1 THE GOLOD SHAFAREVICH COUNTER–EXAMPLE WITHOUT HILBERT SERIES Alon Regev Department of Mathematicsal Sciences Northern Illinois University DeKalb, Illinois 60115 9 0 E-Mail: [email protected] 0 2 n Amitai Regev a J 1 Department of Theoretical Mathematics 1 The Weizmann Institute of Science ] Rehovot 76100, Israel A E-Mail: [email protected] R . h t a Abstract: LetF beanarbitraryfield. TheGolod–Shafarevichexampleofafinitelygenerated m nil F-algebra which is infinite dimensional – is revisited. Here we offer a rather elementary [ treatment of that example, in which induction replaces Hilbert series techniques. This note 1 also contains a detailed exposition of the construction of that example. v 6 2 4 1 . 1 0 9 0 : v i X r a 2 1 Introduction Throughout this paper F is a field; d ≥ 2 a natural number, and T = F{x ,...,x } is the 1 d associative non-commutative algebra of the polynomials in d variables. Let T ⊂ T denote ≥1 the polynomials with no constant term. If I ⊆ T is a two sided ideal then the quotient ≥1 algebra T /I is finitey generated. ≥1 Theorem 1.1. [3] There exists a homogeneous two sided ideal I ⊂ T , such that the finitely ≥1 generated algebra A = T /I is both nil and infinite dimensional. ≥1 Definition 1.2. Let H = {f ,f ,...} be a sequence of homogeneouspolynomials. We assume 1 2 that all degf ≥ 2. Let r be the number of elements of H of degree ℓ. Let I = I be the two j ℓ H sided ideal generated in T by H. Call H a “G.S. sequence” if its ideal I and numbers r ≥1 H ℓ satisfy 1. For every polynomial g ∈ T there exists n such that gn ∈ I. ≥1 2. For some ε > 0 satisfying d−2ε > 1, r ≤ ε2(d−2ε)ℓ−2 for all ℓ ≥ 2. ℓ Theorem 1.1 is a corollary of the following two theorems. Theorem 1.3. Let H ⊂ T be a G.S. sequence, then the quotient algebra T /I is both ≥1 ≥1 H nil and infinite dimensional. Theorem 1.4. There exist G.S. sequences. The first step in proving Theorem 1.3 is Theorem 2.1 [1] of Section 2, which applies to any homogeneous ideal I ⊆ T and which establishes the basic inequality (2) below. This ≥ inequality, together with conditions 2 of Definition 1.2, imply Proposition 3.1, which states that A = T /I is infinite dimensional. Note that condition 1 of Definition 1.2 implies that ≥1 this algebra A is nil. In Section 4 we prove the existence of G.S. sequences, see Theorem 4.5, thus proving Theorem 1.4 and completing the proof of Theorem 1.1. All the proofs of Proposition 3.1 known to us, apply the techniques of Hilbert series, see [1], [2], [3], [4], [5] and [6]. In Section 3 below we present a new and rather elementary proof of that proposition, a proof where Hilbert series considerations are replaced by an induction argument. The presentation here of the Golod-Shafarevich theorem is completely elementary. 2 The basic inequality Let T ⊂ T denote the homogeneous polynomials of total degree n, so dim T = dn and n n ∞ T = T . n Mn=0 We denote ∞ T = T . ≥k n Mn=k 3 Thus T ⊆ T are the polynomials with no constant term. ≥1 Let H = {f ,f ,...} be a sequence of homogeneous polynomials. We assume that all 1 2 degf ≥ 2, so H ⊂ T . Let r be the number of elements of H of degree n. Thus j ≥2 n ∞ H = H , where H = {f | degf = n} and |H | = r . n n j j n n n[=2 Let R = span {f ,f ,...