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On group rings of linear groups
Journal of Pure and Applied Algebra 221 (2017) 25–35
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
On group rings of linear groups A.I. Lichtman Department of Mathematics, University of Wisconsin-Parkside, Kenosha, WI 53141, United States
a r t i c l e
i n f o
Article history: Received 1 December 2015 Received in revised form 12 May 2016 Available online 31 May 2016 Communicated by S. Donkin MSC: 16K; 12E15
a b s t r a c t Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number N = N (H). We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field. © 2016 Elsevier B.V. All rights reserved.
1. Introduction 1.1.
The main result of this paper is the following theorem.
Theorem 1. Let H be a finitely generated subgroup of GLn (F ), char(F ) = 0, and let KH be the group ring of H over a field K. i) If char(K) = 0 then there exists a torsion free normal subgroup G of finite index in H and an isomorphic embedding of KH into a semisimple Artinian ring S such that the group ring KG generates an H-invariant division subring Δ ⊆ S and S is isomorphic to a suitable cross product S∼ = Δ ∗ (H/G)
(1.1)
where the isomorphism (1.1) extends the isomorphism KH ∼ = KG ∗ (H/G). ii) If char(K) is finite and greater than N , a number depending on H, then statement i) remains true for KH. The additional information about the normal subgroup G and its group ring is contained in Theorems 2 and 3 below. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2016.05.015 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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Corollary 1. Let H be a finitely generated subgroup of GLn (F ), char(F ) = 0, and let KH be the group ring of H over a field K with char(K) = 0 or char(K) > N = N (H). Then there exists a division K-algebra Δ such that KH is isomorphic to a subring of a matrix ring Δm×m . The corollary follows from Theorem 1 immediately. In fact, S has a finite left dimension m = (H : G) over Δ, hence it has a faithful representation in Δm×m . It is worth remarking that the division ring Δ in the corollary will be commutative only if the group H has an abelian normal subgroup of finite index when char(K) = 0, or a p-abelian subgroup of finite index if char(K) = p. This follows from Corollaries 5.3.8–5.3.9 in Passman’s book [12]. 1.2. The proof of Theorem 1 is given in section 6. It is based on Theorems 2, 3 and 4. We prove Theorem 2 in section 3; we obtain there a torsion free normal subgroup G of finite index in H such that the group ring of G over the ring of p-adic integers Ω has an H-invariant filtration with an associated graded ring isomorphic to the polynomial ring over a prime field Zp . The proof of this theorem makes an essential use of Lazard’s p-valuations (see Lazard [6]). A short description of Lazard’s method is presented in subsection 2.2. The second main step in the proof of Theorem 1 is Theorem 4 (section 5) whose proof is based on the method developed by the author in [9] and [10]; this method is described also in Cohn [3], section 2.6. We apply Theorem 2 and Theorem 4 to construct the division ring Δ in Theorem 1 in the case when char(K) = 0. To prove statement ii) of Theorem 1 we need Theorem 3 and Corollary 3 (section 4) which states that the existence of a p-valuation in a group G implies that there exists a filtration and a valuation in the group ring Zp G over the prime field Zp with associated graded ring isomorphic to the polynomial ring over Zp . Our arguments show in fact that the conclusions of Theorems 1 and 2 remain true for an arbitrary, not necessarily finitely generated, subgroup H ⊆ GLn (T ) if T is a finitely generated commutative domain of characteristic zero. 2. Preliminaries 2.1. Lemma 1. Let T be a finitely generated commutative domain of characteristic zero and of transcendence degree n, and t1 , t2 , · · · , tn be a system of elements in T algebraically independent over Z. Then there exists a natural number N such that for every prime p > N and natural number c > N the powers of the ideal Ap,c generated by the elements p, t1 − c, t2 − c, · · · , tn − c define a p-adic valuation ρp,c of T such that i) ρp,c (p) = 1, and ii) if J(T ) = {r ∈ T |ρp,c (r) > 0}, then the quotient ring T /J(T ) is a finite field with characteristic p. Proof. Since T is finitely generated we can extend the system t1 , t2 , · · · , tn to a system of elements t1 , t2 , · · · , tn ; s1 , s2 , · · · , sm which generates T . Let φj [x] (1 ≤ j ≤ m) be the minimal polynomial of sj over Z[t1 , t2 , · · · , tn ]. We consider the field of fractions R of T and its subfield S = Q(t1 , t2 , · · · , tn ) which is the field of rational functions in variables t1 , t2 , · · · , tn over the field Q of rational numbers. We pick an element θ such that R = S(θ) and let ψ[x] be the minimal polynomial of θ; we can assume that all the coefficients of ψ[x] belong to the subring Z[t1 , t2 , · · · , tn ] as well as its discriminant d[t1 , t2 , · · · , tn ]. We pick now an arbitrary prime number p and a natural number c and consider the ideal Ap,c ⊆ Z[t1 , t2 , · · · , tn ] generated by the system of elements p, t1 − c, t2 − c, · · · , tn − c. The quotient ring
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∞ (Z[t1 , t2 , · · · , tn ])/Ap,c is the prime field Zp = Z/pZ and i=1 Aip,c = 0. We claim that the filtration Aip,c (i = 1, 2, · · · ) defines a valuation vp,c of Z[t1 , t2 , · · · , tn ]. In fact, the map ti −→ ti −c (i = 1, 2, · · · , n) defines an automorphism of Z[t1 , t2 , · · · , tn ] so the filtration i Ap,c (i = 1, 2, · · · ) defines a valuation if the powers of the ideal Ap,0 generated by the system of elements p, t1 , t2 , · · · , tn define a valuation vp,0 . But it is well known that the powers of the ideal Ap,0 define a p-adic valuation of Z[t1 , t2 , · · · , tn ]; this follows, for instance, from Lemma 1 in [2], VI. 10 if we apply this lemma and extend the p-adic valuation of Z to a valuation v of Z[t1 , t2 , · · · , tn ], and observe that this valuation v will be defined in fact by the filtration Aip,0 (i = 1, 2, · · · ), i.e., it will coincide with the valuation vp,0 . It is also possible to verify by a straightforward argument that the powers of the ideal Ap,0 define a p-adic valuation of Z[t1 , t2 , · · · , tn ]. Since the coefficients of the polynomials φj [x] (j = 1, 2, · · · , m) and ψ[x] are elements of Z[t1 , t2 , · · · , tn ] we can pick a natural number N such that if p > N, c > N then the ideal Ap,c contains no one of the coefficients of these polynomials or of the polynomial d[t1 , t2 , · · · , tn ]. We extend the valuation vp,c defined in the ring Z[t1 , t2 , · · · , tn ] to its field of fractions S and denote by Svp,c the corresponding valuation ring. Let Rvp,c be the integral closure of Svp,c in R, i.e. the subring formed by all the elements r ∈ R whose minimal polynomials are monic polynomials over Svp,c . We have θ ∈ Rvp,c because ψ[x] is a monic polynomial over Svp,c . It is important that the discriminant d[t1 , t2 , · · · , tn ] is a unit in Rvp,c and we can now apply Theorem 8 in section III.5 of Borevich and Shafarevich [1] to obtain that the valuation vp,c does not ramify in R. Let ρp,c be an extension of vp,c to R. Since all the coefficients of the polynomials φj [x] (j = 1, 2, · · · , m) are units in Sv,c we obtain that sj ∈ Rρp,c and then T ⊆ Rρp,c . We see that ρp,c (u) ≥ 0 for an arbitrary u ∈ T . Since (Z[t1 , t2 , · · · , tn ])/ Ap,c ∼ = Zp we obtain that T /J(T ) is a finite field. Finally, ρp,c (p) = 1 because ρp,c is unramified over vp,c . This completes the proof. 2 We will need the following corollary of Lemma 1 in the proof of Theorem 1. Corollary 2. Let T˜ be the completion of T in the topology defined by the valuation ρp,c . Then for every natural k the ideal T˜k = {x ∈ T˜|ρp,c (x) ≥ k} has finite index in T˜ . Proof. Let Tk = T˜k T . Since T /Tk ∼ = T˜/T˜k it is enough to verify that the quotient ring T /Tk is finite. k Further since Tk ⊇ (J(T )) it is enough to verify that the quotient ring T /(J(T ))k is finite. Since the quotient ring T /J(T ) is a finite field by Lemma 1, we can assume that k > 1 and that the ring T /(J(T ))k−1 is finite, and it is enough to prove that the ideal J(T )k−1 /(J(T ))k is finite. The ring T is Noetherian and so is the ring T /(J(T ))k ; because of this the ideal J(T )k−1 /(J(T ))k is generated by a finite set of elements x1 , x2 , · · · , xr . Since J(T )/(J(T ))k annihilates the ideal (J(T ))k−1 /(J(T ))k this ideal is in fact a finite dimensional vector space over the finite field T /J(T ). Thus it must be finite and the proof is complete. 2 We need the following known fact. Lemma 2. Let T be a commutative domain with unit which is a finitely generated algebra over the field of rational numbers Q, and let Aj (j ∈ J) be the system of all the maximal ideals of T . Then the ring T can be embedded into a suitable ultraproduct of fields ( j∈J Kj )/F where Kj = T /Aj (j ∈ J) is a finite algebraic extension of Q. Proof. We will give a sketch of the proof. It is well known that every quotient ring Kj = R/Aj is an algebraic number field and that j∈J Aj = 0. Let φj be the natural homomorphism T −→ T /Aj and for an arbitrary 0 = r ∈ T let Jr = {j ∈ J|φj (r) = 0}
