_,... ,Xj,... ,Xi,..., xn, t) = 0 fa alH < j. (1)
A stronger definition, which would suffice for our purposes, is to require that
f (X1, ...,xi,...,xi,...,xn,t) = 0; (2)
i.e., we get 0 when specializing Xj to Xi fa any 1 ^ i < j ^ n. We get (2.2) by linearizing (2), and can recover (2) from (2.2) in characteristic = 2.)
Lemma 2.3. Let f (x1,... ,xn,t ) be multilinear and altern ating in x1,...,xn- Then for each 0 ^ k ^ n, (x]_,..., xn, t)) is also alternating in x1,..., xn-
Proof. Let v = 1 + ez where e is a central indeterminate. Obviously f (vx1,... ,vxn,t ) is also alternating m x1,... ,xn. Since
n
f (vxi, . . .,VXn,t ) = ^ (Ж1, . . .,Xn,t )) • £k
k=0
is alternating in x1,..., xn, it follows that each à^X'^ (f (x1,..., xn, t) is alternating in x1,..., xn. □
Remark 2.4.
1. Since CAPn is generated as a T-ideal by polynomials alternating in x1,..., xn, we have
n) ç CAPn and ôkxf (CAPn+1) ç CAPn+1.
2. The results proved for the indeterminate z specialize to ад arbitrary polynomial h, and thus can be formulated for h.
Lemma 2.5. The ^^(f)-operator is functorial, in the sense that if a = (a1,... ,am) g A and h(d) = h'(Й), then 5kXf(f)(a) = ^(/Xa).
Proof. We get the same result in Definition 2.1 bv specializing 2 to h and then to a, as we get bv specializing 2 to h', and then to a. □
This observation is needed in our later specialization arguments.
The following observation, which is rather well known, motivates Proposition 2.10 below. Let V = Cx1 ® • • • ® Cxn and let 2 : V ^ V be a linear transformation from V to V. Let
det(А/ - z) = ^ Ck (z)\k
k=0
be the characteristic ("Cavlev-Hamilton") polynomial of z. Then we have the following formula from [26, Theorem 1.4.12]:
^k,z )(CaPn(^ 1, • • • > xn] y)) ^^ Capn{X\, . . . , ZXi1, . . . , ZXik , . . . , Xn] y) —
— ck (z) ■ Cap,n(xi, ...,Xn; y),
and the coefficients ck (z) are independent of the particular indeterminates x1,...,xn. Proposition 2.10 below displays a similar phenomenon.
2.2. Zubrilin's Proposition
Our goal in this section is Proposition 2.17. Let us define the terms used there. Let C{x, y, i} denote the relatively free algebra C[x,y,t]/CAPn+1. We denote the image of f G C{x,y,t} in C{x,y,t} by f.
Remark 2.6. If A satisfies Capn+1, then any algebra homomorphism p : C{x,y,t} ^ A naturally induces an algebra homomorphism p : C{x,y,t} ^ A given by
m — M ),
since CAPn+1 C ker p.
Remark 2. Let f (X1,... ,xn+{) he multilinea r in X1,...,xn+1 and alternating %Th X1 , . . . , Xfl ' Construct
n+1
\k-
f = f(xi,...,xn+i) = ^(-V)k 1 f (xi,...,xk-i,xk+i,...,xn+i ,xk). (3)
k=1
(All other variables occurring in f are left untouched.) Then f is (n + 1)-alternating in x1,..., xn+1.
Proposition 2.7. Let f (x]_,... ,xn,xn+1) be multilinear in x1,... ,xn,xn+1 and alternating in x1,... ,xn (so f of Equation (3) is (n + 1)-alternating). Then
n
^(-1)J^'n)(f (X1,... ,xn,zn-Xn+1)) = 0 modulo CAPn+1. 3=0
proof. Throughout we work modulo CAVn+v Since /is (n + 1)-alternating, we have
0 = f = f (X2,X3 . . .,Xn+1,X1) - f (X1,X3 . . .,Xn+1,X2) +-----+ (-1)"f (X1,X2, . . .,Xn,Xn+1).
Thus, modulo CAVn+1 the last summand (—1)nf(x1,x2,... ,xn,xn+1) can be replaced bv minus the sum of the other summands:
n
-1
k=1
(-1)"7 (xi,x2,.. .,xn,xn+i) = E (-1)k f (Xi, . . . , Xk— 1 ,Xk+i, . . .,Xn,Xn+1,Xk),
k=1
Given 0 ^ j ^ n, substitute xn+1 ^ zn-^xn+1-, so
(-1)™/(X1,X2, ...,Xn,Zn 3Xn+1) = E (-1)fc f (X1, . . .,Xk-1,Xk+1, . . . .Xn,Zn 3xn+1,xk).
k=1
Applying b^zf2 and summing with sign, we get
n
(-i)n Et-i)'^ (/(®1,..., zn-*n+1) =
3=0
n n
E(-1)J E(-1)fc ^(f (x1,.., *k-1,xk+1, ...,xn, zn-xn+1,xk))
j=0 k=1
n n
E(-1)fc E(-1)^ir,'ra)(/(X1,...,Xk-1 ,xk+1,...,xn,zn-xn+1,xk)). k=1 j=0
Denote gjtk = f (x1,.. .,xk-1,xk+1, ...,xn,zn Jxn+1,xk), and
n
Qx = E(-1)J4TW). w
3=0
It suffices to show that Qk = 0 for each k. Note that in calculating ^^(gj, k) = S<f'a\f (x-\_,...,
xk)), xk is unchanged (since it is the last indeterminate), while for all other XiS (in particular - for xn+1) we substitute Xi^ (z + 1)xi, cf. Remark 1.1. Therefore
r- (X,n) f \ Jx,n) f \ . Ax,n) f \
6), z (9j,k) = ^ [k'](9j,k) + ^ [k''](9j,k)
where
$jXf\ln(gj,k) is the sum of the monomials of ^^(dik) having z-degree j, where xn+1 was replaced by zxn+1;
and
^jXzn\k"](9j>k) '1S the sum the monomials of (gj,k) having z-degree j, where xn+1 was unchanged.
It is not difficult to see that for j > 0,
^W]! (X~1, ■ ■ ■,xk—1,^k+1, ■■ .,Xn,Zn-j Xn+1,Xk) =
5j-i!Z,lk'']f (X1, ■ ■ ■,%k—1,Xk+1, ■ ■ ■,Xn,Zn-3+1Xn+1,Xk),
namely
c(x,n) ( ) _ x(x,n) ( )
°j,z,\k'](9j,k) _ °j-1,z,\k''](3j-1,k)■ It also follows from the definitions that S0XX'za\k'](90,k) _ ^n'z[k"}(9nk) _ 0- Hence
nn
E(-Dj ¿(Xzi'Mk)_ E(-Dj )_ j=0 j=1
n n— 1
_ E(-!)j t(—!ziW'](9i-1,k) _ - E(-Dj I
j=1 j=0
and
n n— 1
E(-!)j )_ E(-!)j 4X3)»](^k )■ j=0 j=0
Summing in (4) we get
n n
Qk _ E(-!)jC'nW) - E(-!)j (+ - 0^ j=0 j=0
□
2.2.1. The module M over the relatively free algebra of Capn+1
We need a special sort of alternating polynomials.
Definition 2.8. A polynomial f (x]_, ■ ■ ■, xn; y1, ■ ■ ■, yn; i), where t denotes other possible indeter-minates, is doubly alternating if f is linear and alternating in x1, ■ ■ ■, xn and y1, ■ ■ ■ ,yn.
Our main example is the double Capelli polynomial
DCapn _ h Cap,n(x1, ■ ■ ■ ,x,n;i)t2 Cap,n(y1, ■ ■ ■ ,yn; t')t3 ■ (5)
Here t and t' are arbitrary sets of extra indeterminates. We suppress the indeterminates t, t', and t1,t2,ts from the notation, since we do not alter them.
Definition 2.9. Let M denote the C-submodule of C{x,y,t} consisting of all doubly alternating polynomials (in x1..., xn, and in y1,..., yn).
M denotes the image of M in C{x,y,t}, i.e., the C-submodule of C{x,y,t} consisting of the images of all doubly alternating polynomials (in x1..., xn, and in y1,..., yn).
Remark 3. M is a C{t}-submodule of C{x,y,t}, namely C{t}M C M. Indeed, let h e C{i} and f e M. If either h or f is in CAPn+1 then hf e CAPn+1; hence the product hf = hf is well defined. Moreover, if f = f (x1,..., xn, y1,..., yn, t) is doubly alternating in the x's and in the y's, and h e C{t }, then hf is doubly alternating in the x's and in the y's.
2.2.2. The Zubrilin action
The theory hinges on the following amazing result, which we prove in Section 2.7 below. (This is also proved in [7, Theorem 4.82], but more details are given here.)
Proposition 2.10. Let f (x1,...,xn; y1,... ,yn) be doubly alternating in x1,...,xn and in y 1,..., yn (perhaps involving additional indeterminates). Then for any polynomial h,
i7\f) = ^k,kn)(f) modulo CAVn+1; (6)
namely,
f = Yj f \vij modul° CAVn+1.
Before proving Proposition 2.10 we deduce some of its consequences. Remark 2.11. It follows from Proposition 2.10 that S^fHf) - S^if) e CAPn+i whenever f e M, so working modulo CAPn+1 we can suppress x m the notation, writing à^hif ) for <^Xhn)(/).
2.2.3. Commutativity of the operators modulo CAPn+1 MM
Lemma 2.12. (i) ¿g induces a weU-defined map : M ^ M given by = 5—(f).
(ii) produces the same result using the indeterminates x or y.
Proof, (i) If f (x-]_,..., xn, y1,..., yn) and g(x1,..., xn, y1,..., yn) are doubly alternating polynomials, with f = g, then f — g e CAPn+1, so by Remark 2.4(1), S^^f — g) e CAPn+1 and hence
^kh—f — 9) = 0. Therefore we have
0=¿¿¡JT- 9)=—) - ^=§{:i(f) - (9),
proving that b^h is well-defined.