}, so F 1 2 ∞ ∞ R = R = R , where R = spanH . n n n n Mn=0 Mn=2 Note that dim R ≤ r . Since all degf ≥ 2, we have r = r = 0 and R ⊆ T ⊆ T . n n j 0 1 ≥2 ≥1 Since T = TT , we have R ⊆ TT . Let I = hf ,f ,...i be the two–sided ideal generated ≥1 1 1 1 2 in T by the sequence H, so that I = TRT ⊆ T ⊆ T . ≥2 ≥1 Let A = T /I be the corresponding algebra. Since I is generated by homogeneous polyno- ≥1 mials of degrees ≥ 2, ∞ ∞ I = I = I n n Mn=0 Mn=2 where I = I ∩T . Let B ⊆ T be a complement vector space of I : n n n n n T = I ⊕B , (1) n n n and denote b = dim B . Since I = I = 0, hence B = T = F and B = T , so n n 0 1 0 0 1 1 b = dim F = 1 and b = dim T = d. Denote B = B , so 0 F 1 F 1 n≥0 n L T = I ⊕B. With these notations we prove Theorem 2.1. [1] Let d ≥ 2, T = F{x ,...,x } = T , I ⊆ T a homogeneous two-sided 1 d n n ideal, I = ⊕nIn. Write Tn = In⊕Bn and let bn = dimLBn. Recall that I = hf1,f2,...i where the f ’s are homogeneous of degrees ≥ 2, and let r be the number of f ’s of degree ℓ (thus j ℓ j r = r = 0). Then for all n ≥ 2 0 1 n−2 b ≥ db − r b (2) n n−1 n−j j Xj=0 Proof. (Following [6]). Recall that R = span {f ,f ,...} and that T = I ⊕ B. We first F 1 2 show that I = IT +BR. (3) 1 Note that T = T ⊕F = TT ⊕F, hence ≥1 1 I = TRT = TR(TT ⊕F) = (TRT)T +TR = IT +TR. (4) 1 1 1 4 Now T = I ⊕B, R ⊆ TT and IT = I, hence 1 TR = (I ⊕B)R = IR+BR ⊆ ITT +BR = IT +BR. (5) 1 1 Since IT +IT = IT , by (4) and (5) 1 1 1 I = IT +TR ⊆ IT +(IT +BR) ⊆ IT +BR. (6) 1 1 1 1 Since I ⊇ IT ,BR, we conclude that I = IT +BR, proving (3). 1 1 Taking the n–th homogeneous component of (3) yields n n I = I T + B R = I T + B R . (7) n n−1 1 n−k k n−1 1 n−k k Xk=0 Xk=2 Note that dim (I T ) ≤ (dim I )(dim T ) = (dim I )d and dim (B R ) ≤ b r . n−1 1 n−1 1 n−1 n−k k n−k k Taking dimensions on both sides of (7) we obtain n dim I ≤ (dim I )d+ b r . (8) n n−1 n−k k Xk=2 Substituting j = n−k in the above sum gives n−2 dim I ≤ (dim I )d+ b r (9) n n−1 j n−j Xj=0 By (1), dn = dim T = dim I +dim B = dim I +b , so dim I = dn −b and similarly n n n n n n n dim I = dn−1 −b . Substituting this into (9) yields the desired inequality eq1. n−1 n−1 = 3 Infinite dimensionality of the algebra A T /I ≥1 At the heart of the Golod-Shafarevich construction is the following proposition. Proposition 3.1. [1] Let b , r be the sequences in Theorem 2.1, hence in particular satis- n ℓ fying (2). Let ε > 0 such that d−2ε > 1. If r ≤ ε2(d−2ε)ℓ−2 for all ℓ ≥ 2, then all b ≥ 1 ℓ n (In fact, it follows that the b ’s grow exponentially), and therefore the algebra A = T /I is n ≥1 infinite-dimensional. The proofs of Proposition 3.1 known to us, all apply the techniques of Hilbert series ( see [1], [2], [3], [4], [5] and [6]). In this note we deduce Proposition 3.1 from Proposition 3.2. We then give Proposition 3.2 a rather elementary proof, in which induction replaces Hilbert series techniques Proposition 3.2. Let b , r be the sequences in Theorem 2.1, hence in particular satisfying n ℓ (2). Let v > 0 be a real number satisfying the following condition: There exist real numbers c,u > 0 satisfying (a) For all n ≥ 0, r ≤ cun, and n+2 (b) vd−c ≥ v (so in particular vd > c) v+u Then, for all n ≥ 0, b ≥ (d−v)n. (10) n 5 Before proving Proposition 3.2 we show that it implies Proposition 3.1. We want to prove that under the assumptions of Proposition 3.1, for all n, b ≥ 1. By the assumptions of n Proposition 3.1, r ≤ ε2(d−2ε)k−2 for all k ≥ 2. Choose v = ε, c = ε2 and u = d−2ε. Then k r ≤ cun for all n ≥ 0, and n+2 vd−c ε(d−ε) = = ε = v . v +u d−ε Thus v satisfies the hypothesis of Proposition 3.2, so b ≥ (d−v)n, and (d−v)n ≥ 1 since n d−v = d−ε > d−2ε > 1. The proof of Proposition 3.2. Proof. We first prove by induction that for all n ≥ 0, n vb ≥ cun−jb . (11) n+1 j Xj=0 n = 0: Check that vb ≥ cb namely, that vd ≥ c, which is given by assumption (b). 