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Lemma 6.4 and its Corollary in [11] imply that the system of subsets Jr (r ∈ T ) generates an ultrafilter F on J and that the ring T embeds into the ultraproduct ( j∈J Kj )/F. 2 2.2. We give here a description of some concepts and results of Lazard’s paper [6] which will be used in the proof of Theorems 2 and 3. Let G be a torsion free group and φ be a function from G to Z ∞ where Z is the set of natural numbers, and φ(x) = ∞ iff x = 1 (Lazard considered a more general situation, when the set of the values of the function φ is a set of real numbers greater that (p − 1)−1 ). The function φ is a p-valuation on G if the following conditions hold for arbitrary x, y ∈ G: φ(xy −1 ) ≥ min{φ(x), φ(y)}
(2.1)
φ([x, y]) ≥ (φ(x) + φ(y))
(2.2)
φ([xp ]) = φ(x) + 1
(2.3)
We will need the case when the strong form of condition (2.2) holds φ([x, y]) > (φ(x) + φ(y))
(2.4)
Gi = {x ∈ G|φ(x) ≥ i}
(2.5)
Let
We obtain a series of normal subgroups Gi (i ∈ Z) with unit intersection and φ(x) = max{i ∈ C|x ∈ Gi }. Condition (2.3) implies that if x ∈ Gi \Gi+1 then xp ∈ Gi+1 \Gi+2 . Further, if condition (2.4) holds then [Gi , Gj ] ⊆ Gi+j+1
(2.6)
Every factor Gi /Gi+1 is a vector space over the prime field Zp and it is proven in [6] that the map x −→ xp defines a monomorphism ψi from Gi /Gi+1 into Gi+1 /Gi+2 ; we point out that this can be proven easily in the case when we have relation (2.4). Let gr G (in Lazard’s notation) be the Lie algebra over the prime field Zp associated to the series (2.5). The algebra gr G can be considered as an algebra over the ring of polynomials Zp [t] where the action of t on the Zp -subspace Gi /Gi+1 is defined by the rule: t • a = ap (a ∈ Gi /Gi+1 ). Let U (gr G) be the universal enveloping algebra of the Lie algebra gr G over Zp [t]. Now let x ∈ Gi \Gi+1 , y ∈ Gj \Gj+1 , x ˜, y˜ be homogeneous components of x and y and assume that relation (2.4) holds. Then [˜ x, y˜] = 0 and we see that relation (2.4) implies that the algebra gr G is abelian. Let ΩG be the group ring of G over the ring of p-adic integers Ω. Series (2.5) defines in this ring the following filtration ΩG = L0 ⊇ L1 ⊇ · · ·
(2.7)
where Li (i = 1, 2, · · · ) is generated over Ω by all the elements pn (g1 − 1)(g2 − 1) · · · (gm − 1)
(2.8)
(n + φ(g1 ) + φ(g2 ) + · · · + φ(gm )) ≥ i
(2.9)
with
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We define now in ΩG a function w as follows: if r = 0 then w(r) is the maximal i such that r ∈ Li , and w(0) = ∞. This function is a pseudovaluation of KG i.e. w(r1 + r2 ) ≥ min{w(r1 ), w(r2 )}
(2.10)
w(r1 r2 ) ≥ w(r1 ) + w(r2 )
(2.11)
and
Let grw (ΩG) be the graded ring associated to this pseudovaluation. The main content of Theorem 2.3.3 and its corollaries in [6] is that the graded algebra grw (ΩG) is isomorphic to the universal enveloping algebra of the Lie algebra gr G over Zp [t]; the pseudovaluation w is in a fact a valuation, i.e., relation (2.11) is replaced by w(r1 r2 ) = w(r1 ) + w(r2 )
(2.12)
and w(g − 1) = φ(g) for every g ∈ G. Since gr G is an abelian Lie algebra over Zp [t], its universal envelope is isomorphic to the polynomial ring Zp [t, E] where E is a basis of the Lie algebra gr G over Zp [t]. Let g˜ be the homogeneous component of the element g in the algebra gr G and g − 1 the homogeneous component of the element (g − 1) in grw (ΩG). The elements g − 1 ((g − 1) ∈ ΩG) generate in grw (ΩG) a Lie algebra over Zp and the correspondence τ : g − 1 −→ g˜ defines an isomorphism between this Lie algebra and the Zp -Lie algebra gr G. Further, the homogeneous component p˜ of p in grw (ΩG) is transcendental over Zp − 1 ((g − 1) ∈ ΩG) generate a subalgebra isomorphic to the and the element p˜ together with the elements g Zp [t]-algebra gr G; it is shown in [6] that this isomorphism extends the isomorphism τ if we define τ (˜ p) = t. 3. Lazard’s p-valuation in linear groups We will derive in this section Theorem 2. Let F be a field of characteristic zero and let H be a subgroup of GLn (F ) generated by a finite number of matrices h1 , h2 , · · · , hm
(3.1)
We pick a finitely generated subring T ⊆ F with unit which contains all the entries of the matrices (3.1) and of their inverses; hence H ⊆ Tn×n . We consider now the system of valuations ρi (i ∈ I) obtained in Lemma 1 for T and pick an arbitrary valuation ρ from this system; we recall that there exists a prime number p with ρ(p) = 1 and with the quotient ring T /J(T ) is a finite extension of the prime field Zp . Let T˜ be the completion of T and let T˜k = {r ∈ T˜|ρ(r) ≥ k} (k = 1, 2, · · · ). Let T˜n×n be the matrix ring of degree n over T˜, and (T˜k ) be the ideal consisting of all matrices whose entries are in T˜k . Let Uk (k = 1, 2, · · · ) be the set of all the matrices u ∈ T˜n×n such that (u − 1) ∈ (T˜k+1 ), i.e., Uk = 1 + (T˜k+1 ). We obtain in U1 = U a series U = U1 ⊇ U2 ⊇ · · · Since and
∞ k=1
(3.2)
∞ T˜k = 0 we obtain that k=1 Uk = 1. We define now a weight function ω(x) on U : ω(1) = ∞
ω(x) = k if x ∈ Uk \Uk+1 This condition is equivalent to the condition
(3.3)
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(x − 1) ∈ (T˜k+1 )\(T˜k+2 )
(3.4)
Let x ∈ Uk , that is, x = 1 + a, where a ∈ (T˜ )k+1 . Then x−1 = (1 − a + a2 + · · · ) ∈ Uk . Similarly, we obtain from the relation Uk = 1 + (T˜k+1 ) that Uk is multiplicatively closed and is invariant in GLn (T˜). Corollary 2 yields that the index of every ideal T˜k is finite and we obtain from this that all the quotient rings (T˜n×n )/(T˜k ) ∼ = (T˜/T˜k )n×n are finite, and every subgroup Uk = 1 + (T˜k+1 ) has finite index in GLn (T˜). Let x, y ∈ U with ω(x) = k, ω(y) = l. Then a straightforward computation shows that (yx − xy) ∈ ˜ (T )k+l+2 and 1 − x−1 y −1 xy = x−1 y −1 (yx − xy) ∈ (T˜k+l+2 )
(3.5)
We see from the last equation that ω([x, y]) > ω(x) + ω(y)
(3.6)
Further, if x = 1 + a, y = 1 + b with a ∈ (T˜k ), b ∈ (T˜l ) then xy −1 = (1 + a)(1 − b + b2 · · · ) and we obtain that ω(xy −1 ) ≥ min{(ω(x), ω(y))}
(3.7)
Now once again let ω(x) = k, so x = 1 + a, where a ∈ (T˜k+1 )\(T˜k+2 ). We have ai ∈ (T˜i(k+1) ) for 1 ≤ i ≤ p; further ρ(p) = 1 and pa ∈ (T˜k+2 )\(T˜k+3 ). We obtain from this and the binomial expansion of (1 + a)p that (1 + a)p ∈ (T˜k+2 )\(T˜k+3 ). Hence ω(xp ) = ω(x) + 1
(3.8)
Relations (3.6)–(3.8) show that the weight function ω(x) on U is a p-valuation in the group U . The restriction of ω(x) to the group G = H U is a p-valuation in G and we obtain a series G = G1 ⊇ G2 ⊇ · · ·
(3.9)
where Gk = H Uk (k = 1, 2, · · · ). Since every term of the series Uk (k = 1, 2, · · · ) is invariant in the group GLn (T˜) and has finite index in it we obtain that the series Gk (k = 1, 2, · · · ) is H-invariant and all its terms have finite indices in G. We obtain now from Lazard’s Theorem 2.3.3 (see subsection 2.2): Theorem 2. Let H be a finitely generated linear group over a field F of characteristic zero. Then there exists a number N , depending on H, such that for every prime number p > N , there exists a torsion free normal subgroup G of finite index (H : G), depending on p, with a p-valuation ω whose values are natural numbers, with an H-invariant Lazard series (3.9) and an H-invariant filtration Ωp G = L0 ⊇ L1 ⊇ · · ·
(3.10)
with zero intersection and the graded ring gr(ΩG) isomorphic to a polynomial ring Zp [X] over Zp . Remark. The proof of Theorem 2 yields that not only the group G but every subgroup Gk in series (3.9) satisfies all the conclusions of Theorem 3 and that ω(g) ≥ k if g ∈ Gk .
