Arch. Math., Vol. 51,496-499 (1988) 0003-889X/88/5!06-0496 $ 2.30/0 �9 1988 Birkh/iuser Verlag, Basel

Completed group algebras without zero divisors

By

ANDREAS NEUMANN

Introduction. One of the major results concerning the zero divisor problem is the following theorem, proved by Farkas and Snider [3] in 1976: Let G be a torsion free polycyclic-by-finite group and k a field of characteristic 0. Then the group ring kG has no zero divisors.

In the category of pro-p-groups an analogon of this theorem can be proved, the poly-p-adic groups playing the role of the polycyclics. By poly-p-adic we understand the existence of a subnormal series of closed subgroups

I=Go<~GI <=... <Gn=G

with factor groups Gi+ ~/Gi isomorphic to either ;gp (the p-adic integers) or Cv (the cyclic group of order p). For these groups (provided they are torsion-free) not only the group ring ZpG but also the much larger completed group algebra of G (definition see below) does not have zero divisors.

Moreover the result can be extended to all those torsion free pro-p-groups which admit the structure of an analytic manifold over Qp with analytic group operations (such groups are called analytic). They can be characterized in a purely algebraic manner by a result of Lubotzky and Mann [7]: A pro-p-group is analytic if and only if the rank (cardinality of a minimal system of topological generators) of it's open subgroups is bounded by a finite number.

By intersection the given p-adic series of a poly-p-adic group induces in every closed subgroup a p-adic series of same or shorter length; hence these groups are analytic.

To define the suitable algebra for the topological setting (following [6, II.2.2]), let G be a pro-p-group represented as projective limit of its finite quotients:

G = lira GIN.

For N < M the maps GIN ~ G/M induce maps Zp [G/N] ~ 7Z, p [G/M], so the group rings of the quotient groups form a projective system. Call its projective limit

Z e ~G; = lim Zp [G/N]

the completed group algebra of G over Zp. If the ;gp[G/N] are topologized as free Zp-modules of finite rank, then •p ~G~ with the topology of the projective limit can be considered as a completion of Zp [G] (topologized by the augmentation ideals of the open

Vol. 5I, 1988 Completed group algebras without zero divisors 497

normal subgroups of G), since the epimorphisms 2~p [G] ~ Zp [G/N] induce an embedding of Zp[G] into Zp ~G~ as a dense subalgebra.

Theorem 1. Let G be a torsion-free analytic pro-p-group. Then 2~p ~G~ has no zero divisors.

The main tool in proving this theorem will be the one also employed by Farkas and Snider [3]:

Theorem (Walker [9]). Let R be a noetherian ring satisfying

(1) R is semiprime. (2) All f initely generated projective R-modules are stably free. (3) R has finite global dimension.

Then R has no zero divisors.

A module P is said to be stably free, if there exist finitely generated free modules F and F' with F �9 P ~ F'. By [5] all projective modules over a local ring are free, so condition (2) is satisfied by local rings.

The group ring of a finite p-group over 2~p is local, therefore 2~ v [G~ is local for any pro-p-group G, being the projective timit of local rings. If G is analytic, ~p ~G~ is noethe- rian ([6, V.2.2.4]).

The first hypothesis of Walker's theorem follows easily from the fact that group rings are semiprime in general: Let k be a field of characteristic 0 and G any group. For x = Z xgg ~ kG define tr(x) = x 1 . For a nilpotent dement x we have tr(x) = 0 ([10, 2.3.3]). Clearly this remains true if k is replaced by a commutative ring R without zero divisors (embed R in it's quotient field). Now let I be an ideal of RG with 12 = 0 and let x = Z xog be an element of L For all g ~ G , xg -1 is in I and therefore nilpotent. Hence t r(xg-1) = xg = 0 for all g, so x = 0 and RG is semiprime.

Now let J be a nontrivial ideal of Zp [G~, G a pro-p-group. For some open normal subgroup N of G the image of J in Zp[G/N] is not zero, and because the latter is semiprime, the image's square can't be zero. Hence j2 4:0 and 7Zp [FG~ is semiprime.

Let R be a ring. For an R-module A the projective dimension of A is the least integer n such that Ext~ + 1 (A, B) = 0 for all R-modules B. The global dimension of R (gl.dim R) is the supremum of the projective dimensions of all R-modules. If R is noetherian, the global dimension can equivalently be defined by right or left R-modules (for proof see [4, p. 581).

In order to apply Walker's theorem, only hypothesis (3) remains to be verified, so the proof of Theorem 1 reduces to

Theorem 2. Let G be a torsion free analytic pro-p-group. Then gl.dim ~p ~G~ is finite.