(ii) The assertion follows from Remark 2.11, which shows that S<——(f) = 5<——(f). □
Lemma 2.13. Let f = f (x]_,..., xn; y1,..., yn) be doubly alternating in x1,..., and 'my1,..., yn (and perhaps involving other indeterminates). Let 1 ^ k,£ ^ n. Then for any h]_,h2 e C{i},
i(n) tin) ( i) = c(n) tin) ( i) m
°k,h10IM (J )= 0l,h2 "k,h1 (J ).
PROOF. Equation (7) claims that modulo CAPn+l, (i)
5kM 8\M (/) = 6£M SkM (/) and
(Ü)
— №)^(f) -d
m
z(y,n) Ay,n) ( f) — x(y,n) Ay,n) ( f) °k,h1 °e,h2 (J ) — 0i,h2 "k,hi (J )'
The middle equivalence (ii) is an obvious equality. The first and third equivalences are similar, and we prove the first. By Proposition 2.10, by (ii), and again by Proposition 2.10, modulo CAVn+l we can write
s(x,n) ¡-(x,n) ( f ) r(x,n)Jy,n)(f) Ay,n)s(x,n) (f) ¡-(x,n) ¡-(x,n) ( f)
°k,h1 °e,h2 (J) — 0k,h1 °e,h2 (J) — 0i,h2 °k,h1 (J) — °e,h2 °k,hi (J)-Note that in the last step, Lemma 2.3 was applied (to (/)). □
2.2.4. The ideal In,A C A] and the annihilator of M
Definition 2.14. For each a e A let £1>a, ■ ■ ■ ,£n,a be n corresponding new commuting variables, and construct _ A[£1)tt, ■ ■ ■, £n)Q, | a e A]. Let In,A ^ be the ideal generated in
bv the elements
an + (1,aan—1 + ••• + in,a, a e A,
namely
In,A _ (an + £1,aan—1 + ■ ■ ■ + in,a | a e A).
Remark 2.15. In view of Proposition 2.10, the map 5^1 • M ^ M of Lemma 2.12 yields an action of the C-algebra C{i} ] on M, given bv ik,hf _ ¿k™/)(/)-
Working with the relatively free algebra, our next goal is to prove that In ■ M _ 0. For that we shall need the next result.
Proposition 2.16. (Zubrilin) Assume that a multilinear polynomial g(x1, ■ ■ ■ ,xn) is alternating in x1, ■ ■ ■, xn. Then, modulo CAT„,+]_,
n
E(-!)khn—k5inl(9) - 0 k=0
for any h e C|i}. In particular, if g is doubly alternating, then (again modulo CAVnn+1)
n
E(-!)khn—k^(9) - 0■ k=0
Proof. First we take h to be an indeterminate 2. Let f (x-\_, ■ ■ ■ ,xn+1) _ xn+1g(x1, ■ ■ ■ ,xn). Bv Proposition 2.7, C{x,y,t} satisfies the identity
n
E(-1)j ^zU (X1, ■■■, ®n, Zn—j Xn+1)) - 0-j=0
Note that in computing (x1,..., xn, zn 3xn+1)), the last indeterminate is xn+1 and is
unchanged, cf. Remark 1.1, so
^(f (X1, . . . , Xn, Z'n~3 Xn+1)) — Zn~3 Xn+15ijnn)zg(X1, ... ,Xn). Using Proposition 2.7, we have that
E(-!)J zn~3 Xn+1^^(g(x1,.. .,Xn)) G CAT n+1-3=0
The proof now follows by substituting xn+1 ^ 1 and z ^ h e C{i}. □
As a consequence we can now prove the key result: Proposition 2.17. Let M be the module given by Definition 2.9. Then, ■ M _ 0.
PROOF. We prove that ■ M _ 0, by showing for any doubly alternating polynomial
f (x1, ■ ■ ■, xn, y1, ■ ■ ■, yn) e M and h e C{t}, that
(hn + 6 , h hn~1 + ■ ■ ■ + Cn , h)f - 0 (mod CAT n+1)-
It follows from the action (k,hf _ $kh(f) and from Proposition 2.16 that modulo CAVn+1,
n
(hn + i1,hhn~1 + ■ ■ ■ + Uh)f _ E(-1)khn~k(f) - 0^
k,
k=0
□
2.3. The ideal Obstn(A) C A
In order to utilize these results about integrality, we need another concept. We define Obstn(A) _ A n In,A, viewing A c A[^n>A]•
Remark 4.
1. Let
A _ A[(n,A]/In,A, (8)
with f : ^ A the natural homomorphism, and fr : A ^ A be the restriction of f to
A. Then
ker(fr) _ A n I,n,A _ Obstn(A)^
2. Note that for every a e A, f (a) is n-integral (i.e., integral of degree n) over Cand thus over the center of A. Indeed, apply the homomorphism f to the element
an + ii,aan~1 + ■■■ + in,a (e In,A)
to get
an + £1,aan~1 + ■ ■ ■ + Ua _ (an + £1,aan~1 + ■ ■ ■ + £n>a) + In,A _ 0^
Lemma 2.18. ker(fr) also is the intersection of all kernels ker(g) of the following maps g:
g : A ^ B, where B is a C-algebra, and g : A ^ B is a homomorphism such that for any a e A, g(a) is n-integral over the center of B.
Proof. Denote the above intersection ng ker(g) as Obst^(A). Then Obst^(A) C Obstra(A) since ker(fr) is among these ker(g). To show the opposite inclusion we prove
Claim: For such g : A ^ B, ker(g) D A n In,A = Obstra(A).
Extend g to g* : ^ B as follows: g*(a) = a if a e A, while g*(£i,a) = We claim that
g*(In,A) = 0- Indeed, let
r = an + (1,aan-1 + ■■■ + Ua
be one of the generators of In,A-
By assumption there exist p1,a,..., ftn,a in the center of B satisfying
g(a)n + P1,a9(a)n-1 + ■ ■ ■ + pn,a = 0. (9)
Hence,
g*(r) = g(a)n + pltag(a)n-1 + ■ ■ ■ + pn,a = 0. This shows that as claimed, g*(In,A) = 0.
Finally, if a e AnIn,A then g(a) = g*(a) = 0. Hence a e ker(g), so ker(g) D AnIn,A = Obstra(A).
□
Corollary 2.19. If every a e A is n-integral (over the base field), then Obstra(A) = 0.
PROOF. The assumption implies that in the above, the identity map id = g : A ^ A satisfies the condition of Lemma 2.18. Hence 0 = ker(g) D Obstra(A), and the proof follows. □
This corollary explains the notation Obst„(A): it is the obstruction for each a e A to be n-integral. The next result technically is not needed, but helps to show how Obst behaves.
Lemma 2.20. Obstra-1(A) D Obstra(A).
Proof. Represent
Obstra-1(A) = nh ker(h) and Obstra(A) = ng ker(g),
with the respective conditions of n - 1 integrity and of n integrality. Take a e A and h : A ^ B with h(a) being n - 1 integral over the center of Then h( a) is also n integral over the center of
Hence every ker(h) in Obstra-1(A) also appears in the intersection Obstra(A) = ng ker(g), and □
2.4. Reduction to finite modules
The reduction to finite modules is done using Shirshov's theorem.
Proposition 2.21. Let A = C{a1,..., ag} have PI degree d over the base ring C. Then the affine algebra A/ Obstra(A) can be embedded in an algebra which is finite over a central affine subalgebra.
Proof.
Let B C A be the subset of the words in the alphabet a1,..., ag of length ^ d. By Shirshov's
h
W = {bk1 ■■■ bkhh \ bi e B, any ki ^ 0}
spans A over the base ring C.
Similarly to construct A[(n,B] C
A[^n,B] = A[{lib,...,{ntb \ b e B],
and let In,B be the ideal
In,B _ (bn + C1,bbn—1 + ■ ■ ■ + Cn,b I b e B) < A[Cn,B]
Denote
A' _ A[(,n,B]/In,B■ (10)
We show that A' is finite over an aifine central subalgebra and thus is Noetherian.
Given a e A, denote a' _ a + In,B e A', and similarlv £'ib _ (i,b + In, b- Then for every b e B, b' is n-integral over C [£n b]> where
C [Cn, b ] _ C [£ ,6, b I b e B] C cent er^')
Hence the finite subset
W' _ {b'k ■■■ b' khh | h e B, ki < n - 1} (c A1) (11)
spans A' over C[£n b]• ^hus A' is finite over the affine central subalgebra C[£n b] C center(A') and thus is Noetherian.
Restricting the natural map g : A[{n , b] ^ & _ B]/In,B to A, we have
gr : A ^ A (a^ a' _ a + In, b) (12)
which satisfies
ker(gr) _ A n In,b C A n In,A _ Obstn(13)
Let
A _ A/ Obstn(A), (14)
and for a e A denote a _ a + Obstn(A) e A. We then have the corresponding subset B _ {b I b e B} C A, as well as the set of commutative variables £n ^ and the ideal In ^.
Let A* _ A[in^]/IntS. , ,
Replacing A bv A and (i,B bv g, we clearly have the natural homomorphism
9 : Mn,B] ^ Mn,B]/In,B :_ A*,
with restriction
gIA _ gr : A ^ A*■
Note that each a e A is n-integral over the center of A, implying, by Corollary 2.19, that Obstn(A) _ 0. Then, as in (13), we deduce that
ker(a) C Obstn(i) (_0)
Hence a embeds A _ A/ Obstn(A) into A*. Note that A* is a finite algebra over the affine central subalgebra Q C A* generated by the finitely many central elements (k ^ + Injj. Denote b* _t> + In g. Then, as in (11), the finite subset
W * _ {bf1 ■■■ b*hkh I bi e B, ki < n - 1} (C A*)
spans A* over Q. □
2.5. Proving that Obstra(A) ■ (CAVn(A))2 = 0
In this section we show how Proposition 2.17 implies that Obstra(A) ■ (CAVn(A))2 = 0, thereby completing the proof of Razmvslov's Theorem. For this, we need to specialize down to given algebra A, requiring a new construction, the relatively free product, which enables us to handle A together with polynomials. Since, to our knowledge, this crucial step, which is needed one way or another in every published proof of the BKR theorem, has not yet appeared in print in full detail, we present two proofs, one faster but more ad hoc (since we intersect with A and bypass certain difficulties), and the second more structural.