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4. Valuations in modular group ring KG defined by p-valuations of G Let F be a group. A series of normal subgroups F = F1 ⊇ F2 ⊇ · · ·
(4.1)
is a p-series in F if [Fi , Fj ] ⊆ Fi+j and Fip ⊆ Fip (see Lazard [7] or Passman [12]). We will assume in this ∞ section that i=1 Fi = 1 and denote by Lp (F, Fi ) the restricted Lie algebra associated to p-series (4.1) and by Up (Lp (F, Fi )) its universal p-envelope. The algebra Lp (F, Fi ) =
∞
Fi /Fi+1
(4.2)
i=1
is obtained by the classical construction of Lazard [7]. If K is a field of characteristic p then p-series (4.1) defines a filtration An (KF ) (n = 0, 1, · · · ) in the ∞ group ring and n=0 Aj (KF ) = 0 (see Passman [12]), and we obtain a pseudovaluation v defined by this filtration; if the graded ring gr(KF ) associated to this filtration is a domain then v is a valuation in KF . Theorem 3. Let G be a group with a p-valuation φ whose values are natural numbers which are greater than 1 if p > 2. Then there exists a p-series G∗j (j = 1, 2, · · · ) with unit intersection such that the algebra Lp (G, G∗j ) is a free restricted abelian Lie algebra. Proof. We have in G the Lazard series defined by the function φ Gi = {x ∈ G|φ(x) ≥ i} (i = 1, 2, · · · )
(4.3)
Series (4.3) may begin with a term Gi0 with i0 > 1, but in this case we can define G1 = G2 = · · · = Gi0 and so assume that the first term of the series is G1 . We pick an arbitrary non-negative number k and define on G a new weight function f as f (g) = pφ(g)+k , or explicitly, f (g) = pi+k if g ∈ Gi \Gi+1
(4.4)
G∗pi+k = {g ∈ G|f (g) ≥ pi+k }
(4.5)
and let
Let f (g1 ) = pi1 +k , f (g2 ) = pi2 +k with i1 ≥ i2 . Then φ(g1 g2 ) ≥ i1 and f (g1 g2 ) ≥ pi1 +k
(4.6)
which means that g1 g2 ∈ G∗pi1 +k . Similarly, gi−1 , gi−1 ⊆ G∗pi1 +k ; we see that G∗pi1 +k is a subgroup. 1 2 Further, since [g1 , g2 ] ∈ Gi1 +i2 , f ([g1 , g2 ]) ≥ pi1 +i2 +k > pi1 +k + pi2 +k = f (g1 ) + f (g2 )
(4.7)
Let f (g) = pi+k . Since φ(g p ) = φ(g) + 1, we obtain that f (g p ) = pi+k+1
(4.8)
G = G∗1 = G∗2 = · · · = G∗p1+k ⊇ G∗p1+k +1 = G∗p1+k +2 = · · · = G∗p2+k ⊇ · · ·
(4.9)
We consider now the following series in G:
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We point out that the only terms in series (4.9) where a subgroup G∗j may properly contain the subgroup are the terms with j = pi+k . We obtain from relation (4.8) that (G∗j )p ⊆ G∗pj if j = pi+k ; hence this relation holds for an arbitrary term G∗j (j = 1, 2, · · · ). Similarly relation (4.7) implies that [G∗j1 , G∗j2 ] ⊆ G∗j1 +j2 for arbitrary j1 , j2 . We see that (4.9) is a p-series in G. We consider now the algebra Lp (G, G∗j ) associated to this series. Let g1 ∈ G∗j1 \G∗j1 +1 , g2 ∈ G∗j2 \G∗j2 +1 . Since [g1 , g2 ] ∈ G∗j1 +j1 +1 via relation (4.7) we obtain that the homogeneous components g˜1 and g˜2 of the elements g1 , g2 commute in the algebra Lp (G, G∗j ). We see that this algebra is abelian. Relation (4.8) yields that if g ∈ G∗j1 \G∗j1 +1 then g p ∈ G∗pj1 \G∗pj1 +1 . This means that the homogeneous component g˜ of this element is not nilpotent. It is known that an arbitrary restricted abelian Lie algebra R without nilpotent elements must be free abelian. In fact, let R[p] is the subalgebra generated by all the elements r[p] (r ∈ R) and let E be a system of elements of R which forms a basis of the quotient algebra R/R[p] , then a straightforward argument shows that E freely generates R. So the algebra Lp (G, G∗j ) is a free abelian restricted algebra and the proof is complete. 2 G∗j+1
Corollary 3. Let H be a finitely generated linear group over a field F of characteristic zero, let G ⊆ H be a normal subgroup of H obtained in Theorem 2, and let K be a field of finite characteristic p. Then the subgroups G∗j (j = 1, 2, · · · ) obtained by (4.5) are H-invariant as well as the ideals Aj (KG∗ ) (j = 0, 1, · · · ) of the filtration defined by this series. This filtration defines a valuation v with associated graded ring isomorphic to a polynomial algebra K[X]. Proof. Since series (4.3) is H-invariant via Theorem 2 the proof of Theorem 3 implies that the subgroups G∗j (j = 1, 2, · · · ) are H-invariant and so are the ideals Aj (KG) (j = 0, 1, · · · ). The graded ring associated to the filtration Aj (KG) (j = 0, 1, · · · ) is isomorphic to the p-envelope Up (Lp (G, G∗j )); this follows from a version of Quillen’s theorem [13] proven in Lichtman [8]. Theorem 3 yields that the algebra Lp (G, G∗j ) is a free restricted abelian algebra. Hence its p-envelope is isomorphic to a suitable polynomial ring K[X] and the assertion follows. 2 5. The proof of Theorem 4 We need the following Theorem 4 in the proof of Theorem 1. Theorem 4. Let R ∗ W be a crossed product of a domain R with a group W . Assume that there exists a non-negative W -invariant filtration Ri (i = 0, 1, · · · ) such that the associated graded ring gr(R) is an Ore domain. Then there exists a W -invariant division ring D generated by R such that the embedding R −→ D extends to an embedding of R ∗ W into a suitable crossed product D ∗ W . We will apply in the proof below the method developed in [9], section 2.1 and [10], section 4 (see also Cohn [3], section 2.6) in order to construct a division ring D which contains an isomorphic copy of R and is generated by it. This division ring D will be W -invariant because all the rings which will be considered ˜ is a system of elements in R ∗ W which during its construction are W -invariant, and we will prove that if W forms a W -basis for the crossed product R ∗ W then these elements will form a basis for a suitable crossed product D ∗ W . We also point out that a simple example of this kind is the case when R is an Ore domain with a ring of fractions D; in this case D is W -invariant and we obtain an immediate embedding of R ∗ W into D ∗ W . Proof of Theorem 4. Let v(r) be the valuation function defined in R by the filtration Ri (i = 0, 1, · · · ). We consider the Laurent polynomial ring (R ∗ W )[t, t−1 ] in a central variable t and extend the valuation v of R to the subring R[t, t−1 ] by the rule
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v(
m
33
ri ti ) = min{v(ri ) + i}
(5.1)
i
i=1
We obtain a valuation in R[t, t−1 ] which extends the valuation v of R; we will keep for this extended valuation the same notation v. Let T = {x ∈ R[t, t−1 ] | v(x) ≥ 0} be the ring of v-integers of R[t, t−1 ]. Clearly, the ring (R ∗ W )[t, t−1 ] is isomorphic to the crossed product of R[t, t−1 ] with W , T is a W -invariant subring of R[t, t−1 ] and T ⊇ R; we see that R ∗ W is a subring of a suitable crossed product T ∗ W . For every natural number n the epimorphism T −→ T /(t)n = Tn extends to an epimorphism of suitable crossed products (T ∗ W ) −→ (T ∗ W )/(t)n ∼ = Tn ∗ W . We consider now the inverse system of rings Tn ∗ W (n = 1, 2, · · · ) and the natural epimorphisms φn1 ,n2 : Tn1 ∗ W −→ Tn2 ∗ W for n1 > n2 ; the kernel of the epimorphism φn1 ,n2 is the ideal (t)n2 of the ring Tn1 ∗ W . Let M = T \(tT ) be the complement of the ideal tT , and let Mn be the image of M in the quotient ring T /(tT )n . Then (see [10], Lemma 4.3 and Proposition 4.1) the quotient ring T /(tT ) is isomorphic to the ring gr(R) and Mn is a right denominator set of regular elements in the quotient ring Tn = T /(tT )n (n = 1, 2, · · · ). Since the set M is W -invariant in T its image Mn is W -invariant in Tn ∗ W . A straightforward argument now implies that Mn remains a right denominator set of regular elements in Tn ∗ W and we can consider the ring of fractions of Tn ∗ W with respect to Mn ; this ring of fractions is isomorphic to a suitable crossed product (Tn Mn−1 ) ∗ W . A routine argument shows that every epimorphism φn1 ,n2 extends now to an epimorphism of the rings of fractions (Tn1 Mn−1 ) ∗ W −→ (Tn2 Mn−1 ) ∗ W and we obtain an inverse system 1 2 of rings (Tn Mn−1 ) ∗ W (n = 1, 2, · · · ); we also see that there is a natural embedding of the inverse system Tn ∗ W (n = 1, 2, · · · ) in this inverse system Tn ∗ W (n = 1, 2, · · · ). Let T˜ = lim Tn and S = lim Tn Mn−1 be ←−