1 0 The inductive step: Assume (11) holds for some n and show it holds for n+1, namely show n+1 vb ≥ cun+1−jb . n+2 j Xj=0 By the induction hypothesis (which is (11)), combined with assumption (b), n vd−c ·b ≥ vb ≥ cun−jb . n+1 n+1 j v +u Xj=0 By (a) all cun−j ≥ r , hence n+2−j n n (vd−c)b ≥ (vcun−j +ucun−j)b ≥ (vr +ucun−j)b . n+1 j n+2−j j Xj=0 Xj=0 This implies that n n vdb ≥ v r b +u cun−jb +cb , n+1 n+2−j j j n+1 Xj=0 Xj=0 so n n vdb −v r b ≥ u cun−jb +cb . n+1 n+2−j j j n+1 Xj=0 Xj=0 Combined with (2) (with n+2 replacing n, and multiplying by v) we have n n n+1 vb ≥ vdb −v r b ≥ u cun−jb +cb = cun+1−jb , n+2 n+1 n+2−j j j n+1 j Xj=0 Xj=0 Xj=0 6 that is, n+1 vb ≥ cun+1−jb . n+2 j Xj=0 This completes the inductive step and proves (11) for all n ≥ 0. Note that by (11) and by assumption (a), n n vb ≥ c·un−j ·b ≥ r b , n+1 j n+2−j j Xj=0 Xj=0 and therefore n vb ≥ r b . (12) n+1 n+2−j j Xj=0 We now show that (2) and (12) together imply that for all n, b ≥ (d−v)b , (13) n+2 n+1 which implies (10). By (12) n db −(d−v)b = vb ≥ r b , n+1 n+1 n+1 n+2−j j Xj=0 or equivalently n db − r b ≥ (d−v)b . n+1 n+2−j j n+1 Xj=0 Together with (2) this implies that n b ≥ db − r b ≥ (d−v)b . n+2 n+1 n+2−j j n+1 Xj=0 This proves (13) and completes the proof of Proposition 3.2. 4 The construction of G.S. sequences Fix some ε > 0 satisfying d−2ε > 1. Following Definition 1.2 we now construct a sequence of homogeneous polynomials, of degrees ≥ 2, f ,f ,... ∈ T ⊂ T ⊂ T = F{x ,...,x }, 1 2 ≥2 ≥1 1 d having the properties: (i) For every g ∈ T there exist a power n such that gn ∈ I, where I = hf ,f ,...i is the ≥1 1 2 two–sided ideal generated by the f′s, and j (ii) Let r be the number of f′s of degree ℓ then r ≤ ε2(d−2ε)ℓ. ℓ j ℓ+2 Let A = T /I, then A is generated by the d elements x + I,...,x + I, hence is finitely ≥1 1 d generated. By (i) here it is nil, and by (ii) together with Proposition 3.1 it is infinite dimensional. 7 4.1 Preparatory remarks Let q and n be positive integers. Define I(q,n) = {(i ,...,i ) | 1 ≤ i ,...,i ≤ q}, and 1 n 1 n define J(q,n) ⊆ I(q,n) via J(q,n) = {(i ,...,i ) | 1 ≤ i ≤ i ≤ ··· ≤ i ≤ q}. 1 n 1 2 n Remark 4.1. Note that |I(q,n)| = qn. Also, |J(q,n)| = n+q−1 ; for example, the corre- q−1 spondence (cid:0) (cid:1) 1 ≤ j ≤ ··· ≤ j ≤ q ←→ 1 ≤ j < j +1 < j +2 < ··· < j +n−1 ≤ n+q −1 1 n 1 2 3 n gives a bijection between J(q,n) and the set of n-subsets of an n+q −1-set. Note that n+q −1 |J(q,n)| = ≤ (n+q −1)q−1, (14) (cid:18) q −1 (cid:19) and the right hand side is a polynomial in n (of degree q −1). Given π ∈ Sn and i = (i1,...,in) ∈ I(q,n), define π(i) = (iπ−1(1),...,iπ−1(n)), then σ(π(i)) = (σπ)(i), hence I(q,n) is the disjoint union of the corresponding orbits. Given i ∈ I(q,n), let O = orbit(i) = {π(i) | π ∈ S }, denote the orbit of i under the S action. Then i n n I(q,n) = O , j [ j∈J(q,n) a disjoint union. If j = (j ,...,j ) ∈ J(q,n) then j = (1,...,1,2,...,2,...) and we denote 1 n j = (1µ1,...,qµq), where k appears µ times in j. Then S ×···×S fixes j, and k µ1 µq n! |O | = . j µ !