HOMOLOGY STABILITY FOR UNITARY GROUPS

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1 HOMOLOGY STABILITY FOR UNITARY GROUPS B. MIRZAII AND W. VAN DER KALLEN Abstract. In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results. Our motivation for this work has been to prove homology stability for unitary groups that is as strong as in the linear case [4]. For example we want homology stability of these groups over any commutative finite dimensional noetherian ring. This work is the continuation of our previous work in this direction [3]. After that work, we found that our method, with a little modification, can also be applied in this more general setting. Thus our aim is now to prove that homology stabilizes of the unitary groups over rings with finite unitary stable rank. To do so, as in [3], we prove that the poset of isotropic unimodular sequences is highly connected. Homology stability for symplectic groups and orthogonal groups will appear as a special case. Our approach is similar to [3], especially section 6. We first compare unimodular sequences of R n with a suitable hyperbolic basis of R n, and we will use that to prove that certain posets are highly connected. For this we need the higher connectivity of posets of unimodular sequences due to second author. The higher connectivity of the poset of isotropic unimodular sequences then follows inductively as in [3]. We conclude with the homology stability theorem. 1. Posets of unimodular sequences Let R be an associative ring with unit. A vector (r 1,..., r n ) R n is called unimodular if there exist s 1,..., s n R such that Σ n i=1 s ir i = 1, or equivalently if the submodule generated by this vector is a free summand of the right R-module R n. We denote the standard basis of R n by e 1,..., e n. If n m, we assume that R n is the submodule of R m generated by e 1,..., e n R m. We denote the set of all unimodular elements of R n by Um(R n ). We say that a ring R satisfies the stable range condition (S m ), if m 1 is an integer so that for every unimodular vector (r 1,..., r m, r m+1 ) R m+1, there exist t 1,..., t m in R such that (r 1 + t 1 r m+1,..., r m + t m r m+1 ) R m is unimodular. We say that R has stable rank m, we denote it with sr(r) = m, if m is the least number such that (S m ) holds. If such a number does not exist we say that sr(r) =. 1

2 B. MIRZAII AND W. VAN DER KALLEN We refer to [3] or [4] for notations such as O(V ). In particular U(R n ) denotes the subposet of O(R n ) consisting of unimodular sequences. Recall that a sequence of vectors v 1,..., v k in R n is called unimodular when v 1,..., v k is basis of a free direct summand of R n. Note that if (v 1,..., v k ) O(R n ) and if n m, it is the same to say that (v 1,..., v k ) is unimodular as a sequence of vectors in R n or as a sequence of vectors in R m. We call an element (v 1,..., v k ) of U(R n ) a k-frame. Theorem 1.1 (Van der Kallen). Let R be a ring with sr(r) < and n m. Let δ be 0 or 1. Then (i) O(R n + e n+1 δ) U(R m ) is (n sr(r) 1)-connected. (ii) O(R n + e n+1 δ) U(R m ) v is (n sr(r) v 1)-connected for all v U(R m ). Proof. See [4, Thm..6].. Hyperbolic spaces and some posets Let there be an involution on R, that is an automorphism of the additive group of R, R R with r r, such that r = r and rs = s r. Let ɛ be an element in the center of R such that ɛɛ = 1. Set R ɛ =: {r ɛr : r R} and R ɛ =: {r R : ɛr = r} and observe that R ɛ R ɛ. A form parameter relative to the involution and ɛ is a subgroup Λ of (R, +) such that R ɛ Λ R ɛ and rλr Λ, for all r R. Notice that R ɛ and R ɛ are form parameters. We denote them by Λ min and Λ max, respectively. If there is an s in the center of R such that s + s R, in particular if R, then Λ min = Λ max. Let e i,j (r) be the n n-matrix with r R in the (i, j) place and zero elsewhere. Consider Q n = Σ n i=1 e i 1,i(1) M n (R) and F n = Q n +ɛ t Q n = Σ n i=1 (e i 1,i(1) + e i,i 1 (ɛ)) GL n (R). Define the bilinear map h : R n R n R by h(x, y) = Σ n i=1 (x i 1y i + ɛx i y i 1 ) and q : R n R/Λ by q(x) = Σ n i=1 x i 1x i mod Λ, where x = (x 1,..., x n ), y = (y 1,..., y n ) and x = (x 1,..., x n ). The triple (R n, h, q) is called a hyperbolic space. By definition the unitary group relative Λ is the group U ɛ n(r, Λ) := {A GL n (R) : h(ax, Ay) = h(x, y), q(ax) = q(x), x, y R}. For more general definitions and the properties of these spaces and groups see []. Example.1. (i) Let ɛ = 1 and let the involution be the identity map id R, then Λ max = R. If Λ = Λ max = R then U ɛ n (R, Λ) =: Sp n(r) is the usual symplectic group. Note that R is commutative in this case. (ii) Let ɛ = 1 and let the involution be the identity map id R, then Λ min = 0. If Λ = Λ min = 0 then U ɛ n (R, Λ) =: O n(r) is the usual orthogonal group. As in the symplectic case, R is necessarily commutative. (iii) Let ɛ = 1 and the involution is not the identity map id R. If Λ = Λ max then U ɛ n (R, Λ) =: U n(r) is the classical unitary group corresponding to the involution.