←−
the inverse limits of the systems Tn and Tn M −1 respectively. We obtain natural embeddings T˜ ⊆ lim(Tn ∗ W )
(5.2)
S ⊆ lim(Tn Mn−1 ∗ W )
(5.3)
T ⊆ T˜ ⊆ S
(5.4)
←−
←−
and
The system of epimorphisms φn : T −→ Tn extends to systems of epimorphisms T˜ −→ Tn ; S −→ Tn Mn−1 (n = 1, 2, · · · ). We also have extensions of these epimorphisms lim(Tn ∗ W ) −→ Tn ∗ W (n = 1, 2, · · · )
(5.5)
lim(Tn Mn−1 ∗ W ) −→ (Tn Mn−1 ∗ W ) (n = 1, 2, · · · )
(5.6)
←−
and ←−
¯ be a system of elements in R ∗ W which forms a W -basis for this crossed product over R. We can Let W ¯ with their images in Tn ∗ W and in (Tn M −1 ) ∗ W and obtain that the system identify the elements of W ¯ of elements W forms also a W -basis for the crossed products Tn ∗ W and (Tn M −1 ) ∗ W . We consider now the subring of the ring lim(Tn Mn−1 ∗ W ) generated by the subring S = lim(Tn Mn−1 ) ←− ←− ¯ . The existence of the system of epimorphisms (5.6) implies by a routine and the system of elements W ¯ are left linearly independent over S, that they normalize S, and that the argument that the elements of W group of automorphisms of S generated by them is W . We see that the subring of the inverse limit of the ¯ is isomorphic to a suitable crossed product S ∗ W . Similarly, system (Tn Mn−1 ∗ W ) generated by S and W
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¯ is isomorphic the subring of this inverse limit generated by the subring lim Tn and the system of elements W ←− to a suitable crossed product T˜ ∗ W . We obtain also embeddings R ∗ W ⊆ T˜ ∗ W ⊆ S ∗ W
(5.7)
R ⊆ T˜ ⊆ S
(5.8)
which extend the embeddings
We obtain from section 2.1 in Lichtman [9] or section 4.3 in Lichtman [10] (see also Cohn [3], Theorem 2.6.2) that S is a local domain with radical tS and S/(tS) is isomorphic to the division ring of fractions of the Ore domain T /(tT ) ∼ = gr(R). Further, the subset {t, t2 , · · · } of S is a central subsemigroup in S ∗ W and the ring of fractions of S with respect to this subsemigroup is a division ring Q(S). We consider now the ring of fractions of S ∗ W with respect to the subsemigroup {t, t2 , · · · } and obtain an embedding (S ∗ W ) ⊆ (Q(S) ∗ W ). The division subring D generated by R is invariant with respect to all the automorphisms generated by the elements w ∈ W and it is contained in Q(S). We obtain from this that (R ∗ W ) ⊆ (D ∗ W ) and the proof is complete. 2 6. The proof of Theorem 1 We observe first that if we prove the statements about the existence of the normal subgroup G of finite index, about the division ring generated by the group ring KG, and about isomorphism (1.1) then this will imply that the crossed product S ∼ = Δ ∗ (H/G) is a semisimple Artinian ring because the order of H/G does not divide the characteristic of Δ (see Jacobson [4], 4.18, Theorem 48). Lemma 3. Let H be a group, let G be a normal subgroup of finite index, in H, and let Kj (j ∈ J) be a family of fields. Assume that for every group ring Kj H (j ∈ J) there exists an isomorphic embedding of Kj H into a ring Sj such that the group ring Kj G generates an H-invariant skew field Δj ⊆ Sj such that Sj ∼ = Δj ∗ (H/G)
(6.1)
where the embedding Δj ⊆ Sj extends the embedding Kj G ⊆ Kj H and isomorphism (6.1) extends the isomorphism Kj H ∼ = (Kj G) ∗ (H/G). Let F be an arbitrary ultrafilter on the set J and let K be an arbitrary subfield of the ultraproduct ( j∈J Kj )/F. Then Theorem 1 holds for the group ring KH. Proof. We have embeddings KH ⊆ (
j∈J
Kj H)/F ⊆
(Δj ∗ (H/G))/F
(6.2)
j∈J
It is known that the ultraproduct of division rings is a division ring; hence Δ(J) = ( j∈J Δj )/F is a division ring. Further, it is well known and it is easy to verify that the ultrapower ( j∈J (H/G))/F of the finite group H/G is isomorphic to H/G, and thus we obtain that the ultraproduct j∈J (Δj ∗(H/G))/F is isomorphic to a suitable crossed product ΔJ ∗(H/G). We obtain an embedding KH ∼ = (KG) ∗(H/G) ⊆ (ΔJ ∗(H/G)) with J KG embedded into Δ . We denote now by Δ the division subring of ΔJ generated by KG, and by S the ¯ 1 = 1, h ¯ 2, · · · , h ¯ k is a system of coset subring of ΔJ ∗(H/G) generated by KH and Δ. We have Δ ⊆ S and if h representatives for the group H/G then these elements belong to S and normalize the division subring Δ; they are left linearly independent over Δ because they are linearly independent over ΔJ . This implies that the subring generated by these elements and Δ is isomorphic to Δ ∗ (H/G) and that S ∼ = Δ ∗ (H/G). This completes the proof. 2
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Proof of Theorem 1. Statement i). We consider first the case when K is an algebraic number field. Pick a prime number p from the set of prime numbers obtained in Theorem 2 such that the ring of p-adic integers Ω contains the field K. Theorems 2 and 4 imply that there exists a ring S0 which contains ΩH and such that the group ring ΩG generates a division subring Δ0 and S0 ∼ = Δ0 ∗ (H/G). We consider the division subring Δ ⊆ Δ0 generated by KG and the subring S ⊆ S0 generated by KH and Δ; a straightforward argument yields that S ∼ = Δ ∗ (H/G). This completes the proof for the special case when K is an algebraic number field. Now consider the case when K is a finitely generated field, i.e. K is the field of fractions of a finitely generated domain Q(x1 , x2 , · · · , xk ). Lemma 2 implies that K can be embedded into a suitable ultraproduct of algebraic number fields Kj (j ∈ J) and the assertion follows from Lemma 3. Finally, let K be an arbitrary field of characteristic zero. Let Kγ (γ ∈ Γ) be the system of all the finitely generated subfields of K. The same argument as in 1.L.7 of Kegel and Wehrfritz [5] yields that the field K can be embedded into a suitable ultraproduct ( γ∈Γ Kγ )/F and the assertion follows now from Lemma 3. This completes the proof of statement i). Statement ii). This follows immediately from Corollary 3 and Theorem 4 by the same argument as in statement i) if p = 2; if p = 2 we can assume (see the remark after the proof of Theorem 2) that the p-values of the elements of G are greater than 1 and once again the same argument as in i) completes the proof. 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Z.I. Borevich, I.R. Shafarevich, Number Theory, Academic Press, 1966. N. Bourbaki, Algebra II, Chaps. 4–7, Springer, Heidelberg, 1990. P.M. Cohn, Skew Fields, Theory of General Division Rings, Cambridge University Press, Cambridge, 1995. N. Jacobson, The Theory of Rings, Amer. Math. Soc. Math. Surveys, Amer. Math. Soc., New York, 1943. O.H. Kegel, B.A.F. Wehrfritz, Locally Finite Groups, North-Holland/American Elsevier, 1973. M. Lazard, Groups analitiques p-adiques, Publ. Math. IHÉS 26 (1965) 389–603. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Éc. Norm. Super. 3 (1954) 101–190. A.I. Lichtman, Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings, I, J. Algebra 88 (1984) 1–37. A.I. Lichtman, Valuation methods in division rings, J. Algebra 177 (1993) 870–898. A.I. Lichtman, Valuation methods in group rings and in skew fields, I, J. Algebra 257 (2002) 106–167. A.I. Lichtman, Nilpotent and soluble subgroups of linear groups over fields of fractions of enveloping algebras and group rings, I, Contemp. Math. 93 (1989) 247–283. D.S. Passman, The Algebraic Structure of the Group Rings, Wiley, 1977. D.G. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968) 711–718.