···µ ! 1 q For example, (1,1,1,2,2,2,2)= (13,24) ∈ J(2,7), and |O | = 7!/(3!4!) = 35. (13,24) 4.2 The order-symmetric polynomials Definition 4.2. Let j = (j ,...,j ) ∈ J(q,n) then define “the order-symmetric polynomial” 1 n s (y ,...,y ) = y ···y , a homogeneous polynomial of degree n in y ,...,y . j 1 q i1 in 1 q iX∈Oj Note that we can also write 1 s (y ,...,y ) = · y ···y . j 1 q µ !···µ ! jπ−1(1) jπ−1(n) 1 q πX∈Sn Recall that d ≥ 2. Given 0 < c ∈ N, denote q = d + d2 + ··· + dc, then q is the number of monomials of degree between 1 and c in the non-commuting variables x ,...,x . Let 1 d {M ,...,M } be the set of these monomials. Given 0 < n ∈ N, and j ∈ J(q,n), denote 1 q h (x) = h (x ,...,x ) = s (M ,...,M ). (15) j j 1 d j 1 q With these notations we prove 8 Lemma 4.3. Let 0 < c ∈ N, q = d+d2+···+dc, with {M ,...,M } the monomials of degrees 1 q between 1 and c. Let 0 < n ∈ N and let I ⊆ F{x ...,x } be a two-sided ideal containing 1 d h (x) for all j ∈ J(q,n), where h (x) are given by (15). Let g = g(x ,...,x ) ∈ T be a j j 1 d ≥1 polynomial of degree ≤ c. Then gn ∈ I. Proof. Since M ,...,M are all the non-constant monomials (in x ,...,x ) of degrees ≤ c, 1 q 1 d and since degg ≤ c, we can write q g = α M with α ∈ F. i i i Xi=1 Then gn = (α M )···(α M ) = (α ···α )(M ···M ). i1 i1 in in i1 in i1 in 1≤i1X,...,in≤q 1≤i1X,...,in≤q Since the α ’s commute, for any π ∈ S , α ···α = α ···α , and it follows that i n i1 in iπ−1(1) iπ−1(n) (α ···α )(M ···M ) = (α ···α )· M ···M = i1 in i1 in j1 jn i1 in 1≤i1X,...,in≤q j∈XJ(q,n) iX∈Oj = (α ···α )·s (M ,...,M ) = (α ···α )·h (x ,...,x ) j1 jn j 1 q j1 jn j 1 d X X j∈J(q,n) j∈J(q,n) by (15). Thus gn is a linear combination of the polynomials {h (x ,...,x ) | j ∈ J(q,n)}. j 1 d By assumption all these h (x) are in I, hence gn ∈ I. j Remark 4.4. We saw in Remark 4.1 that |J(q,n)| ≤ (n+q−1)q−1. The right hand side is a polynomial in n (of degree q −1), hence it grows slower than any exponential function in n, in particular, slower than ε2 ·αn, provided ε > 0 and α > 1. This implies the following: Let d ≥ 2,ε > 0 such that d−2ε > 1, then there exists n large enough such that |J(q,n)| ≤ ε2 ·(d−2ε)n−2. (16) 4.3 The construction We now prove Theorem 4.5. G.S. sequences exist. Namely: let d ≥ 2, ε > o such that d−2ε > 1. Then there exist a sequence f ,f ... ∈ T of homogeneous polynomials of degrees ≥ 2 satisfying 1 2 conditions 1 and 2 of Definition 1.2. Proof. The construction is inductive, starting with the empty sequence. The induction assumption is that by the k-th step we have chosen integers c ≤ c′ and a sequence of k k homogenouspolynomialsf ,...,f ofdegreesbetween2andc′. Weassumethatf ,...,f 1 mk k 1 mk satisfy the following condition(c ): k 1. For any ℓ ≤ c′, the number r of the elements f of degree ℓ satisfies r ≤ ε2·(d−2ε)ℓ. k ℓ i ℓ+2 9 2. For any g ∈ T of degree ≤ c there exists some power n such that gn ∈ I , where ≥1 k k I = hf ,...,f i. k 1 mk The inductive step. In the next step (step k + 1) we choose integers c ≤ c′ satisfying c′ < c , and k+1 k+1 k k+1 construct another block of polynomials f ,...,f having degrees c′ < degf < c′ , mk+1 mk+1 k j k+1 thus obtaining the sequence f ,...,f ,f ,...,f . 