MATHEMATICS

B. B. VENKOV

COHOMOLOGY OF GROUPS OF UNITS IN DIVISION ALGEBRAS

(Presented by Academician L. S. Pontryagin on 18 XI 1960)

Let \(\mathfrak A\) be a division algebra of rank \(n>1\) over the field of rational numbers \(Q\), let \(J=[\tau_1,\ldots,\tau_n]\) be a maximal order in \(\mathfrak A\), and let \(G\) be the group of units in \(J\) of norm \(+1\). The aim of the present note is to prove the following periodicity property of the cohomology of the group \(G\):

For any \(G\)-module \(A\) and all sufficiently large \(k\) there is an isomorphism

\[ H^k(G,A)\approx H^{n-1+k}(G,A) \tag{1} \]

(more precisely, see Theorem 1).

1. The proof of (1) is based on Grothendieck’s theory of \(G\)-bundles \((^1)\). Let \(X\) be a topological space with a group of transformations \(G\); let \(Y\) be the quotient space \(X/G\), and let \(f:X\to Y\) be the canonical map. Denote by \(C^{X(G)}\) (respectively, \(C^Y\)) the abelian category of \(G\)-bundles of abelian groups over \(X\) (respectively, bundles of abelian groups over \(Y\)). Denote by \(\Gamma^G\) the functor assigning to a \(G\)-bundle \(A\) the group of its \(G\)-invariant sections over \(X\), and by \(f_*^G\) the functor from \(C^{X(G)}\) to \(C^Y\) given by the formula \((f_*^G A)V=(A(f^{-1}(V)))^G\). The right derived functors of \(\Gamma^G\) will be denoted by \(H^p(X;G,A)\), and the right derived functors of \(f_*^G\) by \(\mathscr H^p(G,A)\). The functor \(\Gamma^G\) can be decomposed in two ways into a composition of functors \(\Gamma^G=(\Gamma_X)^G=\Gamma_Y(f_*^G)\). This gives two spectral functors converging to \(H^n(X;G,A)\), whose second terms are respectively equal to \(I_2^{pq}=H^p(Y,\mathscr H^q(G,A))\) and \(II_2^{pq}=H^p(G,H^q(X,A))\). Suppose three \(G\)-bundles \(A,B,C\) and a \(\cup\)-product \(A\otimes B\to C\), compatible with the \(G\)-structures on \(A,B,C\), are given. The product structure can be extended in a natural way to the derived functors of the functors \(\Gamma^G\) and \(f_*^G\). This gives mappings
   \[ H^p(X;G,A)\otimes H^q(X;G,B)\to H^{p+q}(X;G,C) \]
   and
   \[ \mathscr H^p(G,A)\otimes \mathscr H^q(G,B)\to \mathscr H^{p+q}(G,C), \]
   possessing the usual properties of products. In what follows we shall be especially interested in the case when \(G\) is discrete on the space \(X\); in this case, for any point \(y\in Y\),

\[ \mathscr H^p(G,A)_{(y)} \mathrel{\mathop{\approx}^{\theta_y}} H^p(G_x,A), \tag{2} \]

where \(x\in f^{-1}(y)\), and \(G_x\) is the stabilizer of the point \(x\). This isomorphism carries the product structure in \(\mathscr H^*(G,)_{(y)}\) to the ordinary multiplication in the cohomology groups of groups.

The product \(A\otimes B\to C\) is also carried in a natural way to the spectral sequences \(I\) and \(II\). Namely, mappings
\[ I_r^{pq}(A)\otimes I_r^{p'q'}(B)\to I_r^{p+p',q+q'}(C) \]
are defined, compatible with the structures of differential groups on \(I_r\) and compatible with the product structure

in the limiting term. Moreover, on \(I_2\) this product coincides, up to sign, with the natural product present in \(H^{*}(Y,\mathcal H^{*}(G))\). The situation is analogous for the second spectral sequence as well.