P r o o f. Let JgL be the category of topological G-modules (or, equivalently, Zp [G~- modules) with underlying abelian group a pro-p-group and G-linear topology (i. e. having a basis of the neighbourhoods of zero consisting of G-submodules). By [6, III.3.2.2 and V.2.2.8] any analytic pro-p-group has a subgroup H of finite index and an integer n such

Archlv der MathemaUk 51 32

498 A. NEUMANN ARCH. MATH.

that H"(H, M) = 0 for every M s -J/gL. Since l ips Jr H"(H, lip)= 0 and cd(H) < n, the cohomological dimensions defined as usual for profinite groups by discrete torsion G-modules with continuous G-action. By [8] in this situation a torsionfree finite extension doesn't change the cohomologieal dimension, so cd(G) is finite.

Consider a resolution

. . . ~ X2-~ XI -~ ~p-~ O

of Zp as (trivial) module over Zp ~G~ with finitely generated free modules, By [6, V.3.2.7] there is a natural equivalence H " ( G , M ) ~ - g x t ~ ( Z p , M) for all M e J'd L and n ~ N (where gxt denotes the Ext-functor in the category J//L)- To get rid of topology one has to observe that all (abstract) 2gp ~G~-homomorphisms 2gp ~ G ) k ~ lip are continous, so Hom~(X~, lip) ~ Hom(X~, lip) and in the given situation gxt"(Z v, lip) ~ Ext"(;g v, Fp) for all n. Since cd(G) is finite, the Ext-groups are zero for sufficiently large n.

If Ext"(Zp, lip) = 0, it follows that Ext "+ ~ (lip, lip) = 0: Consider the exact sequence

p being the multiplication by p and ~ the natural projection. Treating this sequence with E x t ( - , l i v ) we get the long exact sequence

. . . . , Exff (•p, lFp) , Ext" + 1 (lip, lip) ~' , Ext" +1 (2gv, lip)___.. ~ o

Now the vanishing of the outer two groups implies the vanishing of the middle one. In the case of a local ring, however, the residue class field turns out to be a test module

for the global dimension by a theorem of Boratinsky (see also [1 et al.]), provided the radical satisfies the Artin Rees condition.

An ideal J in a ring R is meant to be Artin Rees, if for any left ideal L of R there exists an integer k with jk C~ L ~= JL. Our case R = 2g v [G~, R/J ~ IF v guarantees the existence of such k: Since L as an ideal of a noetherian ring is finitely generated, so is L/JL as module over R/J and L/JL must be finite. Therefore the images of the jk C~ L in L/JL become stationary with increasing k. Now ~ (f i ~ L + JL) = L c~ (~ ( f i + JL) = L c~ J-L, where JL denotes the closure of JL in the J-adic topology. Since G is finitely generated, the J-adic topology on 7/p ~G~ coincides with the topology induced by the projective limit representation, which is compact ([7, II.2.2.2]). In a compact noetherian ring R every left ideal is closed, being the image of the compact module R" under the continuous map (r~, r2,..., r,) -~ ~ rixi, where {Xl, x2 , . . . , x,} is a set of generators of the given left ideal. It follows J L = JL and the stationary point of the images of the f i a L in L/JL is zero.

Now we can apply Boratinsky's theorem and get the equivalence

gl.dim 2g, [[G~ __< n.*~ Ext~+~a~(lip, lip) = 0

which proves the two theorems.

A c k n o w 1 e d g e m e n t . I would like to thank Professor P. P laumann for his valu- able advice and encouragement.

Vot. 51, 1988 Completed group algebras without zero divisors 499

References

[1] K. A. BROWN, C. R. HAJARNAVIS and A. B. MCEACHARN, Noetherian rings of finite global dimension. Proc. London Math. Soc. (3) 44, 349 371 (1982).

[2] M. BORATYNSKY, A change of rings theorem and the Artin-Rees-property. Proc. Amer. Math. Soc. 53, 307-310 (1975).

[3] D. R. FARKAS and R. L. SMDER, Ko and noetherian group rings. J. Algebra 42, 192-198 (1976).

[4] J. P. JANS, Rings and homology. New York 1964. [5] I. KAPLANS~:V, Projective modules. Ann. of Math. 68, 372-377 (1958). [6] M. LAZA~D, Groupes analytiques p-adiques. PuN. Math. IHES 26, 389-603 (1965). [7] A. LUBOTZKV and A. MANN, Powerful p-groups I + II. J. Algebra 105, 484-515 (1987). [8] J. P. SERRE, Sur la dimension cohomologique des groupes profinis. Topology 3, 413-420

(1965). [9] R. WALKER, Local rings and normalizing sets of elements. Proc. London Math. Soc. (3) 24,

27-45 (1972). [10] D. S. PASSMAN, The Algebraic Structure of Group Rings. New York 1977.

Anschrift des Autors:

Andreas Neumann Mathematisches Institut Universit/it Erlangen Bismarckstr. 1�89 D-8520 Erlangen

Eingegangen am24.7.1987

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