1 mk mk+1 mk+1 The construction starts with choosing any ck+1 > c′k. Let q = qk+1 = d+ d2 + ···+ dck+1, then choose n large enough such that c′ < n and |J(q,n)| < ε2(d−2ε)n−2. (17) k By (16) such n exists. We choose c′ = n·c . Denote m = |J(q,n)|+ m , then let k+1 k+1 k+1 k f ,f ,...,f be the |J(q,n)| polynomials mk+1 mk+2 mk+1 h (x ,...,x ) = s (M ,...,M ), j ∈ J(q,n), j 1 d j 1 q given by (15). Note that for j ∈ J(q,n), degs (y ,...,y ) = n and all 1 ≤ degM ≤ c , j 1 q i k+1 hence c′ < n ≤ degs (M ,...,M ) ≤ n·c = c′ . (18) k j 1 q k+1 k+1 Form now the new sequence from the above two blocks of f′s: j f ,...,f ,f ,...,f . 1 mk mk+1 mk+1 We show that this last sequence satisfies condition(c ). We also show that r calculated k+1 ℓ in step k for ℓ ≤ c′, remains unchanged when calculated in step k +1. Hence the numbers k r are well defined for the resulting infinite sequence f ,f ,... This will complete the proof ℓ 1 2 of the theorem. Note that hf ,f ,...,f i ⊆ hf ,...,f ,f ,...,f i = I . By Lemma mk+1 mk+2 mk+1 1 mk mk+1 mk+1 k+1 4.3, for any polynomial g ∈ T of degree ≤ c , gn ∈ hf ,f ,...,f i, hence ≥1 k+1 mk+1 mk+2 mk+1 gn ∈ I , which is part 2 of condition(c ). k+1 k+1 By condition(c ) the degrees of the polynomials in the block f ,...,f are ≤ c′. By k 1 mk k (17) c′ < n, and by (18) the degrees of f ,...,f are between n and c′ = c n. k mk+1 mk+1 k+1 k+1 Recall that for for each degree ℓ, r is the number of elements of degree ℓ in the sequence ℓ f ,...,f ,f ,...,f . Thus for ℓ ≤ c′ there is no contribution to r from the second 1 mk mk+1 mk+1 k ℓ block. Therefore these r ’s remain unchanged when the process of constructing the f ’s ℓ j continues. If ℓ ≤ c′, it is given (by the induction assumption) that r ≤ ε2 ·(d−2ε)ℓ−2. k ℓ Note that similarly, for ℓ > c there is no contribution to r from the first block, so r is k ℓ ℓ calculated on the second block only. If ℓ > c′, we may assume that ℓ ≥ n (since there are no f ’s with degree between c′ +1 and k j k n), then by (17) r ≤ |J(q,n)| < ε2 ·(d−2ε)n−2 ≤ ε2 ·(d−2ε)ℓ−2. ℓ This completes the inductive step. 10 We summarize: We are given d ≥ 2, ε > 0 such that d−2ε > 1. We have constructed the infinite sequence {f ,f ,...} of homogeneous polynomials in x ,,...,x of degrees ≥ 2. Let I = hf ,f ,...i ⊆ 1 2 1 d 1 2 F{x ,...,x } denote the ideal generated by the f ’s. Let r denote the number of f ’s of 1 d j ℓ j degree ℓ. These polynomials satisfy that r ≤ ε2 · (d − 2ε)ℓ−2 for all ℓ ≥ 2. Then, by ℓ Proposition 3.1 the algebra T /I is infinite dimensional. Obviously, it is finitely generated. ≥1 But by construction, for every g ∈ T there exist n such that gn ∈ I, hence T /I is nil. ≥1 ≥1 References [1] E. S. Golod, On nil-algebrasand finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–276; English transl., Amer. Math. Soc. Transl. (2) 48 (1965), 103–106. MR 28 #5082. [2] E. S.Golod, Some problems ofBurnside type, Proc.I.C.M., Moscow, 1966,pp.284–289. ˇ [3] E. S. Golod and I. R. Safareviˇc, On class field towers, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272; English transl., Amer. Math. Soc. Transl. (2) 48 (1965), 91–102. MR 28 #5056. [4] I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, (1969). [5] L. 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