An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Author:
Marian F. Anton

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2327-2340

MSC (2000):
Primary 57T10, 20J05; Secondary 19D06, 55R40

DOI:
https://doi.org/10.1090/S0002-9947-03-03255-0

Published electronically:
January 27, 2003

MathSciNet review:
1973992

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Abstract: Conjecturally, for an odd prime and a certain ring of -integers, the stable general linear group and the étale model for its classifying space have isomorphic mod cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if is regular and certain homology classes for vanish. We check that this criterion is satisfied for as evidence for the conjecture.

**1.**Anton, M.F.:*On a conjecture of Quillen at the prime*, J. Pure Appl. Algebra**144**(1999), 1-20. MR**2000m:19003****2.**Anton, M.F.:*Etale approximations and the mod cohomology of*, Cohomological methods in homotopy theory (Bellaterra, 1998), 1-10, Progr. Math., 196, Birkhäuser, Basel, 2001. MR**2002h:11119****3.**Dwyer, W.G.:*Exotic cohomology for*, Proc. Amer. Math. Soc.**126**(1998), 2159-2167. MR**2000a:57092****4.**Dwyer, W.G. and Friedlander, E.M.:*Algebraic and etale -theory*, Trans. Amer. Math. Soc.**292**(1985), 247-280. MR**87h:18013****5.**Dwyer, W.G. and Friedlander, E.M.:*Topological models for arithmetic*, Topology**33**(1994), 1-24. MR**95h:19004****6.**Lichtenbaum, S.:*Values of zeta-functions, étale cohomology, and algebraic -theory*. Algebraic -theory, II: ``Classical'' algebraic -theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 489-501. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973. MR**53:10765****7.**Mitchell, S.A.:*Units and general linear group cohomology for a ring of algebraic integers*, Math. Zeit.**228**(1998), 207-220. MR**99k:19002****8.**Quillen, D.:*On the cohomolgy and -theory of the general linear groups over a finite field*, Annals of Math.**96**(1972), 552-586. MR**47:3565****9.**Voevodsky, V.:*The Milnor conjecture*, preprint MPI, 1997.

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Additional Information

**Marian F. Anton**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom and IMAR, P.O. Box 1-764, Bucharest, Romania 70700

Address at time of publication:
Department of Mathematics, University of Kentucky, 715 POT, Lexington, Kentucky 40506-0027

Email:
Marian.Anton@imar.ro

DOI:
https://doi.org/10.1090/S0002-9947-03-03255-0

Keywords:
Etale model,
linear group,
cohomology,
invariants

Received by editor(s):
May 1, 2002

Received by editor(s) in revised form:
November 14, 2002

Published electronically:
January 27, 2003

Article copyright:
© Copyright 2003
American Mathematical Society