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
On group rings of linear groups A.I. Lichtman Department of Mathematics, University of Wisconsin-Parkside, Kenosha, WI 53141, United States
a r t i c l e
i n f o
Article history: Received 1 December 2015 Received in revised form 12 May 2016 Available online 31 May 2016 Communicated by S. Donkin MSC: 16K; 12E15
a b s t r a c t Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number N = N (H). We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field. © 2016 Elsevier B.V. All rights reserved.
1. Introduction 1.1.
The main result of this paper is the following theorem.
Theorem 1. Let H be a finitely generated subgroup of GLn (F ), char(F ) = 0, and let KH be the group ring of H over a field K. i) If char(K) = 0 then there exists a torsion free normal subgroup G of finite index in H and an isomorphic embedding of KH into a semisimple Artinian ring S such that the group ring KG generates an H-invariant division subring Δ ⊆ S and S is isomorphic to a suitable cross product S∼ = Δ ∗ (H/G)
(1.1)
where the isomorphism (1.1) extends the isomorphism KH ∼ = KG ∗ (H/G). ii) If char(K) is finite and greater than N , a number depending on H, then statement i) remains true for KH. The additional information about the normal subgroup G and its group ring is contained in Theorems 2 and 3 below. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2016.05.015 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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Corollary 1. Let H be a finitely generated subgroup of GLn (F ), char(F ) = 0, and let KH be the group ring of H over a field K with char(K) = 0 or char(K) > N = N (H). Then there exists a division K-algebra Δ such that KH is isomorphic to a subring of a matrix ring Δm×m . The corollary follows from Theorem 1 immediately. In fact, S has a finite left dimension m = (H : G) over Δ, hence it has a faithful representation in Δm×m . It is worth remarking that the division ring Δ in the corollary will be commutative only if the group H has an abelian normal subgroup of finite index when char(K) = 0, or a p-abelian subgroup of finite index if char(K) = p. This follows from Corollaries 5.3.8–5.3.9 in Passman’s book [12]. 1.2. The proof of Theorem 1 is given in section 6. It is based on Theorems 2, 3 and 4. We prove Theorem 2 in section 3; we obtain there a torsion free normal subgroup G of finite index in H such that the group ring of G over the ring of p-adic integers Ω has an H-invariant filtration with an associated graded ring isomorphic to the polynomial ring over a prime field Zp . The proof of this theorem makes an essential use of Lazard’s p-valuations (see Lazard [6]). A short description of Lazard’s method is presented in subsection 2.2. The second main step in the proof of Theorem 1 is Theorem 4 (section 5) whose proof is based on the method developed by the author in [9] and [10]; this method is described also in Cohn [3], section 2.6. We apply Theorem 2 and Theorem 4 to construct the division ring Δ in Theorem 1 in the case when char(K) = 0. To prove statement ii) of Theorem 1 we need Theorem 3 and Corollary 3 (section 4) which states that the existence of a p-valuation in a group G implies that there exists a filtration and a valuation in the group ring Zp G over the prime field Zp with associated graded ring isomorphic to the polynomial ring over Zp . Our arguments show in fact that the conclusions of Theorems 1 and 2 remain true for an arbitrary, not necessarily finitely generated, subgroup H ⊆ GLn (T ) if T is a finitely generated commutative domain of characteristic zero. 2. Preliminaries 2.1. Lemma 1. Let T be a finitely generated commutative domain of characteristic zero and of transcendence degree n, and t1 , t2 , · · · , tn be a system of elements in T algebraically independent over Z. Then there exists a natural number N such that for every prime p > N and natural number c > N the powers of the ideal Ap,c generated by the elements p, t1 − c, t2 − c, · · · , tn − c define a p-adic valuation ρp,c of T such that i) ρp,c (p) = 1, and ii) if J(T ) = {r ∈ T |ρp,c (r) > 0}, then the quotient ring T /J(T ) is a finite field with characteristic p. Proof. Since T is finitely generated we can extend the system t1 , t2 , · · · , tn to a system of elements t1 , t2 , · · · , tn ; s1 , s2 , · · · , sm which generates T . Let φj [x] (1 ≤ j ≤ m) be the minimal polynomial of sj over Z[t1 , t2 , · · · , tn ]. We consider the field of fractions R of T and its subfield S = Q(t1 , t2 , · · · , tn ) which is the field of rational functions in variables t1 , t2 , · · · , tn over the field Q of rational numbers. We pick an element θ such that R = S(θ) and let ψ[x] be the minimal polynomial of θ; we can assume that all the coefficients of ψ[x] belong to the subring Z[t1 , t2 , · · · , tn ] as well as its discriminant d[t1 , t2 , · · · , tn ]. We pick now an arbitrary prime number p and a natural number c and consider the ideal Ap,c ⊆ Z[t1 , t2 , · · · , tn ] generated by the system of elements p, t1 − c, t2 − c, · · · , tn − c. The quotient ring
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∞ (Z[t1 , t2 , · · · , tn ])/Ap,c is the prime field Zp = Z/pZ and i=1 Aip,c = 0. We claim that the filtration Aip,c (i = 1, 2, · · · ) defines a valuation vp,c of Z[t1 , t2 , · · · , tn ]. In fact, the map ti −→ ti −c (i = 1, 2, · · · , n) defines an automorphism of Z[t1 , t2 , · · · , tn ] so the filtration i Ap,c (i = 1, 2, · · · ) defines a valuation if the powers of the ideal Ap,0 generated by the system of elements p, t1 , t2 , · · · , tn define a valuation vp,0 . But it is well known that the powers of the ideal Ap,0 define a p-adic valuation of Z[t1 , t2 , · · · , tn ]; this follows, for instance, from Lemma 1 in [2], VI. 10 if we apply this lemma and extend the p-adic valuation of Z to a valuation v of Z[t1 , t2 , · · · , tn ], and observe that this valuation v will be defined in fact by the filtration Aip,0 (i = 1, 2, · · · ), i.e., it will coincide with the valuation vp,0 . It is also possible to verify by a straightforward argument that the powers of the ideal Ap,0 define a p-adic valuation of Z[t1 , t2 , · · · , tn ]. Since the coefficients of the polynomials φj [x] (j = 1, 2, · · · , m) and ψ[x] are elements of Z[t1 , t2 , · · · , tn ] we can pick a natural number N such that if p > N, c > N then the ideal Ap,c contains no one of the coefficients of these polynomials or of the polynomial d[t1 , t2 , · · · , tn ]. We extend the valuation vp,c defined in the ring Z[t1 , t2 , · · · , tn ] to its field of fractions S and denote by Svp,c the corresponding valuation ring. Let Rvp,c be the integral closure of Svp,c in R, i.e. the subring formed by all the elements r ∈ R whose minimal polynomials are monic polynomials over Svp,c . We have θ ∈ Rvp,c because ψ[x] is a monic polynomial over Svp,c . It is important that the discriminant d[t1 , t2 , · · · , tn ] is a unit in Rvp,c and we can now apply Theorem 8 in section III.5 of Borevich and Shafarevich [1] to obtain that the valuation vp,c does not ramify in R. Let ρp,c be an extension of vp,c to R. Since all the coefficients of the polynomials φj [x] (j = 1, 2, · · · , m) are units in Sv,c we obtain that sj ∈ Rρp,c and then T ⊆ Rρp,c . We see that ρp,c (u) ≥ 0 for an arbitrary u ∈ T . Since (Z[t1 , t2 , · · · , tn ])/ Ap,c ∼ = Zp we obtain that T /J(T ) is a finite field. Finally, ρp,c (p) = 1 because ρp,c is unramified over vp,c . This completes the proof. 2 We will need the following corollary of Lemma 1 in the proof of Theorem 1. Corollary 2. Let T˜ be the completion of T in the topology defined by the valuation ρp,c . Then for every natural k the ideal T˜k = {x ∈ T˜|ρp,c (x) ≥ k} has finite index in T˜ . Proof. Let Tk = T˜k T . Since T /Tk ∼ = T˜/T˜k it is enough to verify that the quotient ring T /Tk is finite. k Further since Tk ⊇ (J(T )) it is enough to verify that the quotient ring T /(J(T ))k is finite. Since the quotient ring T /J(T ) is a finite field by Lemma 1, we can assume that k > 1 and that the ring T /(J(T ))k−1 is finite, and it is enough to prove that the ideal J(T )k−1 /(J(T ))k is finite. The ring T is Noetherian and so is the ring T /(J(T ))k ; because of this the ideal J(T )k−1 /(J(T ))k is generated by a finite set of elements x1 , x2 , · · · , xr . Since J(T )/(J(T ))k annihilates the ideal (J(T ))k−1 /(J(T ))k this ideal is in fact a finite dimensional vector space over the finite field T /J(T ). Thus it must be finite and the proof is complete. 2 We need the following known fact. Lemma 2. Let T be a commutative domain with unit which is a finitely generated algebra over the field of rational numbers Q, and let Aj (j ∈ J) be the system of all the maximal ideals of T . Then the ring T can be embedded into a suitable ultraproduct of fields ( j∈J Kj )/F where Kj = T /Aj (j ∈ J) is a finite algebraic extension of Q. Proof. We will give a sketch of the proof. It is well known that every quotient ring Kj = R/Aj is an algebraic number field and that j∈J Aj = 0. Let φj be the natural homomorphism T −→ T /Aj and for an arbitrary 0 = r ∈ T let Jr = {j ∈ J|φj (r) = 0}