Both approaches are taken in the context of varieties in universal algebra, by taking the free product of A with the free associative algebra, and then modding out the identities defining its variety.
2.5.1. The relatively free product
Definition 2.22. The free product A *c B of C-algebras A and B is their coproduct in the category of algebras.
(For C-algebras with 1, there are canonical C-module maps
A ^ A ® 1 c A ®G B, B ^ 1 ® B c A ®G B,
viewed naturally as C-modules, so A *c B can be identified with the tensor algebra of A ®c B, as reviewed in [29, Example 18.38].)
Although the results through Theorem 2.32 hold over any commutative base ring C, it is easier to visualize the situation for algebras over a field F, in which case we have an explicit description oiA[£ntA] * F{x; y; t}:
Fix a base Ba = {1} U B0 of A over F, and let B be the monomials in the {{ra,a : a e A} with coefficients in Ba- (For algebras without 1, we take B = Bo.) Thus B is an F-base of and * F{x; y; i} is the vector space having base comprised of all elements of the form
b0h1 b1h2b2 ■ ■ ■ hmbm where m ^ 0, b0,bm e B, b1,..., bm-1 e B\ {1}, and the hi are nontrivial words in the indeterminates Xi,yj ,tk. The free product ^[^a] * F {x; y; i} becomes an algebra via juxtaposition of terms. In other words, given
9j = bj,ohj'1bj'1hj'2bj'2 ■ ■ ■ hj,mjbj,mj for j = 1,2, we write b1,m1 b2,0 = a1 + ^k akbk for ak e F and bk ranging over B \ {1}, and define 9192 =a1b1,oh1'1b1,1 hi^ha ■ ■ ■ (hi>m,1 h2,^2,^2^2,2 ■ ■ ■ h2>m2
+ E akbl'0h1'lbl'1h1'2bl'2 ■ ■ ■ h1,m1 bkh2,1 b2,1h2,2b2,2 ■ ■ ■ h2,m2 .
(15)
k
For example, if b1b2 = 1 + b3 + b4, then
(b2hl,lbl)(b2h2,lb2) = b2(hl,lh2,l)b2 + hhl,lb3 h2,lb2 + b2hl,lb4h2,lb2.
2.5.2. The relatively free product of A and C{x; y; i} modulo a T-ideal
Even for algebras over an arbitrary base ring C, we can describe the free product of a C-algebra with C{x; y; i} by going over the same construction and mimicking the tensor product. Namely we
form the free C-module M having base comprised of all elements of the form a0h1a1h2a2 ■ ■ ■ hmam, h1a1h2a2 ■ ■ ■ hmam, a0h1a1h2a2 ■ ■ ■ hm, and h1a1h2a2 ■ ■ ■ hm where m ^ 0, a0,... ,am e A, and the hi are nontrivial words in the indeterminates Xi,yj,tk.
The free product A * C{x; y; t} is M/N, where N is the submodule generated by all
a0hlalh2 ■ ■ ■ di ■ ■ ■ hmam + a0hlalh2 ■ ■ ■ a'i ■ ■ ■ hmam — a0hlalh2 ■ ■ ■ (a,i + a'i) ■ ■ ■ hmam,
(ch1) — ch1,
ca0h1a1h2 ■ ■ ■ a,i — a0h1a1h2 ■ ■ ■ (ca{) ■ ■ ■ hmam, a,i e A, c e C; A * C{x; y; i} becomes an algebra via juxtaposition of terms, i.e., given
9j = ^,j,0hj, 1^j,1 hj,2^j,2 ■ ■ ■ hj,mj aj,mj
for j = 1,2, we define
9192 = calflhllallhl^al^ ■ ■ ■ (hl^h2:1)0,2,^2,,2^2;.2 ■ ■ ■ h2>m2 (16)
when a1,m1 a2,0 = c e C, or
9l92 = &la.1'0h1'la.1'1h1'2a.1,2 ■ ■ ■ h1,m1 (Ol,m1 &2,0)h2,1^2,1h2,2^2,2 ■ ■ ■ h2,m2 (17)
when al,m10,2,0 e
We write A(x; y; t) fa the fee pro duct A * C {x; y; i}.
We have the natural embedding C{x; y; i} ^ A(x; y; t). For g e C{x; y; t}, we write g for its natural image in A * C{x; y; i}.
Definition 2.23. Suppose X is a T-ideal of C{x; y; t}, for which X c ld(A). The relatively free
product A(x; y; t)x of A rnd C{x; y; i} ^^ulo X is defined as (A *c C{x; y; t})/X, where X is the two-sided ideal X(A *c C{x; y; i}) consisting of all evaluations on A * C{x; y; i} of polynomials X
We can consider A(x; y; t)x as the ring of (noncommutative) polynomials but with coefficients from A interspersed throughout, token modulo the relations in X.
This construction is universal in the following sense: Any homomorphic image of A(x; y; t) satisfying these identities (from X) is naturally a homomorphic image of A(x; y; t)x- Thus, we have:
Lemma 2.24. (i) For any g1,...,gk ,h1,...,h]tm A(x; y; t), there is a natural endomorphism A(x; y; t) ^ A(x; y; t) which fixes A and all U and sends Xi ^ gi, yi ^ hi.
(ii) For any g1,... ,gk ,h]_,... ,hk in A(x; y; t)x, there is a natural endomorphism
A(x; y; t)x ^ A(x; y; t)x,
which fixes A and all ti and sen ds Xi^ gi, yi^ hi.
Although difficult to describe explicitly, the relatively free product is needed implicitly in all known proofs of the Braun-Kemer-Razmvslov Theorem in the literature. From now on, we assume that X contains CAPn+1, so that we can work with M.
Let Ma denote the image of M under substitutions to A, i.e., the C-submodule of C{x,y,t} consisting of the images of all doubly alternating polynomials (in x1 ... ,xn, and in y1,...,yn). In view of Lemma 2.13, the natural action of Obst„(A) on Ma respects multiplication by the ¿^-operators.
Proposition 2.25. Obstra(A)m^ = 0.
proof. If a G Obstn(A), then aM G X, in view of Lemmas 2.5 and 2.12 and Proposition 2.17, so is 0 modulo X. □
Corollary 2.26. If b G A belongs to the T-ideal generated by doubly alternating polynomials, then Obstn(A)6 — 0.
Proof. The element b belongs to the linear combinations images of Ma under specializations Xj, ^ a^ □
By Step 7 of Section 1.2, this will complete the proof of the nilpotence of Jac(A) when C is a field, or more generally of any nil ideal when C is Noetherian, once we complete the proof of Proposition 2.10.
2.6. A more formal approach to Zubrilin's argument
Rather than push immediately into A, one can perform these computations first at the level of polynomials and then specialize. This requires a bit more machinery, since it requires adjoining the commuting indeterminates (n,A to the free product, but might be clearer conceptually.
Note that C{i}[£n,c{i}] _ R ®c C{t}■
Lemma 2.27. M becomes an C{t}[(n,c{t}]-module via the action given as follows: Order the £k,h as ^j _ ^ h- for 1 ^ j < t For a letter (j _ £k ,h., define
b f _ 4n A, (II
and, inductively,
^f _ ^(?¡—1f)■ For a monomial h _ {j3 ■ ■ ■ of degree d _ d1 + ■ ■ ■ + dj, define
hf _ ef (ej—i1 ■■■cl1 f)
inductively on j. Finally, define
J2(cihi)f _Y1 ai(hif)
where Ci e C and hi are distinct monomials. Proof.
The action is clearly well-defined, so we need to verify the associativity and commutativitv of the action. It is enough to show that (hjh^)/ _ hj(h^)/ for any two monomials hi and h[. But this follows inductively from induction on their length, plus the fact that £j (£j' f) _ £j' (£j f) fa any £j and £j' ■ □
Let us continue to take X _ CAT„,+]_■
Remark 2.28. Clearly A^^a] *c C{i} c A[£n,A] *c C{x; y; t} in the natural way, and then
X(A[£n,A] *c C{i}) _ (A[^n,A] *c C{i}) nX(A^a] *c C{x; y; t})
since we are just restricting the indeterminates x, y, t to the indeterminates t. It follows from Noether's Isomorphism Theorem that
F :_ (A[£n,^] *c C{t})/X(A[Cn,A] *c C{t}),
can be viewed naturally in (A[£n,^] *c C{x; y; t})/X(A[£n,^] *c C{x; y; t}). Viewing M c C{x; y; i} c *c C{x; y; t}, we define
M' _ (A[£n,A] *c C{t})M c A[inA] *c C{x; y; t}, (18)
and its image in (A[£n,^] *c C{x; y; t})/X(A[(n,A] *c C{x; y; t}), which we call M (intuitively consisting of terms ending with images of doubly alternating polynomials), which acts naturally by right multiplication on F. To understand how M works, we look at the Capelli polynomial acting on A *c C{x; y; i} fa an arbitrary algebra A satisfving Capn+1.
There is a more subtle action that we need. Mi can be viewed as an E-module where R _ C[C n^^+x]' v*a the crucial Lemma 2.12. But as above, M is an A * C{¿}-module where the
n, { X }
algebra multiplication is induced from (15) (viewing M c C{x; y; t}), implying M is an A * C{¿}-modul define
module annihilated by CAPn+1^^\.d Mi thereby becomes an ] * C{i}-module, where we
n, { X }
ik,h f _ S^lf
for h e C{x; y; t} and f e M, by means of the action given in Lemma 2.27, also cf. Remark 2.15. Our main task is to identify these two actions when they are specialized to A.
2.6.1. The specialization argument
Having in hand the module M on which x] acts, we can specialize the assertion of
1, ^^ { X ity }
Proposition 2.17 down to A once we succeed in matching the actions of A^cjX^}] and when specializing to A.
Remark 2.29. CAPk(A[£n,^]) _ CAPk(A)[£n,^], since Capk is multilinear.