Let us also note the following naturality property of the functors under consideration. Let \(\Pi\) be a subgroup of the group \(G\); let \(Y'\) be the quotient space of \(X\) by \(\Pi\); and let \(I'\) and \(II'\) be the spectral sequences \(I\) and \(II\), considered with respect to \(\Pi\). The inclusion \(\Pi \subset G\) induces a homomorphism of spectral sequences \(I\to I'\) and \(II\to II'\), which on \(I_2\) and \(II_2\) coincides with the natural homomorphisms, for example on \(II_2\) with the restriction homomorphism.

1. We return to the proof of (1). Thus, \(\mathfrak A\) is a division algebra, \(J=[\tau_1,\ldots,\tau_n]\) is its maximal order; \(G\) is the group of proper units in \(J\) (norm \(+1\)). Denote by \(x\to A_x\), \(x\in\mathfrak A\), the left regular representation of the algebra \(\mathfrak A\) in the basis \(\tau_1,\ldots,\tau_n\). Since \(\mathfrak A\) has no zero divisors, all matrices \(A_x\), \(x\in\mathfrak A\), \(x\ne1\), do not have \(+1\) as an eigenvalue and, in particular, the group \(G\) acts without fixed points on the space \(X=\mathfrak A\otimes R-(0)\) (\(R\) is the field of real numbers).

Lemma. Let \(\pi\) be a finite subgroup of order \(s\) of the group \(G\); then

\[ H^n(\pi,Z)=Z/sZ. \]

Moreover, in \(H^n(G,Z)\) there is an element \(\xi\) such that, for all finite subgroups \(\pi\subset G\), \(i(G,\pi)\xi\) is a generator of the cyclic group \(H^n(\pi,Z)\) (\(i(G,\pi)\) is the restriction homomorphism).

Let \(II\) and \(II'\) be the spectral sequences for the pairs \((G,X)\) and \((\pi,X)\) with constant sheaf of coefficients \(Z\). The space \(X\) is a homological sphere \(S^{n-1}\), and therefore in these spectral sequences there is only one nontrivial differential \(d_n^{p,n-1}: II_n^{p,n-1}\to II_n^{p+n,0}\) \((d_n^{\prime p,n-1}: II_n^{\prime p,n-1}\to II_n^{\prime p+n,0})\). The first part of the lemma follows from the fact that the group \(\pi\) is discrete on \(X\) (since it is finite), and the transformations from \(G\) preserve the orientation of the space \(X\); moreover \(d_n^{\prime 0,n-1}(1)\) is a generating element for \(H^n(\pi,Z)\). As the element \(\xi\in H^n(G,Z)\) one may take \(d_n^{0,n-1}(1)\); indeed, if we denote by \(h\) the homomorphism of spectral sequences \(II\to II'\) induced by the inclusion \(\pi\subset G\), then we shall have

\[ h d_n^{0,n-1}(1)=d_n^{\prime 0,n-1}(1), \]

while on \(II_2^{p,0}\) \(h\) coincides with the restriction homomorphism.

Denote by \(\smile: H^p(G,Z)\otimes H^q(G,A)\to H^{p+q}(G,A)\) the product induced by the product \(Z\otimes A\to A,\; 1\otimes a\to a\).

Theorem 1. Let \(G\) be the group of proper units in \(\mathfrak A\); let \(n\) be the rank of \(\mathfrak A\) over the field of rational numbers \(Q\); and let \(A\) be an arbitrary \(G\)-module. In the group \(H^n(G,Z)\) there is an element \(\xi\) such that for any \(k\ge n(n+1)/2\) multiplication by \(\xi\) is an isomorphism

\[ H^k(G,A)\approx H^{n+k}(G,A). \]

Proof. We shall prove the following more general assertion. Let \(G\) be a discrete subgroup of the group \(GL(n,R)\), consisting of matrices of determinant \(+1\), and suppose that the assertion of the lemma holds for \(G\); then Theorem 1 is true for \(G\).