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Lemma 6.4 and its Corollary in [11] imply that the system of subsets Jr (r ∈ T ) generates an ultrafilter F on J and that the ring T embeds into the ultraproduct ( j∈J Kj )/F. 2 2.2. We give here a description of some concepts and results of Lazard’s paper [6] which will be used in the proof of Theorems 2 and 3. Let G be a torsion free group and φ be a function from G to Z ∞ where Z is the set of natural numbers, and φ(x) = ∞ iff x = 1 (Lazard considered a more general situation, when the set of the values of the function φ is a set of real numbers greater that (p − 1)−1 ). The function φ is a p-valuation on G if the following conditions hold for arbitrary x, y ∈ G: φ(xy −1 ) ≥ min{φ(x), φ(y)}
(2.1)
φ([x, y]) ≥ (φ(x) + φ(y))
(2.2)
φ([xp ]) = φ(x) + 1
(2.3)
We will need the case when the strong form of condition (2.2) holds φ([x, y]) > (φ(x) + φ(y))
(2.4)
Gi = {x ∈ G|φ(x) ≥ i}
(2.5)
Let
We obtain a series of normal subgroups Gi (i ∈ Z) with unit intersection and φ(x) = max{i ∈ C|x ∈ Gi }. Condition (2.3) implies that if x ∈ Gi \Gi+1 then xp ∈ Gi+1 \Gi+2 . Further, if condition (2.4) holds then [Gi , Gj ] ⊆ Gi+j+1
(2.6)
Every factor Gi /Gi+1 is a vector space over the prime field Zp and it is proven in [6] that the map x −→ xp defines a monomorphism ψi from Gi /Gi+1 into Gi+1 /Gi+2 ; we point out that this can be proven easily in the case when we have relation (2.4). Let gr G (in Lazard’s notation) be the Lie algebra over the prime field Zp associated to the series (2.5). The algebra gr G can be considered as an algebra over the ring of polynomials Zp [t] where the action of t on the Zp -subspace Gi /Gi+1 is defined by the rule: t • a = ap (a ∈ Gi /Gi+1 ). Let U (gr G) be the universal enveloping algebra of the Lie algebra gr G over Zp [t]. Now let x ∈ Gi \Gi+1 , y ∈ Gj \Gj+1 , x ˜, y˜ be homogeneous components of x and y and assume that relation (2.4) holds. Then [˜ x, y˜] = 0 and we see that relation (2.4) implies that the algebra gr G is abelian. Let ΩG be the group ring of G over the ring of p-adic integers Ω. Series (2.5) defines in this ring the following filtration ΩG = L0 ⊇ L1 ⊇ · · ·
(2.7)
where Li (i = 1, 2, · · · ) is generated over Ω by all the elements pn (g1 − 1)(g2 − 1) · · · (gm − 1)
(2.8)
(n + φ(g1 ) + φ(g2 ) + · · · + φ(gm )) ≥ i
(2.9)
with
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We define now in ΩG a function w as follows: if r = 0 then w(r) is the maximal i such that r ∈ Li , and w(0) = ∞. This function is a pseudovaluation of KG i.e. w(r1 + r2 ) ≥ min{w(r1 ), w(r2 )}
(2.10)
w(r1 r2 ) ≥ w(r1 ) + w(r2 )
(2.11)
and
Let grw (ΩG) be the graded ring associated to this pseudovaluation. The main content of Theorem 2.3.3 and its corollaries in [6] is that the graded algebra grw (ΩG) is isomorphic to the universal enveloping algebra of the Lie algebra gr G over Zp [t]; the pseudovaluation w is in a fact a valuation, i.e., relation (2.11) is replaced by w(r1 r2 ) = w(r1 ) + w(r2 )
(2.12)
and w(g − 1) = φ(g) for every g ∈ G. Since gr G is an abelian Lie algebra over Zp [t], its universal envelope is isomorphic to the polynomial ring Zp [t, E] where E is a basis of the Lie algebra gr G over Zp [t]. Let g˜ be the homogeneous component of the element g in the algebra gr G and g − 1 the homogeneous component of the element (g − 1) in grw (ΩG). The elements g − 1 ((g − 1) ∈ ΩG) generate in grw (ΩG) a Lie algebra over Zp and the correspondence τ : g − 1 −→ g˜ defines an isomorphism between this Lie algebra and the Zp -Lie algebra gr G. Further, the homogeneous component p˜ of p in grw (ΩG) is transcendental over Zp − 1 ((g − 1) ∈ ΩG) generate a subalgebra isomorphic to the and the element p˜ together with the elements g Zp [t]-algebra gr G; it is shown in [6] that this isomorphism extends the isomorphism τ if we define τ (˜ p) = t. 3. Lazard’s p-valuation in linear groups We will derive in this section Theorem 2. Let F be a field of characteristic zero and let H be a subgroup of GLn (F ) generated by a finite number of matrices h1 , h2 , · · · , hm
(3.1)
We pick a finitely generated subring T ⊆ F with unit which contains all the entries of the matrices (3.1) and of their inverses; hence H ⊆ Tn×n . We consider now the system of valuations ρi (i ∈ I) obtained in Lemma 1 for T and pick an arbitrary valuation ρ from this system; we recall that there exists a prime number p with ρ(p) = 1 and with the quotient ring T /J(T ) is a finite extension of the prime field Zp . Let T˜ be the completion of T and let T˜k = {r ∈ T˜|ρ(r) ≥ k} (k = 1, 2, · · · ). Let T˜n×n be the matrix ring of degree n over T˜, and (T˜k ) be the ideal consisting of all matrices whose entries are in T˜k . Let Uk (k = 1, 2, · · · ) be the set of all the matrices u ∈ T˜n×n such that (u − 1) ∈ (T˜k+1 ), i.e., Uk = 1 + (T˜k+1 ). We obtain in U1 = U a series U = U1 ⊇ U2 ⊇ · · · Since and
∞ k=1
(3.2)
∞ T˜k = 0 we obtain that k=1 Uk = 1. We define now a weight function ω(x) on U : ω(1) = ∞
ω(x) = k if x ∈ Uk \Uk+1 This condition is equivalent to the condition
(3.3)
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(x − 1) ∈ (T˜k+1 )\(T˜k+2 )
(3.4)
Let x ∈ Uk , that is, x = 1 + a, where a ∈ (T˜ )k+1 . Then x−1 = (1 − a + a2 + · · · ) ∈ Uk . Similarly, we obtain from the relation Uk = 1 + (T˜k+1 ) that Uk is multiplicatively closed and is invariant in GLn (T˜). Corollary 2 yields that the index of every ideal T˜k is finite and we obtain from this that all the quotient rings (T˜n×n )/(T˜k ) ∼ = (T˜/T˜k )n×n are finite, and every subgroup Uk = 1 + (T˜k+1 ) has finite index in GLn (T˜). Let x, y ∈ U with ω(x) = k, ω(y) = l. Then a straightforward computation shows that (yx − xy) ∈ ˜ (T )k+l+2 and 1 − x−1 y −1 xy = x−1 y −1 (yx − xy) ∈ (T˜k+l+2 )
(3.5)
We see from the last equation that ω([x, y]) > ω(x) + ω(y)
(3.6)
Further, if x = 1 + a, y = 1 + b with a ∈ (T˜k ), b ∈ (T˜l ) then xy −1 = (1 + a)(1 − b + b2 · · · ) and we obtain that ω(xy −1 ) ≥ min{(ω(x), ω(y))}
(3.7)
Now once again let ω(x) = k, so x = 1 + a, where a ∈ (T˜k+1 )\(T˜k+2 ). We have ai ∈ (T˜i(k+1) ) for 1 ≤ i ≤ p; further ρ(p) = 1 and pa ∈ (T˜k+2 )\(T˜k+3 ). We obtain from this and the binomial expansion of (1 + a)p that (1 + a)p ∈ (T˜k+2 )\(T˜k+3 ). Hence ω(xp ) = ω(x) + 1
(3.8)
Relations (3.6)–(3.8) show that the weight function ω(x) on U is a p-valuation in the group U . The restriction of ω(x) to the group G = H U is a p-valuation in G and we obtain a series G = G1 ⊇ G2 ⊇ · · ·
(3.9)
where Gk = H Uk (k = 1, 2, · · · ). Since every term of the series Uk (k = 1, 2, · · · ) is invariant in the group GLn (T˜) and has finite index in it we obtain that the series Gk (k = 1, 2, · · · ) is H-invariant and all its terms have finite indices in G. We obtain now from Lazard’s Theorem 2.3.3 (see subsection 2.2): Theorem 2. Let H be a finitely generated linear group over a field F of characteristic zero. Then there exists a number N , depending on H, such that for every prime number p > N , there exists a torsion free normal subgroup G of finite index (H : G), depending on p, with a p-valuation ω whose values are natural numbers, with an H-invariant Lazard series (3.9) and an H-invariant filtration Ωp G = L0 ⊇ L1 ⊇ · · ·
(3.10)
with zero intersection and the graded ring gr(ΩG) isomorphic to a polynomial ring Zp [X] over Zp . Remark. The proof of Theorem 2 yields that not only the group G but every subgroup Gk in series (3.9) satisfies all the conclusions of Theorem 3 and that ω(g) ≥ k if g ∈ Gk .