We write VCAPn fa the C{i}-submodule of C{x; y; i} generated by DCapn, cf. (5), and DCAPn fa its image in C{x; y; i^. This is a set of doubly alternating polynomials in x1,...,xn and y1,..., yn, with variables U interspersed arbitrarily.
Lemma 2.30. Any specialization p : C{x; y; i} ^ A (together with its accompanying specialization p : C{x; y; i} ^ A) gives rise naturally to a map
$ : A[(n,A]VCAPn ^ M
given by
Yj^kMhj. )fi ^Yj ai(P(^k,hji fi) ^^khlfi)
where % e VCAPn.
proof. We need to show that this is well-defined, which follows from the functorialitv property given in Lemma 2.5. Namely, if p(hji) _ p(hj.), then p('hji) _ p(hj.) and
) _ E wtfk^.ti _ E w^kx ft _ E w^kx ft.
tilt
□
The objective of this lemma was to enable us to replace by A in our considerations,
ker $ contains all b^hf - £k,hf (cf- Remark 2.11) as well as (hn -Yin—o hk£k,h)f, where h ranges over all words and f e VCAPn, so we see that the Zubrilin integrality relations are passed on.
Lemma 2.31. If i is an infinite set of noncommuting indeterminates whose cardinality N is at least that of A then fa any given evaluation w in VCAPn(A *c C{x; y; t}), there is a map
(pw : C{x; y; t} ^ A *c C{x; y; t},
sending VCAPn to VCAPn(A *c C{x; y; t}), such that w is in the image of <pw.
Proof. Note that A *c C{x; y; i} has cardinality N. Setting aside indeterminates
{tg : g e A *c C{x; y; i}},
we still have N indeterminates left over, to map onto our original set t of N indeterminates. But any evaluation w of VCAPn on A *c C{x; y; i} can be written as
w = g Capjxl, ...,xn; gu .. .,gn)g' Capra (yl, ...,yn; hl,.. .,hn)g", (19)
for suitable g,g',g",gi,hj e A *c C{x; y; i^. Defining pw by sending Xi ^ Xi, yj ^ yj, and sending
the appropriate tg ^ g,tg'^- g', tg» ^ g", tgi ^ gi, and thj ^ hj, we have an element in p-1(w). □
Clearly tpw(CAPn+1) C CAPn+1(A *c C{x; y; t}), so, when Capra+1 e X, pw induces a map
0w : C{x; y; t} ^ (A *c C{x; y; t})x,
which sends M ^ M.
Although we do not see that CAPn+1 need be mapped onto CAPn+1(A *c C{x; y; t}), Lemma 2.31 says that it is "pointwise" onto, according to any chosen point, and this is enough for our purposes.
Theorem 2.32. Obstra(A) ■ (CAPn(A))2 = 0, for any PI-algebra A = C{a1, ...,ae} satisfying the Capelli identity Capra+1.
Proof.
We form the free algebra C{x; y; i} by taking a separate indeterminate tj for each element of A[(n,A]VCAPn. We work with A[(n,A]VCAPn, viewed in the relatively free product A := *c C{x; y; t})x, where X = CAP „++]_( A^n^ *c C{x; y; t}). In view of Lemma 2.30, the
relation
In^t} ■ MM = 0 (mod CAPn+l(A[Ca,A] *c c{x; y; t}))
restricts to the relation I VCAPn = 0 (mod CAPn+1 *c C {x; y; t})). But the
various specializations of Lemma 2.31 cover all of VCAPn(A). Hence Lemma 2.5 applied to Proposition 2.17 and Lemma 2.30 implies In,AVCAPn(A) = 0, and thus
Obstra(A) ■ (CAPn(A))2 C InAVCAPn(A) = 0.
□
2.7. The proof of Proposition 2.10
Now we present the proof of the crucial Proposition 2.10, stating that for a doubly alternating polynomial f = f (xi, ... ,xn,yl,... ,yn,t),
& (f) = ¿if (f) modulo CAPn+l.
2.7.1. The connection to the group algebra of Sn
We begin with the basic correspondence between multilinear identities and elements of the group algebra over Sn.
Vn _ Vn(x]_,..., xn) denotes the C-module of multilinear polynomials in x1,..., xn, i.e.,
Vn _ spanc{xa<1)Xa<2) ■ ■ ■ xa<n) I a e Sn}.
Definition 2.33. We identify Vn with the group algebra C[5n], by identifying a permutation a e Sn with its corresponding monomial (in x1,x2, ■ ■ ■, xn):
a o Ma (X1, ...,Xn) _ xa<1) ■ ■■ Xfj(n) ■
Any polynomial xa<i) ■ ■ ■ xa<n) corresponds to an element a e C[5n], and conversely, Y & corresponds to the polynomial
(Ea°a) X1 ■■■ xn _ Ea°x<7<1) ■ ■ ■ x<7<n).
Here is a combinatorial identity of interest of its own.
Consider two disjoint sets XnY _ 0, each of cardinalitv n, and the symmetric group S2n _ SXuY acting on X U Y. For each subset Z C X we define an element P(Z) e C[S2n] as follows:
P(Z)_ ^ sgn(a) ■ a.
a(Z)CY
In particular
Proposition 2.34.
P(0) — ^ sgn(a) ■ a.
J2(-1)lZlP(Z)_ E sgn(a) ■ a. (20)
zcx a(X)=X
Proof. Let a e S2n and let aa (resp. ba) be the coefficient of a on the l.h.s. (resp. r.h.s.) of (20). We show that aa _ ba.
Let Z(a) _ a—1(Y) be the largest subset Z C X such that a(Z) C Y. Note that a(X) _ X if and only if Z(a) _ 0. Therefore
ba _ sgn(ff) if Z(a) _ 0 and ba _ 0 if Z(a) _ 0, (21)
since P(0) _ Y sgn(^) ■ a. We claim that
aa _ sgn(a) ■ £ (-1)|Z|.
ZcZ(a)
To show this, recall that
l.h.s _ £ (-1)izi £ sgn(a) ■ a.
zcx a(Z) cy
In P(Z) the coefficient of a is sgn(a) if Z C Z(a) (since then a(Z) C Y), and is zero if Z C Z(a) (since if a(Z) C Y then a(Z U Z(a)) C Y, contradicting the maximalitv of Z(a)). It follows that as claimed,
_ sgn(a) ■ £ (-1)izi.
ZCZ(a)
It is well known that Cz(a)(—1)^Z 1 = 1 when. Z(a) = 0 and = 0 otherwise. Therefore
aa = sgn(ff) if Z (a) = 0 and aa = 0 if Z (a) = 0. (22)
The proof now follows by comparing (21) with (22). □
Lemma 2.35. Let f (x]_,..., xn, y1,... ,yn,t) be doubly alternating. Then
f (xi, ...,xn,yl,.. .,yn,t) = f (yl, ...,yn,Xl,.. .,xn,t) modulo CAPn+l.
Proof. Let X = {x1,.. .,xn} and Y = {y1,.. .,yn}. Then IX| = IY| = n mid X n Y = 0, and we identify S2n = sxuk- Let M = {x^,...,Xik} C X, with 1 ^ i1 < ■■■ < ik ^ n, and N = {yh,. ..,yjk} C F, wit h 1 ^ j1 < ■■■ < jk ^ n. Thus, IM | = IN | = k ^ n. M will play the role of Z in Proposition 2.34. We consider permutations a e S2n with a(M) = N. Define the permutation
TMN = (Xh,Vh) ■■■ (xik,Vjk).
Since M n N = 0, tmn has order 2 in S2n, and satisfies sgn(rMW) = (—1)k. If M = X then N = Y and sgn(rMW) = sgn(rxy) = (—1)"". Moreover tmn(M) = N and tmn(N) = M.
Next, we define
Tmn = E sgn(^) ■ ft e C[S„].
K(M )=N
Let p = tmn ■ ft, so that p(M) = M. Then ft = tmn ■ P and
Tmn = sgn(rMw) ■ tMn ■ I E sgn(p) ■ p
\p(M )=M
But by Proposition 2.34,
£ (—1)^IP(M)= £ sgna ■ a. (23)
MCX a(X)=X
If M C X, then P(M) is alternating on 2n — IM| ^ n + 1 indeterminates, and hence is 0 modulo CAPra+1. Thus, modulo CAPn+1, the left hand side of (23) equals the unique summand with M = X, which is
(—1)n Y sgn(a) ■ a = (—1)rasgn(rXy) ■ tXy ■ I E sgn(a) ■ a
a(X)=Y \a(Y)=Y
= TXY ■ I E sgn(a) ■ a
\a(Y )=Y
Since a(X) = X if and only if a(Y) = Y, it follows that
E sgn(a) ■ a = E sgn(a) ■ a = txy ■ I E sgn(a) ■ , modu\oCAPn+l.
a(Y)=Y a(X)=X \a(Y)=Y
Now we identify elements in C [52n] with polynomials multil inear in x1,..., xn, y1,... ,yn. Taking a monomial h(x1,..., xn, y 1,..., yn; t) multilinear in x1,..., xn, y1,...,yn, we define
f (X1, ■■■, Xn,V1, ■■■, yn;^) :_ I E sgn(a) ■ ct) h.
V(Y)=Y J
Then
txy ■ I E sgn(CT) ■ ct) h _ f(y1,...,yn,X1,...,Xn;t).