3 HOMOLOGY STABILITY FOR UNITARY GROUPS 3 (iv) If Λ = Λ max = R then U ɛ n (R, Λ) = {A GL n(r) : h(ax, Ay) = h(x, y) for all x, y R} = {A GL n (R) : t AF n A = F n }. (v) If Λ = Λ min = 0 then U ɛ n (R, Λ) = {A GL n(r) : q(ax) = q(x) for all x R} = {A GL n (R) : t AQ n A = Q n }. Let σ be the permutation of the set of natural numbers given by σ(i) = i 1 and σ(i 1) = i. For 1 i, j n, i j, and every r R define I n + e i,j (r) if i = k 1, j = σ(i), r Λ I n + e i,j (r) if i = k, j = σ(i), r Λ E i,j (r) = I n + e i,j (r) + e σ(j),σ(i) ( r) if i + j = k, i j I n + e i,j (r) + e σ(j),σ(i) ( ɛ 1 r) if i σ(j), i = k 1, j = l I n + e i,j (r) + e σ(j),σ(i) (ɛr) if i σ(j), i = k, j = l 1 where I n is the identity element of GL n (R). It is easy to see that E i,j (r) Un ɛ (R, Λ). Let EU ɛ n(r, Λ) be the group generated by the E i,j (r), r R. We call it elementary unitary group. A nonzero vector x R n is called isotropic if q(x) = 0. This shows automaticly that if x is isotropic then h(x, x) = 0. We say that a subset S of R n is isotropic if for every x S, q(x) = 0 and for every x, y S, h(x, y) = 0. If h(x, y) = 0, then we say that x is perpendicular to y. We denote by S the submodule of R n generated by S, and by S the submodule consisting of all the elements of R n which are perpendicular to all the elements of S. From now, we fix an involution, an ɛ, a form parameter Λ and we consider the triple (R n, h, q) as defined above. Definition. (Transitivity condition). Let r R and define Cr ɛ(rn, Λ) = {x Um(R n ) : q(x) = r mod Λ}. We say that R satisfies the transitivity condition (T n ), if EU ɛ n (R, Λ) acts transitively on Cr(R ɛ n, Λ), for every r R. It is easy to see that e 1 + re Cr(R ɛ n, Λ). Definition.3 (Unitary stable range). We say that a ring R satisfies the unitary stable range condition if R satisfies the conditions (S m ) and (T m+1 ). We say that R has unitary stable rank m, we denote it with usr(r), if m is the least number such that (S m ) and (T m+1 ) are satisfied. If such a number does not exist we say that usr(r) =. Clearly sr(r) usr(r). Remark.4. Our definition of unitary stable range is a little different than the one in [, Chap. VI 4.6]. In fact usr(r) + 1 = m + 1 = USR m where USR m is the unitary stable rank as defined in []. Example.5. Let R be a commutative noetherian ring where the dimension d of the maximal spectrum Mspec(R) is finite. If A is a finite R-algebra then usr(a) d + (see [5, Thm.8], [, Thm ]). Lemma.6. Let R be a ring with usr(r) <. Assume n usr(r) + k + 1 and (v 1,..., v k ) U(R n ). Then there is a hyperbolic basis {x 1, y 1,..., x n, y n } of R n such that v 1,..., v k x 1, y 1,..., x k, y k.