Denote by \(\mathfrak P_n\) the space of real positive quadratic forms in \(n\) variables, and by \(X\) the discriminant surface in \(\mathfrak P_n\), i.e. the set of \(f\in\mathfrak P_n\) with \(\det f=1\). From our assumptions on \(G\) it follows that \(G\) acts discretely on \(X\). Denote

by \(Z\) and \(A\) the constant sheaves over \(X\) with stalks \(Z\) and \(A\), and consider the spectral sequences \(I(Z)\) and \(I(A)\) for the pair \((G,X)\). First of all it is clear that they converge to \(H^*(G,Z)\) and \(H^*(G,A)\), since the space \(X\) is contractible to a point. The homomorphism
\(\varphi: H^q(G,Z)\to \Gamma_Y(\mathscr H^q(G,Z))\), obtained from the spectral sequence \(I\), is constructed as follows: if \(\eta\in H^q(G,Z)\), then for any point \(y\in Y\), \(\varphi\) on the stalk over the point \(y\) coincides with the restriction homomorphism, if \(\mathscr H^q(G,Z)_{(y)}\) is identified with \(H^q(G_x,Z)\), \(x\in f^{-1}(y)\), by means of the isomorphism (2),
\[ (\varphi\eta)(y)=\theta_y^{-1} i(G,G_x)\eta . \]
Thus, if \(\eta\in H^q(G,Z)\), then the function
\(\psi(\eta): y\mapsto \theta_y^{-1}i(G,G_x)\eta\) belongs to \(I_2^{0,q}(Z)\) and \(d_r^{0,q}\psi=0\) for \(r=2,3,\ldots\). Let \(\xi\in H^n(G,Z)\) be an element satisfying the conclusion of the lemma; \(\psi(\xi)\in I_r^{0,n}(Z)\), \(r\ge 2\); let us see how the spectral sequence \(I(A)\) behaves under multiplication by \(\psi(\xi)\). We shall prove that multiplication by \(\psi(\xi)\) is an isomorphism
\[ I_r^{p q}(A)\longrightarrow I_r^{p,q+n}(A),\qquad q\ge \frac{n(n+1)}{2} \tag{3} \]
for all \(r\ge 2\). Since \(d_r^{0,n}\psi(\xi)=0\) for \(r\ge 2\), it is enough to prove this for \(r=2\). In this case the mapping (3) becomes the homomorphism
\[ H^p(Y,\mathscr H^q(G,A))\longrightarrow H^p(Y,\mathscr H^{q+n}(G,A)), \]
induced by multiplication of the coefficients by \(\psi(\xi)\); therefore it suffices to prove that multiplication by \(\psi(\xi)\) is an isomorphism of sheaves
\[ \mathscr H^q(G,A)\longrightarrow \mathscr H^{q+n}(G,A),\qquad q>0. \]
Let \(y\in Y\); then the transformation \(\theta_y\) carries \(\alpha_y\) into multiplication by \(i(G,G_x)\xi\),
\[ H^q(G_x,A)\xrightarrow{\ \theta_y\alpha_y\theta_y^{-1}\ } H^{q+n}(G_x,A),\qquad q>0. \]
By the lemma, \(G_x\) is a finite group with
\(H^n(G_x,Z)=Z\backslash(\operatorname{ord}G_x)Z\), but then \(G_x\) is a periodic group in the sense of Cartan–Eilenberg\(^2\), and since \(i(G,G_x)\xi\) is a generator of \(H^n(G_x,Z)\), it follows from \(^2\) that multiplication by this element is an isomorphism for any module \(A\). Thus (3) is an isomorphism.

The space \(Y\) is a cell complex of dimension
\(n(n+1)/2-1\); hence \(I_\infty^{p,q}=0\) for \(p\ge n(n+1)/2\). Together with the isomorphisms (3) for \(r=\infty\), this shows that for \(k\ge n(n+1)/2\) multiplication by \(\xi\) is an isomorphism of the groups associated with the filtered groups \(H^k(G,A)\) and \(H^{k+n}(G,A)\), and consequently these groups themselves are isomorphic. The theorem is proved.

Remark. It follows from the proof that \(H^k(G,A)=0\) for
\(k\ge n(n+1)/2\), if \(A\) is a vector space over a field of characteristic \(0\) (or relatively prime to the orders of the finite subgroups of the group \(G\)).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
12 XI 1960

REFERENCES

\(^1\) A. Grothendieck, Tohoku Math. J., 9, 2/3, 119 (1957)
\(^2\) H. Cartan, S. Eilenberg, Homological Algebra, Princeton, 1956.