V(Y)=Y J
Again, since a(X) _ X if and only if a(Y) _ Y, it follows that f (x]_,..., xn, y1,... ,yn;t) is doubly alternating, and we have proved that
f (x1,...,xn,y1,...,yn;£) - f (y1,..., yn, X1,. ..,Xn;t) modulo CAVn+1,
as desired. □
2.7.2. Proof of Proposition 2.10
We may assume that h is a new indeterminate z. Recall that 4X)(/^^ ■■■,Xn,y1 ,■■■,Уn, t)) _
_ E f (X1,...,x,n,y1,...,yn,t )IXiu^zXiu; u _1,■■■,k,
i 1<"<ik ^n
and
(f (X1, ...,Xn,y1,...,yn, t )) _
_ E f (X1,...,Xn,y1,...,yn,t )Iyiu^zyiu; u _1,...,k. 1^11<-< ik^n
Let z' _ 1 + ez, e being a central indeterminant. Then clearly
k,
k=o
f (z'x1,.. .,z'xn,y1, ■■■ ,yn ,t) _ E £k ■ S<k,f(f (x1, ...,Xn,y1,.. ■ ,Vn,t)) (24)
and
n
f (X1,.. .,Xn,z'y1,.. ■,z'yn,t) _ ^ ek ■ 5iZn\f (X1, ...,Xn,y1,.. .,yn,t ))■ (25)
k=o
By Equations (24) and (25) it is enough to show that
f(z'x1,...,z'xn,y1,...,yn,t ) - f(X1,...,Xn,z'y1,...,z'yn,t ) modulo CAPn+1.
Let
§1(X1, ...,Xn,y1,.. .,yn,t) _ f (z'x1, ■ ■ .,z'Xn,y1, ■ ■ ■ ,yn,t )
and
92(X1, ...,Xn ,y1,.. .,yn ,t ) _ f (X1, ■ . .,Xn,z'y1,.. .,z'yn,t )■
We have to show that
gl = g2 modulo CAPn+l.
Denote x^ = z'xi, y\ = z'yi; i = 1,... ,n. Then
gl(xl,...,xn,yl ,...,yn,t) = f (x'l,...,x'n,yl,...,yn,t) =
= f(yl,...,yn,x'l,...,x'n,t) = = g2(yl, ...,yn,Xl,.. .,xn,t) = = g2(xl,...,xn,yl,...,yn,t) modulo CAPn+l. The congruences follow from Lemma 2.35 since both ^d g2 are doubly alternating.
3. Proof of Kemer's "Capelli Theorem,"
To complete the proof of Theorem 1.1, it remains to present an exposition of Kemer's "Capelli Theorem," that any affine PI algebra over a field F satisfies a Capelli id entity Capra for large enough n. This is done by abstracting a kev property of Capra, called spareseness.
Definition 3.1. A multilinear polynomial g = ^ a<jxa{1)... xa(d) is a sparse identity of A if, for any monomial f (x1,... ,Xd]t) we have
f(Xa(l),.. .,xa(dy,t) e id(A).
See [7, §2.5.2] for more detail. The major example of a sparse identity is the Capelli identity. One proves rather quickly that any sparse identity implies a Capelli identity, so it remains to show that any affine PI algebra over a field satisfies a sparse identity. There are two possible approaches, both using the classical representation theory of Sn. One proof relies on "the branching theorem," which requires characteristic 0, and the other relies more on the structure of the group algebra F [5n], also with the technique of "pumping" polynomial identities, and works in arbitrary characteristic.
3.1. Affine algebras satisfying a sparse identity
Sparse identities work well with the left lexicographic order < .If b1 < ■ ■ ■ < bm and 1 = a e Sm, then (b1,... ,bm) < (ba(1),... ,ba(m)). Any sparse identity over a field yields a powerful sparse reduction procedure. Namely, we may assume a(1) = 1; given a1,... ,aa in A, we can replace any term f (a1,..., ad) by
— E f (Xa(1) . . . Xa(d), Xd+1, ...,Xn). a=1
(The analogous assertion also holds for Cd )
Lemma 3.2. Let A = C{a1,a2...} be a PI algebra, satisfying a sparse multilinear identity p = J2aesd Xa(1) ■ ■ ■ xa(d) of degree d, with d ^ n, and let M(x]_,..., xn; y ) be a monomial multilinear in x1,..., xn and perhaps involving extra indeterminates We consider A = M (v1,...,vn; y), where v1,..., vn are words in the generators a1,a2,... and y is an arbitrary specialization of y in A Assume that k of the ^satisfy ^ d (length as words in a1,a2,...). If 1 ^ then A is a linear combination of monomials A' = M (v^,... ,v'n; y) where at m ost l — 1 of the wor ds v\ have length ^ d.
This clearly implies that A is spanned by monomials A' = M(v[,v'2,...), with at most d — 1 of the v\ having length ^ d. Proof.
Claim: If ^ |,..., Ivid | ^ d, then A = M (v1,...,vn; y) is a linear combination of terms A' = M(v'1,...,v'n;y) satisfying
(Iv'll,..., KI) < (IviI,..., Ivn I).
The above Claim implies the existence of descending sequences of monomials, under the left lexicographic order. Such a descending sequence must stop. When it stops we have a corresponding monomial having strictly fewer words for which Iv¡I ^ d. Therefore proving the above Claim will prove the lemma. We now prove the Claim.
We rewrite A = M (v^ ,...,Vid; y), where i1 < i2, ■■■ < id; then we may assume that i1 = 1,... ,id = d. We write Vi = WiUi where IuiI = d — i, 1 ^ i ^ d. The sparse identity p implies that A is a linear combination of terms Aa = M (w1ua(1),..., wh ua(d); y) = M (vl,... ,v'd; y) where 1 = a e Sd- (A itself corresponds to a = 1.) To see this, we rewrite A = M(w1u1,... ,WdUd; y) as
N(u1,... ,Ud; W). The sparse identity p implies that N(u1,... ,ua; W) is a linear combination of elements of the form
N(ua(l),..., ua(d); W) = M(WlUa(l),..., WdUa(d); y), 1 = a e Sd.
Denote WiUa(i) = v'^ 1 ^ i ^ But then (I^lI,..., I^^I) < (I^1I,..., I Vd I) for such a = 1. This proves the Claim, and completes the proof of the lemma. □
Although we did not apply Shirshov's Height Theorem, the main argument here is similar. Note also that Lemma 3.2 applies to any PI algebra, not necessarily affine. In the next theorem, due to Kemer, we do assume that A is affine.
Theorem 3.3. Let A = C {a1,... ,ar} be an affine PI algebra over a commutative ring C, satisfying a sparse identity p of degree d, and let n ^ rd + d. Then A satisfies the Capelli identity Capra[^; y],
PROOF. We may assume that r ^ 2, since otherwise A is commutative. Consider
Capn(vl, ...,vn; wi,.. . ,wn)
where Vi,Wi e A By Lemma 3.2 we may assume that at most d — 1 of the Vi have length ^ d (as words in the generators a1..., ar). at 1 east n — (d — 1) of th e Vi have leng th ^ d — 1. The
number of distinct words of length g is ^ rq. the number of words of length ^ d — 1 is
ipd _ 1
^ 1 + r + r2 +-----+ rd-1 =-< rd (since r ^ 2.
r — 1
But we have at least n — (d — 1) such words appearing in v1,...,vn, and n — (d — 1) > r' (since by assumption n ^ rd + d). ft follows that there must be repetitions among v1,...,vn, so Capn(v1,... ,vn; w1,..., wn) = 0 □
3.2. Actions of the group algebra
It remains to prove the existence of sparse identities for affine Pi-algebras. For this, we turn to the representation theory of Sn. After a brief review of actions of Sn on Young diagrams, we treat
.d
the characteristic 0 case, cf. Kemer [15]. The characteristic p > 0 proof, which requires some results about modular representations but bypassing branching, is done in §3.2 and §4.
Given a,n e Sn, by convention we take an(i) _ n(a(i)). The product aw corresponding (by Definition 2.33) to the monomial
Ma7T _ Xa7T(1) ■ ■ ■ Xa7i(n)
can be interpreted in two ways, according to left and right actions of Sn on Vn, described respectively as follows:
Let a,n e 5!. Let yi _ xa(i). Then
(i) 0M7(X1..., Xn) :_ Ma7 _ M7(xa(1), ■ ■ ■, xa(n)) and
(ii) Ma (X1 ..., Xn)K :_ (V1 ■■■ yn)K _ Ma7 _ yn(1) ■ ■ ■ y-7(n).
Thus, the effect of the right action of k on a monomial is to permute the places of the indeterminates according to n.
Extending by linearity, we obtain for any f _ f (x1,..., xn) e Vn that
(i) ap(X1,...,Xn) _ p(xa(1),...,xa(n));
(ii) p(x1,..., xn)w _ q(y1, ■ ■ ■, yn), where q(y1, ■ ■ ■, yn) is obtained from p(x1,..., xn) by place-permuting all the monomials of p ^cording to the permutation n.
For any finite group G and field F, there is a well-known correspondence between the F [G]-modules and the representations of G. The simple modules correspond to the irreducible representations.
Remark 3.4. If p e Id(A), then ap e Id(A) since the left action is just a change of variables. Hence, for any PI-algebra A, the spaces
Id(A) n Fn C Vn
are in fact left ideals of F[5n] (thereby ^fording certain Sn representations), but need not be two-sided ideals. However, we prove below the existence of a nonzero two-sided ideal in Id(A) n Vn, a fact which is of crucial importance in what follows.
Remark 3.5. Let A be a partition. As explained in [7, p. 147], any tableau T of A gives rise to an element
aT _ ^ sgn(q)qp e G [Sn],
where Ctx (resp. ~Rtx ) denotes the set of column (resp. row) permutations of the tableau T\. a?p _ aTat for some aT in ^te base field F. When aT _ 0, which by [29, Lemma 19.59(i)] is always the case when char(F) does not divide n, in particular, when char(F) _ 0, we will call the idempotent eT :_ a--1 aT ^^e Young symmetrizer of the tableau T.
Furthermore, by [29, Lemma 19.59(i)], if aT _ 0 and then F[Sn]aT _ FaT, implying F[S'n]aT (if nonzero) is a minimal left ideal, which we call J\ .Thus, if J\ contains an element corresponding to a nontrivial PI of A, aT itself must correspond to a PI of A.
sx :_ dim J\ is given by the "hook"formula, see for example [30] or [14], where we recall that each "hook"number hX for a, box x is the number of boxes in "hook"formed by taking all boxes to the right of x and beneath x. (In the literature, one writes fx instead of sx, but here we have used f throughout for polynomials.)
Lemma 3.6. Suppose L is a minimal left ideal of a ring R. Then the minimal two-sided ideal of R containing L is a sum of minimal left ideals of R isomorphic to L as modules.