4 4 B. MIRZAII AND W. VAN DER KALLEN Proof. The proof is by induction on k. If k = 1, by definition of unitary stable range there is an E EU ɛ n (R, Λ) such that Ev 1 = e 1 + re. So the base of induction is true. Let k and assume the induction hypothesis. Arguing as in the base of the induction we can assume that v 1 = (1, r, 0,..., 0). Let W = e + Σ n i= e ir. By theorem 1.1, the poset F := O(W ) U(R n ) (v1,...,v k ) is ((n 1) sr(r) k 1)-connected. Since n usr(r) + k + 1 sr(r) + k + 1, it follows that F is not empty. Choose (w, v 1,..., v k ) U(R n ) where w W. Then (w, v 1 wr, v,..., v k ) U(R n ). But (w, v 1 rw) is a hyperbolic pair, so there is an E EU ɛ n(r, Λ) such that Ew = e n 1, E(v 1 wr) = e n by [, Chap. VI, Thm ]. Let (Ew, E(v 1 wr), Ev,..., Ev k ) =: (w 0, w 1,..., w k ) where w i = (r i,1,..., r i,n ). Put u i = w i e n 1 r i,n 1 e n r i,n for i k. Then (u,..., u k ) U(R n ). Now by induction there is a hyperbolic basis {a, b,..., a n, b n } of R n such that u i a, b,..., a k, b k. Let a 1 = e n 1 and b 1 = e n. Then w i a 1, b 1,..., a k, b k. But Ev 1 = w 1 + Ewr = e n + e n 1 r, Ev i = w i for i k and considering x i = E 1 a i, y i = E 1 b i, one sees that v 1,..., v k x 1, y 1,..., x k, y k. Let Z n = {x R n : q(x) = 0} and put U (R n ) = O(Z n ) U(R n ). Lemma.7. Let R be a ring with sr(r) < and n m. Then (i) O(R n ) U (R m ) is (n sr(r) 1)-connected, (ii) O(R n ) U (R m ) v is (n sr(r) v 1)-connected for every v U (R m ), (iii) O(R n ) U (R m ) U(R m ) v is (n sr(r) v 1)-connected for every v U(R m ). Proof. Let W = e, e 4,..., e n and F := O(R n ) U (R m ). It is easy to see that O(W ) F = O(W ) U(R m ) and O(W ) F u = O(W ) U(R m ) u for every u U (R m ). By theorem 1.1, the poset O(W ) F is (n sr(r) 1)- connected and the poset O(W ) F u is (n sr(r) u 1)-connected for every u F. It follows from lemma [4,.13 (i)] that F is (n sr(r) 1)-connected. The proof of (ii) and (iii) is similar to the proof of (i). Lemma.8. Let R be a ring with usr(r) < and let (v 1,..., v k ) U (R n ). If n usr(r) + k + 1 then O( v 1,..., v k ) U (R n ) (v1,...,v k ) is (n usr(r) k 1)-connected. Proof. By lemma.6 there is a hyperbolic basis {x 1, y 1,..., x n, y n } of R n such that v 1,..., v k x 1, y 1,..., x k, y k. Let W = x k+1, y k+1,..., x n, y n R (n k) and F := O( v 1,..., v k ) U (R n ) (v1,...,v k ). It is easy to see that O(W ) F = O(W ) U (R n ). Let V = v 1,..., v k, then x 1, y 1,..., x k, y k = V P where P is a (finitely generated) projective module. Consider (u 1,..., u l ) F \O(W ) and let u i = x i + y i where x i V and y i P W. One should notice that (u 1 x 1,..., u l x l ) U(R n ) and not necessarily in U (R n ). It is not difficult to see that O(W ) F (u1,...,u l ) = O(W ) U (R n ) U(R n ) (u1 x 1,...,u l x l ). By lemma.7, O(W ) F is (n k usr(r) 1)-connected and O(W ) F u

5 HOMOLOGY STABILITY FOR UNITARY GROUPS 5 is (n k usr(r) u 1)-connected for every u F \O(W ). It follows from lemma [4,.13 (i)] that F is (n usr(r) k 1)-connected. 3. Posets of isotropic and hyperbolic unimodular sequences Let IU(R n ) be the set of sequences (x 1,..., x k ), x i R n, such that x 1,..., x k form a basis for an isotropic direct summand of R n. Let HU(R n ) be the set of sequences ((x 1, y 1 ),..., (x k, y k )) such that (x 1,..., x k ), (y 1,..., y k ) IU(R n ), h(x i, y j ) = δ i,j, where δ i,j is the Kronecker delta. We call IU(R n ) and HU(R n ) the poset of isotropic unimodular sequences and the poset of hyperbolic unimodular sequences, respectively. For 1 k n, let IU(R n, k) and HU(R n, k) be the set of all elements of length k of IU(R