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4. Valuations in modular group ring KG defined by p-valuations of G Let F be a group. A series of normal subgroups F = F1 ⊇ F2 ⊇ · · ·
(4.1)
is a p-series in F if [Fi , Fj ] ⊆ Fi+j and Fip ⊆ Fip (see Lazard [7] or Passman [12]). We will assume in this ∞ section that i=1 Fi = 1 and denote by Lp (F, Fi ) the restricted Lie algebra associated to p-series (4.1) and by Up (Lp (F, Fi )) its universal p-envelope. The algebra Lp (F, Fi ) =
∞
Fi /Fi+1
(4.2)
i=1
is obtained by the classical construction of Lazard [7]. If K is a field of characteristic p then p-series (4.1) defines a filtration An (KF ) (n = 0, 1, · · · ) in the ∞ group ring and n=0 Aj (KF ) = 0 (see Passman [12]), and we obtain a pseudovaluation v defined by this filtration; if the graded ring gr(KF ) associated to this filtration is a domain then v is a valuation in KF . Theorem 3. Let G be a group with a p-valuation φ whose values are natural numbers which are greater than 1 if p > 2. Then there exists a p-series G∗j (j = 1, 2, · · · ) with unit intersection such that the algebra Lp (G, G∗j ) is a free restricted abelian Lie algebra. Proof. We have in G the Lazard series defined by the function φ Gi = {x ∈ G|φ(x) ≥ i} (i = 1, 2, · · · )
(4.3)
Series (4.3) may begin with a term Gi0 with i0 > 1, but in this case we can define G1 = G2 = · · · = Gi0 and so assume that the first term of the series is G1 . We pick an arbitrary non-negative number k and define on G a new weight function f as f (g) = pφ(g)+k , or explicitly, f (g) = pi+k if g ∈ Gi \Gi+1
(4.4)
G∗pi+k = {g ∈ G|f (g) ≥ pi+k }
(4.5)
and let
Let f (g1 ) = pi1 +k , f (g2 ) = pi2 +k with i1 ≥ i2 . Then φ(g1 g2 ) ≥ i1 and f (g1 g2 ) ≥ pi1 +k
(4.6)
which means that g1 g2 ∈ G∗pi1 +k . Similarly, gi−1 , gi−1 ⊆ G∗pi1 +k ; we see that G∗pi1 +k is a subgroup. 1 2 Further, since [g1 , g2 ] ∈ Gi1 +i2 , f ([g1 , g2 ]) ≥ pi1 +i2 +k > pi1 +k + pi2 +k = f (g1 ) + f (g2 )
(4.7)
Let f (g) = pi+k . Since φ(g p ) = φ(g) + 1, we obtain that f (g p ) = pi+k+1
(4.8)
G = G∗1 = G∗2 = · · · = G∗p1+k ⊇ G∗p1+k +1 = G∗p1+k +2 = · · · = G∗p2+k ⊇ · · ·
(4.9)
We consider now the following series in G:
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32
We point out that the only terms in series (4.9) where a subgroup G∗j may properly contain the subgroup are the terms with j = pi+k . We obtain from relation (4.8) that (G∗j )p ⊆ G∗pj if j = pi+k ; hence this relation holds for an arbitrary term G∗j (j = 1, 2, · · · ). Similarly relation (4.7) implies that [G∗j1 , G∗j2 ] ⊆ G∗j1 +j2 for arbitrary j1 , j2 . We see that (4.9) is a p-series in G. We consider now the algebra Lp (G, G∗j ) associated to this series. Let g1 ∈ G∗j1 \G∗j1 +1 , g2 ∈ G∗j2 \G∗j2 +1 . Since [g1 , g2 ] ∈ G∗j1 +j1 +1 via relation (4.7) we obtain that the homogeneous components g˜1 and g˜2 of the elements g1 , g2 commute in the algebra Lp (G, G∗j ). We see that this algebra is abelian. Relation (4.8) yields that if g ∈ G∗j1 \G∗j1 +1 then g p ∈ G∗pj1 \G∗pj1 +1 . This means that the homogeneous component g˜ of this element is not nilpotent. It is known that an arbitrary restricted abelian Lie algebra R without nilpotent elements must be free abelian. In fact, let R[p] is the subalgebra generated by all the elements r[p] (r ∈ R) and let E be a system of elements of R which forms a basis of the quotient algebra R/R[p] , then a straightforward argument shows that E freely generates R. So the algebra Lp (G, G∗j ) is a free abelian restricted algebra and the proof is complete. 2 G∗j+1
Corollary 3. Let H be a finitely generated linear group over a field F of characteristic zero, let G ⊆ H be a normal subgroup of H obtained in Theorem 2, and let K be a field of finite characteristic p. Then the subgroups G∗j (j = 1, 2, · · · ) obtained by (4.5) are H-invariant as well as the ideals Aj (KG∗ ) (j = 0, 1, · · · ) of the filtration defined by this series. This filtration defines a valuation v with associated graded ring isomorphic to a polynomial algebra K[X]. Proof. Since series (4.3) is H-invariant via Theorem 2 the proof of Theorem 3 implies that the subgroups G∗j (j = 1, 2, · · · ) are H-invariant and so are the ideals Aj (KG) (j = 0, 1, · · · ). The graded ring associated to the filtration Aj (KG) (j = 0, 1, · · · ) is isomorphic to the p-envelope Up (Lp (G, G∗j )); this follows from a version of Quillen’s theorem [13] proven in Lichtman [8]. Theorem 3 yields that the algebra Lp (G, G∗j ) is a free restricted abelian algebra. Hence its p-envelope is isomorphic to a suitable polynomial ring K[X] and the assertion follows. 2 5. The proof of Theorem 4 We need the following Theorem 4 in the proof of Theorem 1. Theorem 4. Let R ∗ W be a crossed product of a domain R with a group W . Assume that there exists a non-negative W -invariant filtration Ri (i = 0, 1, · · · ) such that the associated graded ring gr(R) is an Ore domain. Then there exists a W -invariant division ring D generated by R such that the embedding R −→ D extends to an embedding of R ∗ W into a suitable crossed product D ∗ W . We will apply in the proof below the method developed in [9], section 2.1 and [10], section 4 (see also Cohn [3], section 2.6) in order to construct a division ring D which contains an isomorphic copy of R and is generated by it. This division ring D will be W -invariant because all the rings which will be considered ˜ is a system of elements in R ∗ W which during its construction are W -invariant, and we will prove that if W forms a W -basis for the crossed product R ∗ W then these elements will form a basis for a suitable crossed product D ∗ W . We also point out that a simple example of this kind is the case when R is an Ore domain with a ring of fractions D; in this case D is W -invariant and we obtain an immediate embedding of R ∗ W into D ∗ W . Proof of Theorem 4. Let v(r) be the valuation function defined in R by the filtration Ri (i = 0, 1, · · · ). We consider the Laurent polynomial ring (R ∗ W )[t, t−1 ] in a central variable t and extend the valuation v of R to the subring R[t, t−1 ] by the rule
A.I. Lichtman / Journal of Pure and Applied Algebra 221 (2017) 25–35
v(
m
33
ri ti ) = min{v(ri ) + i}
(5.1)
i
i=1
We obtain a valuation in R[t, t−1 ] which extends the valuation v of R; we will keep for this extended valuation the same notation v. Let T = {x ∈ R[t, t−1 ] | v(x) ≥ 0} be the ring of v-integers of R[t, t−1 ]. Clearly, the ring (R ∗ W )[t, t−1 ] is isomorphic to the crossed product of R[t, t−1 ] with W , T is a W -invariant subring of R[t, t−1 ] and T ⊇ R; we see that R ∗ W is a subring of a suitable crossed product T ∗ W . For every natural number n the epimorphism T −→ T /(t)n = Tn extends to an epimorphism of suitable crossed products (T ∗ W ) −→ (T ∗ W )/(t)n ∼ = Tn ∗ W . We consider now the inverse system of rings Tn ∗ W (n = 1, 2, · · · ) and the natural epimorphisms φn1 ,n2 : Tn1 ∗ W −→ Tn2 ∗ W for n1 > n2 ; the kernel of the epimorphism φn1 ,n2 is the ideal (t)n2 of the ring Tn1 ∗ W . Let M = T \(tT ) be the complement of the ideal tT , and let Mn be the image of M in the quotient ring T /(tT )n . Then (see [10], Lemma 4.3 and Proposition 4.1) the quotient ring T /(tT ) is isomorphic to the ring gr(R) and Mn is a right denominator set of regular elements in the quotient ring Tn = T /(tT )n (n = 1, 2, · · · ). Since the set M is W -invariant in T its image Mn is W -invariant in Tn ∗ W . A straightforward argument now implies that Mn remains a right denominator set of regular elements in Tn ∗ W and we can consider the ring of fractions of Tn ∗ W with respect to Mn ; this ring of fractions is isomorphic to a suitable crossed product (Tn Mn−1 ) ∗ W . A routine argument shows that every epimorphism φn1 ,n2 extends now to an epimorphism of the rings of fractions (Tn1 Mn−1 ) ∗ W −→ (Tn2 Mn−1 ) ∗ W and we obtain an inverse system 1 2 of rings (Tn Mn−1 ) ∗ W (n = 1, 2, · · · ); we also see that there is a natural embedding of the inverse system Tn ∗ W (n = 1, 2, · · · ) in this inverse system Tn ∗ W (n = 1, 2, · · · ). Let T˜ = lim Tn and S = lim Tn Mn−1 be ←−