We let I\ denote the minimal two-sided ideal of F[5n] containing J\.
We define the codimension Cn(A) = dim (Id(^"ny ) . The characteristic 0 version of the next result is in [24].
Lemma 3.7. Let A be an F-algebra, and let A be a partition of n. If dim J\ > cn(A), then h C ld(A) n Fra.
PROOF. By Lemma 3.6, J\ is a sum of minimal № ideals, with each such minimal left ideal J isomorphic to J\. Thus, dim J = dim J\ > cri(A). Since J is minimal, either J c Id(A) n Vn or J n (Id(A) n Vn) = 0. If J n (Id(A) n Vn) =0 then it follows that
cn(A) = dim Vn/(Id(A) n Vn) ^ dim J > cn(A),
a contradiction. Therefore each J c Id(A) n Vn. I\ C Id(A) n Vn since I\ equals the sum of these □
3.3. The characteristic 0 case [15]
The characteristic 0 case is treated separately here, since it can be handled via the classical representation theory of the symmetric group. By Maschke's Theorem, the group algebra FSn now is a finite direct product of matrix algebras over F. We have the decomposition FSn = 0Xhn I\. Thus, Lemma 3.7 yields at once:
Lemma 3.8. [24] Let char(F) = 0, let A be an F algebra, and let A be a partition of n. If sA > cn(A), then Ix C Id(A) n Vn.
(Here I\ is the sum of those F[Sn]eT for which T is a standard tableau with partition A. These I\ are minimal two sided ideals, each a sum of sA minimal № ideals isomorphic to J\.)
Example 3. Consider the "rectangle" of u rows and v columns. By [20, page 11], the hook numbers of the partition p, = (uv) satisfy
E hx = uv(u + v)/2 = n—+—.
Let us review the proof, for further reference. For any box x in the (1,j) position, the hook has length u + v — j, so the sum of all hook numbers in the first row is
E" .. v(v — 1) ( v — 1\ (U + V — J) = uv + -2- = v(u + —^ ) .
3=1 ^ J
Summing this over all rows yields
iu + 1 v — 1\
2
as desired.
3.3.1. Strong identities
u(u + 1) v — 1 iu + 1 v — 1\ U + V + uv—-— = uv —---1---— = uv—-—,
Definition 3.9. Let A ^e a PI The multilinear polynomial g e Vn is a strong identity of
A if fa every m ^ n we have FSm ■ g ■ FSm C Id(A).
Note that every strong identity is sparse. To obtain strong identities, we utilize the following construction, due to Amitsur.
The natural embedding Sn C Sn+1 (via a(n + 1) _ n + 1 for a e Sn) induces the embedding Vn C Vn+1: f(x1 ,...,xn) = f(x1,... ,xn) ■ xn+1. More generally, for any n < m we have the inclusion Vn C Vmv'rn f (x1, ■■■,x,n) = f (x1, ■■■,x,n) ■ x,n+1 ■■■ xm.
For f (x) _ f (X1, ...,Xn) _ Yaesn a° xv(1) ■ ■ ■ xa(n) e Vn, we define
f *(X1,... , Xn; Xn+1, ■ ■ ■ , X2n—1) — ^ ^ Xa(1)Xn+1Xa(2)Xn+2 ■ ■ ■ Xa(n—1)X2n—1Xa(n) (26)
vesn
_ (f (X1, ..., Xn)Xn+1 ■ ■ ■ X2n—1)v, where e S2n—1 is the permutation
V _ ( 1 2 3 4 ■■■ 2n - 1 ^ (27)
1 V 1 n + 12 n + 2 ■■■ n ) ■ K '
Let L C [xn+1,... ,x2n—1} and denote by fl the polynomial obtaine d from f * by substituting Xj ^ 1 fa all Xj e L. Rename the indeterminates in {x,n+1, ■ ■ ■ ,X2n—1} \ L as {Xn+1, ■ ■ ■ , Xn+q} (where q _ n — 1 — m) and denote the resulting polynomial as f*. Then similarly to (26), there exists a permutation p e Sn+q such that fl _ (fxn+1 ■ ■ ■ xn+q)p.
Note that if 1 e A and f * e Id(A), then also fl e Id(A) for any such L, and in particular f e Id(A). The converse is not true: it is possible that f e Id(A) tat f * <£ Id(A).
Lemma 3.10. Let A be a PI algebra, let I C Vn be a two-sided ideal in Vn, and assume for any f e I that f * e Id(A) (and thus f e Id(A)). Then for any m ^ n,
(FSm)I(FSm) C Id(A).
Proof. Since (FSm)I C Id(A), it suffices to prove:
Claim: If f e ^d -k e Sm, then f* n _ (f (x]_,..., xn)xn+1 ■ ■ ■ xm)K e Id(A).
If f _ a<J a(x1 ■■■ Xn ■■■ X^, the n fl -K _ Ya£Sn ct(^(x1 ■■■ X,n ■■■ Xm)).
Consider the positions of x1,..., xn in the monomial n(x1 ■ ■ ■ xm): There exists t e Sn such that
-k(X1 ■■■ Xn ■■■ Xm) _ goXT (1)g1XT (2)g2 ■ ■ ■ gn—1XT (n)gn _ T (goX1g1X2g2 ■ ■ ■ gn—1Xngn),
where each gj is _ 1 or is a monomial in some of the indeterminates xn+1,... ,xm. It follows that fl-k _ (fr)(goX1g1X2g2 ■■■ 5'n-1£n5'n)- Since f e V,n and t e Sn, fr only permutes the indeterminates x1,... ,xn, and hence (see (26))
fl * _ (fT )(.9oX1g1 X2g2 ■ ■■ gn—1Xngn) _ go((fr )*[x1, ...,Xn; 91, ■■ .,gn—1])gn.
Since I is two-sided, fr e I, hence by assumption (fr)* e Id(A), which by the last equality implies that fn e Id(A). □
3.3.2. Existence of nonzero two-sided ideals I\ Q FSn of identities
Let cn(A) ^ an fa all n. The next lemma yields rectangles p, = (u") h n such that an < s^.
Lemma 3.11. Let 0 < u, v be integers and let ^ be the u x v rectangle ^ = (uv) h u ■ v. Let n = uv. Then n
(û^) ( e) <S>I (where e = 2.718281828...).
In particular, if a ^ ■ 2 then an ^ Proof.
Since the geometric mean is bounded by the arithmetic mean,
1/n 1
, i 1 v^, u +v
hx "
inh*) < ^ Eh*
n^ x 2 '
/
in view of Example 3, and hence
2 \n 1
(—)
\u + v J
<
vu + v) Uxhx' Together with the classical inequality (n/e)ra < n!, this implies that
(V (2\n - (ny (V n! _ ^
+v) V e) ^ e) V u + v) Uxevhx S
□
Remark 3.12. To apply this, we need Regev's estimate [23] of codimensions,
cm(A) < (d — 1)2m,
as explained in [7, Theorem 5.38].
Proposition 3.13. [4] Let A be a PI algebra satisfying an identity of degree d. Choose natural numbers u and v such that wv 2
—-j— — ^ (d — 1)4 . For example, choose u = v ^ e ■ (d — 1)4.
Let n = uv and let ^ = (uv) be the u x v rectangle. Let n ^ m ^ 2n and let A h m be any partition of m which contains p. (uv) C A. Then the elements of the corresponding two-sided ideal I\ C FSm are identities of A I\ C Id(A) n Vm.
PROOF. Since m ^ 2n, (d — 1)2m ^ (d — 1)44a, and by assumption (d — 1)4 ^ ■ 2 . By Lemma 3.11,
(2xra * *
I u+v e
■ f) < and since /j. Q A, we know that s^ ^ sx. Thus, by Remark 3.12,
( uv 2 Y
\u + v e J
cm(A) < (d - 1)2m < (d - 1)4n < ( ^^ ■ <s^ <
and the assertion now follows from Lemma 3.8. □ Corollary 3.14. Hypotheses as in Proposition 3.13, fan ^ m ^ 2n,
0 h Q Id(A).
Ahm
^CA
Consequently, if f G then f * G Id(A) n V2n-\ (see (26)). Also, for any subset L Q {n + 1,..., 2n - 1} fl G Id( A), and in particular f G Id(A).
PROOF. By "branching," the two-sided ideal generated in Vm by is
VmI^Vm _ (FSm)I^(FSm) _ 0 Ix.
Ahm
^CA
Hence, (FSm)I^(FSm) C Id(A) fa any n ^ m ^ 2n — 1, and in particular, if f e and p e Sm then fp e Id(A). (26) concludes the proof. □
By Proposition 3.13 and Lemma 3.10 we have just proved
Proposition 3.15. Every PI algebra in characteristic 0 satisfies non-trivial strong identities. Explicitly, let char(F) _ 0 and let A satisfy an identity of degree d. Let u, v be natural numbers such that ■ 2 ^ (d — 1)4, and let ^ _ (uv) be the u x v rectangle. Then every g e is a strong identity of A The degree of such a strong identity g is uv. We can choose for example u _ v _ \e ■ (d — 1)4l,so deg(g) _ \e ■ (d — 1)4]2 _ e2(d — 1)8.
We summarize:
Theorem 3.16. Every affine PI algebra over a field of characteristic 0 satisfies some Capelli identity. Explicitly, we have the following:
(a) Suppose the F-algebra A satisfies an identity of degree d. Then A satisfies a strong identity of degree
d _ [e(d — 1)4]2 _ e2(d — 1)8.
(b) Suppose A _ F{a1,... ,ar}, and A satisfies an identity of degree d and take d' as in (a). Let n _ rd + d & re (d-1 .Then A satisfies the Capelli i dentitv Capn.