n ) and HU(R n ) respectively. We call the elements of IU(R n, k) and HU(R n, k) the isotropic k-frames and the hyperbolic k-frames, respectively. Define the poset MU(R n ) as the set of ((x 1, y 1 ),..., (x k, y k )) O(R n R n ) such that, (i) (x 1,..., x k ) IU(R n ), (ii) for each i, either y i = 0 or (x j, y i ) = δ ji, (iii) y 1,..., y k is isotropic. We identify IU(R n ) with MU(R n ) O(R n {0}) and HU(R n ) with MU(R n ) O(R n (R n \{0})). Lemma 3.1. Let R be a ring with usr(r) <. If n usr(r) + k then EU ɛ n(r, Λ) acts transitively on IU(R n, k) and HU(R n, k). Proof. The proof is by induction on k. If k = 1, by definition EU ɛ n(r, Λ) acts transitively on IU(R n, 1) and by [, Chap. VI, Thm ] the group EU ɛ n (R, Λ) acts transitively on HU(Rn, 1). The rest is an easy induction and the fact that for every isotropic k-frame (x 1,..., x k ) there is an isotropic k-frame (y 1,..., y k ) such that ((x 1, y 1 ),..., (x k, y k )) is a hyperbolic k-frame [, Chap. I, Cor ]. Lemma 3.. Let R be a ring with usr(r) <, and let n usr(r) + k. Let ((x 1, y 1 ),..., (x k, y k )) HU(R n ), (x 1,..., x k ) IU(R n ) and V = x 1,..., x k. Then (i) IU(R n ) (x1,...,x k ) IU(R (n k) ) V, (ii) HU(R n ) MU(R n ) ((x1,0),...,(x k,0)) HU(R n ) ((x1,y 1 ),...,(x k,y k )) V V, (iii) HU(R n ) ((x1,y 1 ),...,(x k,y k )) HU(R (n k) ). Proof. See [1], the proof of lemma 3.4 and the proof of Thm. 3.. For a real number l, by l we mean the largest integer n with n l. Theorem 3.3. The poset IU(R n ) is n usr(r) 3 -connected and IU(R n ) x is n usr(r) x 3 -connected for every x IU(R n ). Proof. If n usr(r) + 1, the result is clear, so let n > usr(r) + 1. Let X v = IU(R n ) U (R n ) v O( v ), for every v U (R n ), and put X := v F X v where F = U (R n ). It follows from lemma 3.1 that IU(R n ) n usr(r) 1

6 6 B. MIRZAII AND W. VAN DER KALLEN X. So to treat IU(R n ), it is enough to prove that X is n usr(r) 3 - connected. First we prove that X v is n usr(r) v 3 -connected for every v F. The proof is by descending induction on v. If v > n usr(r) 1, then n usr(r) v 3 < 1. In this case there is nothing to prove. If n usr(r) v n usr(r) 1, then n usr(r) v 3 = 1, so we must prove that X v is nonempty. This follows from lemma.6. Now assume v n usr(r) 3 and assume by induction that X w is n usr(r) w 3 - connected for every w, with w > v. Let l = n usr(r) v 3, and observe that n v usr(r) 1 l +. Put T w = IU(R n ) U (R n ) wv O( wv ) where w G v = U (R n ) v O( v ) and put T := w G v T w. It follows by lemma.6 that (X v ) n v usr(r) 1 T. So it is enough to prove that T is l-connected. The poset G v is l-connected by lemma.8. By induction, T w is n usr(r) v w 3 -connected. But min{l 1, l w +1} n usr(r) v w 3, so T w is min{l 1, l w + 1}-connected. For every y T, A y = {w G v : y T w } is isomorphic to U (R n ) vy O( vy ) so by lemma.8, it is (l y + 1)-connected. Let w G v with w = 1. For every z T w we have wz X v, so T w is contained in a cone, call it C w, inside X v. Put C(T w ) = T w (C w ) n v usr(r) 1. Thus C(T w ) T. The poset C(T w ) is l-connected because C(T w ) n v usr(r) 1 = (C w ) n v usr(r) 1. Now by theorems 1.1 and [3, 4.7], T is l-connected. In other words, we have now shown that X v is n usr(r) v 3 -connected. By knowing this one can prove, in a similar way, that X is n usr(r) 3 -connected. (Just pretend that v = 0.) Now consider the poset IU(R n ) x for an x = (x 1,..., x k ) IU(R n ). The proof is by induction on n. If n = 1, everything is easy. Similarly, we may assume n usr(r) x 1 0. Let l = n usr(r) x 3. By lemma 3., IU(R n ) x IU(R (n x ) ) V, where V = x 1,..., x