←−
the inverse limits of the systems Tn and Tn M −1 respectively. We obtain natural embeddings T˜ ⊆ lim(Tn ∗ W )
(5.2)
S ⊆ lim(Tn Mn−1 ∗ W )
(5.3)
T ⊆ T˜ ⊆ S
(5.4)
←−
←−
and
The system of epimorphisms φn : T −→ Tn extends to systems of epimorphisms T˜ −→ Tn ; S −→ Tn Mn−1 (n = 1, 2, · · · ). We also have extensions of these epimorphisms lim(Tn ∗ W ) −→ Tn ∗ W (n = 1, 2, · · · )
(5.5)
lim(Tn Mn−1 ∗ W ) −→ (Tn Mn−1 ∗ W ) (n = 1, 2, · · · )
(5.6)
←−
and ←−
¯ be a system of elements in R ∗ W which forms a W -basis for this crossed product over R. We can Let W ¯ with their images in Tn ∗ W and in (Tn M −1 ) ∗ W and obtain that the system identify the elements of W ¯ of elements W forms also a W -basis for the crossed products Tn ∗ W and (Tn M −1 ) ∗ W . We consider now the subring of the ring lim(Tn Mn−1 ∗ W ) generated by the subring S = lim(Tn Mn−1 ) ←− ←− ¯ . The existence of the system of epimorphisms (5.6) implies by a routine and the system of elements W ¯ are left linearly independent over S, that they normalize S, and that the argument that the elements of W group of automorphisms of S generated by them is W . We see that the subring of the inverse limit of the ¯ is isomorphic to a suitable crossed product S ∗ W . Similarly, system (Tn Mn−1 ∗ W ) generated by S and W
34
A.I. Lichtman / Journal of Pure and Applied Algebra 221 (2017) 25–35
¯ is isomorphic the subring of this inverse limit generated by the subring lim Tn and the system of elements W ←− to a suitable crossed product T˜ ∗ W . We obtain also embeddings R ∗ W ⊆ T˜ ∗ W ⊆ S ∗ W
(5.7)
R ⊆ T˜ ⊆ S
(5.8)
which extend the embeddings
We obtain from section 2.1 in Lichtman [9] or section 4.3 in Lichtman [10] (see also Cohn [3], Theorem 2.6.2) that S is a local domain with radical tS and S/(tS) is isomorphic to the division ring of fractions of the Ore domain T /(tT ) ∼ = gr(R). Further, the subset {t, t2 , · · · } of S is a central subsemigroup in S ∗ W and the ring of fractions of S with respect to this subsemigroup is a division ring Q(S). We consider now the ring of fractions of S ∗ W with respect to the subsemigroup {t, t2 , · · · } and obtain an embedding (S ∗ W ) ⊆ (Q(S) ∗ W ). The division subring D generated by R is invariant with respect to all the automorphisms generated by the elements w ∈ W and it is contained in Q(S). We obtain from this that (R ∗ W ) ⊆ (D ∗ W ) and the proof is complete. 2 6. The proof of Theorem 1 We observe first that if we prove the statements about the existence of the normal subgroup G of finite index, about the division ring generated by the group ring KG, and about isomorphism (1.1) then this will imply that the crossed product S ∼ = Δ ∗ (H/G) is a semisimple Artinian ring because the order of H/G does not divide the characteristic of Δ (see Jacobson [4], 4.18, Theorem 48). Lemma 3. Let H be a group, let G be a normal subgroup of finite index, in H, and let Kj (j ∈ J) be a family of fields. Assume that for every group ring Kj H (j ∈ J) there exists an isomorphic embedding of Kj H into a ring Sj such that the group ring Kj G generates an H-invariant skew field Δj ⊆ Sj such that Sj ∼ = Δj ∗ (H/G)
(6.1)
where the embedding Δj ⊆ Sj extends the embedding Kj G ⊆ Kj H and isomorphism (6.1) extends the isomorphism Kj H ∼ = (Kj G) ∗ (H/G). Let F be an arbitrary ultrafilter on the set J and let K be an arbitrary subfield of the ultraproduct ( j∈J Kj )/F. Then Theorem 1 holds for the group ring KH. Proof. We have embeddings KH ⊆ (
j∈J
Kj H)/F ⊆
(Δj ∗ (H/G))/F
(6.2)
j∈J
It is known that the ultraproduct of division rings is a division ring; hence Δ(J) = ( j∈J Δj )/F is a division ring. Further, it is well known and it is easy to verify that the ultrapower ( j∈J (H/G))/F of the finite group H/G is isomorphic to H/G, and thus we obtain that the ultraproduct j∈J (Δj ∗(H/G))/F is isomorphic to a suitable crossed product ΔJ ∗(H/G). We obtain an embedding KH ∼ = (KG) ∗(H/G) ⊆ (ΔJ ∗(H/G)) with J KG embedded into Δ . We denote now by Δ the division subring of ΔJ generated by KG, and by S the ¯ 1 = 1, h ¯ 2, · · · , h ¯ k is a system of coset subring of ΔJ ∗(H/G) generated by KH and Δ. We have Δ ⊆ S and if h representatives for the group H/G then these elements belong to S and normalize the division subring Δ; they are left linearly independent over Δ because they are linearly independent over ΔJ . This implies that the subring generated by these elements and Δ is isomorphic to Δ ∗ (H/G) and that S ∼ = Δ ∗ (H/G). This completes the proof. 2
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Proof of Theorem 1. Statement i). We consider first the case when K is an algebraic number field. Pick a prime number p from the set of prime numbers obtained in Theorem 2 such that the ring of p-adic integers Ω contains the field K. Theorems 2 and 4 imply that there exists a ring S0 which contains ΩH and such that the group ring ΩG generates a division subring Δ0 and S0 ∼ = Δ0 ∗ (H/G). We consider the division subring Δ ⊆ Δ0 generated by KG and the subring S ⊆ S0 generated by KH and Δ; a straightforward argument yields that S ∼ = Δ ∗ (H/G). This completes the proof for the special case when K is an algebraic number field. Now consider the case when K is a finitely generated field, i.e. K is the field of fractions of a finitely generated domain Q(x1 , x2 , · · · , xk ). Lemma 2 implies that K can be embedded into a suitable ultraproduct of algebraic number fields Kj (j ∈ J) and the assertion follows from Lemma 3. Finally, let K be an arbitrary field of characteristic zero. Let Kγ (γ ∈ Γ) be the system of all the finitely generated subfields of K. The same argument as in 1.L.7 of Kegel and Wehrfritz [5] yields that the field K can be embedded into a suitable ultraproduct ( γ∈Γ Kγ )/F and the assertion follows now from Lemma 3. This completes the proof of statement i). Statement ii). This follows immediately from Corollary 3 and Theorem 4 by the same argument as in statement i) if p = 2; if p = 2 we can assume (see the remark after the proof of Theorem 2) that the p-values of the elements of G are greater than 1 and once again the same argument as in i) completes the proof. 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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