PROOF, (a) is by Proposition 3.15, and then (b) follows from Theorem 3.3, since every strong □
3.4. Actions of the group algebra on sparse identities
Although the method of §3.2 is the one customarily used in the literature, it does rely on branching and thus only is effective in characteristic 0. A slight modification enables us to avoid branching. The main idea is that any sparse identity follows from an identity of the form
f ^ ^ X(j(1)Xn+1 ■ ■ ■ Xa(n)X2n, vesn
since we could then specialize xn+1,..., x2n to whatever we want. Thus, letting vn denote the subspace of V2n generated by the words xa(1)Xn+1 ■ ■ ■ xa(n)x2n, we can identify the sparse identities with F [5n]-subbimodules of Vn inside V2n. But there is an as F [S'n]-bimodule isomorphism f : Vn ^ V^ given by x^^ ■■■ xa(n) ^ xa(1)Xn+1 ■■■ xa(n)x2n. In particular Vn has the same simple Fstructure as Vn and can be studied with the same Young representation theory, although now we only utilize the left action of permutations. Thus, for any PI-algebra A, the spaces
Id(A) n vn C vn
are F[S^-subbimodules of V'n-
Remark 3.17. Again, any tableau T of 2n boxes gives rise to an element
aT = V I E sgn(^)qp I e F[^2n],
gecr., peR-r.
where Ctx (resp. Rtx ) denotes the set of column (resp. row) permutations of the tableau T\.
Thus, F[Sn]a,T (if nonzero) is an F[5n]-submodule, which we call J,\. If J\ contains an element corresponding to a nontrivial PI of A, ar itself must correspond to a PI of A.
We let I\ denote the minimal F[5n]-bisubmodule of F[S2n] containing J\.
Lemma 3.18. Let A be an F-algebra, and let A be a partition of n. If dim J\ > c2n(A) mvd J\ is a simple F[5n]-module, then I\ C Id(A) n V^.
PROOF. Same as Lemma 3.7, noting that I\ is a sum of F[5n]-submodules J\a each isomorphic to J\. Thus, taking such J, one has
a contradiction. Therefore each J C Id(A) n V^, implying I\ C Id(A) n Vn. □
Note that when char(F) = p > 0 the lemma might fail unless J\ is simple. James and Mathas [13, Main Theorem] determined when J\ is simple for p = 2.
One such example is when A is the staircase, which we define to be the Young tableau Tu whose u rows have length u,u — 1,..., 1. This gave rise to the James-Mathas conjecture [21] of conditions on A characterizing when J\ is simple in char act eristic p > 2, which was solved by Favers [9].
4. Kemer's Capelli Theorem for all characteristics
In this section we give a proof of Kemer's "Capelli Theorem" over a field of any characteristic. In fact in characteristic p Kemer proved a stronger result, even for non-affine algebras.
Theorem 4.1. [17] Any PI algebra over a field F of characteristic p > 0 satisfies a Capelli identity Capn for large enough n.
This fails in characteristic 0, since the Grassmann algebra does not satisfy a Capelli identity. The proof of Theorem 4.1 given in [17] is quite complicated; an elementary proof using the "identity of algebraicitv" is given in [7, §2.5.1], but still requires some computations. In the spirit of providing a full exposition which is as direct as possible, we treat only the affine case via representation theory, in which case characteristic p > 0 works analogously to characteristic 0. This produces a much better estimate of the degree of the sparse identity, which we obtain in Theorem 4.4.
In view of Theorem 3.3, it suffices to show that any affine PI algebra satisfies a sparse identity. Although we cannot achieve this through branching, the ideas of the previous section still apply, using [9].
4.1. Simple Specht modules in characteristic p > 0
In order to obtain a p-version of Proposition 3.13 in characteristic p > 2, first we need to find a class of partitions satisfying Faver's criterion.
For a positive integer m, define vp to be the p-adic valuation, i.e., vp(m) is the largest power of p dividing m. Also, temporarily write h(i;j) for hx where x is the box in the i,j position. The
James-Mathas conjecture for p = 2, proved in [9], is that J\ is simple if and only if there do not exist i,j,i',j' for which vp(h^.j)) > 0 wit h vp(h^.j)),vp(h^i.j)),vp(h^.j')) all distinct. Of course this is automatic when each hook number is prime to p, since then every vp(h^.j)) = 0.
Example 4. A wide staircase is a Young tableau Tu whose u rows have all have lengths different multiples of p — I, the first row of length (p — 1)u, the second of length (p — 1)(u — 1), and so forth until the last of length p — 1. The number of boxes is
n = ¿(p — 1)j = (p — 1m u + j.
3=1 V '
When p = 2, the wide staircase just becomes the staircase described earlier. In analogy to Example 3, the dimension of the "wide staircase" Tu can be estimated as follows: We write j = (p — 1)/ + j" for 1 < j" < p — 1. The hook of a box in the (i,j) position has length (u + 1 — i)(p — 1) + 1 — j, and depth u + 1 — j' — i, so the hook number is
(u + 1 — i)(p — 1) + 1 — j + u — j' — i = (u + 1 — i)p — j — j' = (u + 1 + j' — i)p — j'',
which is prime to p. Thus each wide staircase satisfies a stronger condition than Faver's criterion.
The dimension can again be calculated by means of the hook formula. The first p — 1 boxes in the first row have hook numbers
pu — 1,pu — 2,... ,pu — (p — 1),
whose sum is (p — 1)pu — = (2) (2u — 1).
The next p — 1 boxes in the first row have hook numbers
p(u — 1) — 1,p(u — 1) — 2,... ,p(u — 1) — (p — 1),
whose sum is (p — 1)p(u — 1) — (p) = (2)(2u — 3).
2 = (2)
Thus the sum of the hook numbers in the first row is
2
Summing over all rows yields
Q ((2u — 1) + (2u — 3) + ... + 1)= (2) v2.
E ^ = (P) Y, k2 = U(U + 1)(2" + 1) = (P) (2^ + 1)^
(6 \n ( 1 \ n
—,-6T-r ) ( H < fd (where e = 2.718281828 ...).
p(p — 1)(2u + 1)/ V ej J y J
Lemma 4.2. For any integer let ^ be the wide staircase Tu of «rows. Let n = (p — 1)(U)- Then
j)(p — 1)(2u + 1)7 V1
In particular, if a < (2U++i)e, < f^-
proof. We imitate the proof of Lemma 3.11. Since the geometric mean is bounded by the arithmetic mean,
1!/n s
< 1 y^ ^ ^ fp\ (2u + 1)n _ p(p — 1)(2u + 1)n
(\ 1/n
n M < k e i- < (2
xÇ/ / xÇ/ v '
n^ \2J 3 6
xÇ/d x 7
in view of Example 4, together with (n/e)n < n\, implies that
( 6n y /1 \n = / „ r / 6 \n n\ =
\p(p — 1)(2u + 1)^ Ve) v J \p(p — 1)(2u + 1)^ nxGdhx J '
□
Lemma 4.3. Let A be a PI algebra over a field of characteristic p, that satisfies an identity of degree d. Choose a natural number u such that, for n = (p — 1) ("+1),
6n 1 > (d — 1)2.
p(p — 1)(2« + 1) e
Let A h n be any partition of n corresponding to the "wide staircase" Tu. Then the elements of the corresponding F[5n]-bimodule I\ C V'n are sparse identities of A.
Proof. Bv Remark 3.12,
n
^ < -1)4" <vil,—1X2,+1) -i)<SA-□
4.1.1. Existence of Capelli identities
We are ready for a version of Proposition 3.15.
Theorem 4.4. [17] Any PI- algebra A over a field F of characteristic p > 0 satisfies a Capelli identity. Explicitly:
(a) Suppose the F-algebra A satisfies an identity of degree d. Then A satisfies a sparse identity of degree d' = (p — 1)p(M+1), where p('2^+1) ^ (d — 1)2e.
(b) Suppose A = F{a1,... ,ar}, and A satisfies an identity of degree d and take d' as in (a).
Let n = rd + d' « r4e (d-1 .The n A satisfies the Capelli i dentitv Capra.
□
For example, since 2+1 ^ 1, we could take u ^ 2p (d-1) . This concludes the proof of Theorem 4.1 in the affine case.
5. Results and proofs over Noetherian base rings
We turn to the case where C is a commutative Noetherian ring. In general, we say a C-algebra is PI if it satisfies a polynomial identity having at least one coefficient equal to 1. Let us indicate the modifications that need to be made in order to obtain proofs of Theorems 1.6 and 1.7.
The method of proof of Theorem 1(2) (for the case in which the base ring C is a field) was to verify the "weak Nullstellensatz", and a similar proof works for A commutative when C is Jacobson, cf. [26, Proposition 4.4.1]. Thus we have Theorems 1.6 and 1.7 in the commutative case, which provide the base for our induction to prove Theorem 1.3. The argument is carried out using Zubrilin's methods (which were given over an arbitrary commutative base ring.)
It remains to find a way of proving Kemer's Capelli Theorem over arbitrary Noetherian base rings. One could do this directly using Young diagrams, but there also is a ring-theoretic reduction. The following observations about Capelli identities are useful.
Lemma 5.1. (i) Suppose n = n1n2 ■ ■ ■ nt. If A satisfies the identity Caprai x ■ ■ ■ x Caprat, then A satisfies the Capelli identity Capra.
(ii) If I < A and A/I satisfies CapTO for m odd, with Ik = 0, then A satisfies Capfcm .
(iii) If I < A and A/I satisfi es Capm wit h Ik = 0, then A satisfi es Capk(TO+1) .
PROOF, (i) Viewing the symmetric group Sni x ■ ■ ■ x SUm ^ Sn, we partition Sn into orbits under the subgroup Sni x ■ ■ ■ x Snm and match the permutations in Capn.
(ii) This time we note that any interchange of two odd-order sets of letters has negative sign, so we partition Skm into k parts each with m letters.
(iii) Any algebra satisfying CapTO for m even, also satisfies CapTO+1, and m + 1 is odd. □
Thus, it suffices to prove that A satisfies a product of Capelli identities.
Theorem 5.2. Any affine PI algebra over a commutative Noetherian base ring C satisfies some Capelli identity.
PROOF. By Noetherian induction, we may assume that the theorem holds for every affine Pi-algebra over a proper homomorphic image of C.