k. In the above we proved that IU(R (n x ) ) is l-connected and by induction, the poset IU(R (n x ) ) y is n x usr(r) y 3 -connected for every y IU(R (n x ) ). But l y n x usr(r) y 3. So IU(R (n x ) ) V is l-connected by lemma [3, 4.1]. Therefore IU(R n ) x is l-connected. Theorem 3.4. The poset HU(R n ) is n usr(r) 4 -connected and HU(R n ) x is n usr(r) x 4 -connected for every x HU(R n ). Proof. The proof is by induction on n. If n = 1, then everything is trivial. Let F = IU(R n ) and X v = HU(R n ) MU(R n ) v, for every v F. Put X := v F X v. It follows from lemma 3.1 that HU(R n ) n usr(r) 1 X. Thus to treat HU(R n ), it is enough to prove that X is n usr(r) 4 - connected, and we may assume n usr(r) +. Take l = n usr(r) 4 and V = v 1,..., v k, where v = (v 1,..., v k ). By lemma 3., there is

7 HOMOLOGY STABILITY FOR UNITARY GROUPS 7 an isomorphism X v HU(R (n v ) ) V V, if n sr(r) + v. By induction HU(R (n v ) ) is n v usr(r) 4 -connected and again by induction HU(R (n v ) ) y is n v usr(r) y 4 -connected for every y HU(R (n v ) ). So by lemma [3, 4.1], X v is n v usr(r) 4 -connected. Thus the poset X v is min{l 1, l v + 1}-connected. Let x = ((x 1, y 1 ),..., (x k, y k )). It is easy to see that A x = {v F : x X v } IU(R n ) (x1,...,x k ). By the above theorem 3.3, A x is n usr(r) k 3 -connected. But l x + 1 n usr(r) k 3, so A x is (l x + 1)-connected. Let v = (v 1 ) F, v = 1, and let D v := HU(R n ) (v1,w 1 ) HU(R (n 1) ) where w 1 R n is a hyperbolic dual of v 1 R n. Then D v X v and D v is contained in a cone, call it C v, inside HU(R n ). Take C(D v ) := D v (C v ) n usr(r) 1. By induction D v is n 1 usr(r) 4 -connected and so (l 1)-connected. Let Y v = X v C(D v ). By the Mayer-Vietoris theorem and the fact that C(D v ) is l-connected, we get the exact sequence H l (D v, Z) (iv) H l (X v, Z) H l (Y v, Z) 0. where i v : D v X v is the inclusion. By induction (D v ) w is n 1 usr(r) w 4 -connected and so (l w )-connected, for w D v. By lemma [3, 4.1(i)] and lemma 3., (i v ) is an isomorphism, and by exactness of the above sequence we get H l (Y v, Z) = 0. If l 1 by the Van Kampen theorem π 1 (Y v, x) π 1 (X v, x)/n where x D v and N is the normal subgroup generated by the image of the map (i v ) : π 1 (D v, x) π 1 (X v, x). Now by lemma [3, 4.1(ii)], π 1 (Y v, x) is trivial. Thus by the Hurewicz theorem [3,.1], Y v is l-connected. By having all this we can apply theorem [3, 4.7]and so X is l-connected. The fact that HU(R n ) x is n usr(r) x 4 -connected follows from the above and lemma Homology stability From theorem 3.4 one can get the homology stability of unitary groups as Charney proved in [1, Sec. 4]. Here we only formulate the theorem and for the proof we refer to Charney s paper. Theorem 4.1. Let R be a ring with usr(r) <. Then for every abelian group L the homomorphism ψ n : H i (Un ɛ (R, Λ), L) H i(un+ ɛ (R, Λ), L) is surjective for n i + usr(r) + 3 and bijective ( for) n i + usr(r) + 4, A 0 where ψ n : Un ɛ (R, Λ) U n+ ɛ (R, Λ), A. 0 I Proof. See [1, Sec. 4]. References [1] Charney, R. A generalization of a theorem of Vogtmann. J. Pure Appl. Algebra 44 (1987),

8 8 B. MIRZAII AND W. VAN DER KALLEN [] Knus, M. A. Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften, 94. Springer-Verlag, Berlin, [3] Mirzaii, B.; Van der Kallen, W. Homology stability for symplectic groups. [4] Van der Kallen, W. Homology stability for linear groups. Invent. Math. 60 (1980), [5] Vaserstein, L. N. Stabilization of unitary and orthogonal groups over a ring with involution. Math. USSR Sbornik 10 (1970), no. 3, mirzaii@math.uu.nl vdkallen@math.uu.nl