First we do do the case where C is an integral domain, and A = C{a1,..., a¿} satisfies some multilinear PI /. It is enough to assume that A is the relatively free algebra C{x1,..., xn}/I (where I is the T-ideal generated by /). Let F be the field of fractions of C. Then Ap := A ®c F is also a PI-algebra, and thus, by Theorem 4.1 satisfies some Capelli identity f1 = Capn. Thus the image fa of f1 in A becomes 0 when we tensor by F, which means that there is some s £ C for which sf1 = 0. Letting I' denote the T-ideal of A generated by the image of f1, we see that si' = 0. If s = 1 then we are done, so we may assume that s £ C is not invertible. Then A/sA is an affine PI-algebra over the proper homomorphic image C/sC of C, and by Noetherian induction, satisfies some Capelli identity CapTO, so A/(sA n I') satisfies Capmax{mn}. But sA n I' is nilpotent modulo sAI' = Asl' = 0, implying by Lemma 5.1 that A satisfies some Capelli identity.
For the general case, the nilpotent radical N of C is a finite intersection P1 n ■ ■ ■ n Pt of prime ideals. By the previous paragraph, A/Pj A, being an affine PI-algebra over the integral domain C/Pj, satisfies a suitable Capelli identity Capnj, for 1 < j < t, so A/ n (PjA) satisfies Capn, where n = max{m,... ,nt}. But n(PjA) is nilpotent modulo NA, so, by Lemma 5.1, A/NA satisfies a suitable Capelli identity Capn. Furthermore, Nm = 0 for some m, implying again by Lemma 5.1 that A satisfies Cmn. □
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. S.A. Amitsur, A generalization of Hilbert Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957) 649-656.
2. S.A. Amitsur, A note on P.I. rings, Israel J. Math. 10 (1971) 210-211.
3. S.A. Amitsur and C. Procesi, Jacobson rings and Hilbert algebras with polynomial identities, Ann. Mat. Рига Appl. 71 (1966) 67-72.
4. S.A. Amitsur and A. Regev: P.I. algebras and their cocharacters, J. of Algebra 78 (1982) 248-254.
5. S.A. Amitsur and L. Small, Affine algebras with polynomial identities, Supplemento ai Rendiconti del Circolo Matematico di Palermo 31 (1993).
6. A. Belov, L. Bokut, L.H. Rowen, and J.T. Yu, The Jacobian Conjecture, together with Specht and Burnside-type problems, Proc. Groups of Automorphisms in Birational and Affine Geometry, Springer, editors M.Zaidenberg, M. Rich, and M. Reizakis, to appear.
7. A.K. Belov and L.H. Rowen, Computational aspects of Polynomial Identities, A. K. Peters (2005).
8. A. Braun, The nilpotencv of the radical in a finitely generated Pi-ring, J. Algebra 89 (1984), 375-396.
9. M. Favers, Irreducible Specht modules for Hecke algebras of type A, Advances in Math. 193 (2005), 438-452.
10. M. Favers, S. Lvle, S. Martin, p-restriction of partitions and homomorphisms between Specht modules, J. Algebra 306 (2006), 175-190.
11. N. Jacobson, Basic Algebra II, second edition, Freeman and company (1989).
12. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math, Vol. 682, Springer Wring. New York, NY, (1978).
2
Math. Soc. 31 (1999), 457-62.
14. G. D. James and A. Kerber, The Representation Theory of the Symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Weslev, Reading, MA, (1981).
15. A.R Kemer, Capelli identities and nilpotence of the radical of a finitely generated Pi-algebra, Dokl. Akad. Nauk SSSR 255 (1980), 793-797 (Russian). English translation: Soviet Math. Dokl. 22 (1980), 750-753.
16. A.R. Kemer, Ideals of identities of associative algebras, Amer. Math. Soc. Translations of monographs 87 (1991).
17. A.R. Kemer, Multilinear identities of the algebras over a field of characteristic p, Internat. J. Algebra Comput. 5 no. 2, (1995), 189-197.
18. J. Lewin, A matrix representation for associative algebras I and II, Trans. Amer. Math. Soc. 188(2), 293-317 (1974).
19. L'vov, Unpublished (Russian).
20. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press, Oxford, (1995).
21. A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series 15, American Mathematical Society, Providence, RI (1999).
22. Yu.P. Razmvslov, The Jacobson radical in Pi-algebras (Russian), Algebra i Logika 13 (1974), 337-360. English translation: Algebra and Logika 13 (1974), 192-204.
23. A. Regev, Existence of identities in A 0 B, Israel J. Math. 11 (1972), 131-152.
24. A. Regev, The representations of Sn and explicit identities for P.I. algebras, J. Algebra 51 (1978), 25-40.
25. Richard Resco, Lance W. Small and J. T. Stafford, Krull and Global Dimensions of Semiprime Noetherian PI-Rings Transactions of the American Mathematical Society 274, No. 1 (1982), 285-295
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).
27. L.H. Rowen, Ring Theory, Vol. II. Pure and Applied Mathematics, 128. Academic Press, Inc., Boston, MA, (1988).
28. L.H. Rowen, Graduate algebra: Commutative View, AMS Graduate Studies in Mathematics 73, 2006.
29. L.H. Rowen, Graduate algebra: Noncommutative View, AMS Graduate Studies in Mathematics 91, 2008.
30. B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edition, Graduate Texts in Mathematics 203, Springer-Verlag (2000).
31. A.I. Shirshov, On certain non associative nil rings and algebraic algebras (Russian), Mat. Sb. 41 (1957), 381-394.
32. A.I. Shirshov, On rings with identity relations (Russian), Mat. Sb. 43 (1957), 277-283.
33. Small, L.W., An example in PI rings, J. Algebra 17 (1971), 434-436.
34. K.A. Zubrilin, Algebras satisfying Capelli identities, Sbornic Math. 186 no. 3 (1995) 359-370. REFERENCES
1. S.A. Amitsur, A generalization of Hilbert Nullstellensatz, Proc. Amer. Math. Soc. 8 (1957) 649-656.
2. S.A. Amitsur, A note on P.I. rings, Israel J. Math. 10 (1971) 210-211.
3. S.A. Amitsur and C. Procesi, Jacobson rings and Hilbert algebras with polynomial identities, Ann. Mat. Pura Appl. 71 (1966) 67-72.
4. S.A. Amitsur and A. Regev: P.I. algebras and their cocharacters, J. of Algebra 78 (1982) 248-254.
5. S.A. Amitsur and L. Small, Affine algebras with polynomial identities, Supplemento ai Rendiconti del Circolo Matematico di Palermo 31 (1993).
6. A. Belov, L. Bokut, L.H. Rowen, and J.T. Yu, The Jacobian Conjecture, together with Specht and Burnside-type problems, Proc. Groups of Automorphisms in Birational and Affine Geometry, Springer, editors M.Zaidenberg, M. Rich, and M. Reizakis, to appear.
7. A.K. Belov and L.H. Rowen, Computational aspects of Polynomial Identities, A. K. Peters (2005).
8. A. Braun, The nilpotencv of the radical in a finitely generated Pi-ring, J. Algebra 89 (1984), 375-396.
9. M. Favers, Irreducible Specht modules for Hecke algebras of type A, Advances in Math. 193 (2005), 438-452.
10. M. Favers, S. Lvle, S. Martin, p-restriction of partitions and homomorphisms between Specht modules, J. Algebra 306 (2006), 175-190.
11. N. Jacobson, Basic Algebra II, second edition, Freeman and company (1989).
12. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math, Vol. 682, Springer-Verlag, New York, NY, (1978).
2
Math. Soc. 31 (1999), 457-62.
14. G. D. James and A. Kerber, The Representation Theory of the Symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley, Reading, MA, (1981).
15. A.R Kemer, Capelli identities and nilpotence of the radical of a finitely generated Pi-algebra, Dokl. Akad. Nauk SSSR 255 (1980), 793-797 (Russian). English translation: Soviet Math. Dokl. 22 (1980), 750-753.
16. A.R. Kemer, Ideals of identities of associative algebras, Amer. Math. Soc. Translations of monographs 87 (1991).
17. A.R. Kemer, Multilinear identities of the algebras over a field of characteristic p, Internat. J. Algebra Comput. 5 no. 2, (1995), 189-197.
18. J. Lewin, A matrix representation for associative algebras I and II, Trans. Amer. Math. Soc. 188(2), 293-317 (1974).
19. L'vov, Unpublished (Russian).
20. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press, Oxford, (1995).
21. A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series 15, American Mathematical Society, Providence, RI (1999).
22. Yu.P. Razmvslov, The Jacobson radical in Pi-algebras (Russian), Algebra i Logika 13 (1974), 337-360. English translation: Algebra and Logika 13 (1974), 192-204.
23. A. Regev, Existence of identities in A 0 B, Israel J. Math. 11 (1972), 131-152.
24. A. Regev, The representations of Sn and explicit identities for P.I. algebras, J. Algebra 51 (1978), 25-40.
25. Richard Resco, Lance W. Small and J. T. Stafford, Krull and Global Dimensions of Semiprime Noetherian PFRings Transactions of the American Mathematical Society 274, No. 1 (1982), 285-295
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).
27. L.H. Rowen, Ring Theory, Vol. II. Pure and Applied Mathematics, 128. Academic Press, Inc., Boston, MA, (1988).
28. L.H. Rowen, Graduate algebra: Commutative View, AMS Graduate Studies in Mathematics 73, 2006.
29. L.H. Rowen, Graduate algebra: Noncommutative View, AMS Graduate Studies in Mathematics 91, 2008.
30. B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edition, Graduate Texts in Mathematics 203, Springer-Verlag (2000).
31. A.I. Shirshov, On certain non associative nil rings and algebraic algebras (Russian), Mat. Sb. 41 (1957), 381-394.
32. A.I. Shirshov, On rings with identity relations (Russian), Mat. Sb. 43 (1957), 277-283.
33. Small, L.W., An example in PI rings, J. Algebra 17 (1971), 434-436.
34. K.A. Zubrilin, Algebras satisfying Capelli identities, Sbornic Math. 186 no. 3 (1995) 359-370.
Получено 16.05.2020 г. Принято в печать 22.10.2020 г.
