ON THE LINEAR, SYMPLECTIC AND ORTHOGONAL K-GROUPS OF Z/P2Z IN DIMENSIONS 1 THROUGH 5

[摘要] Since Quillen's paper on the K-theory of finite fields in 1972, there have been few explicit computations of higher (> 2) dimensional algebraic K groups. In 1976, Friedlander extended Quillen's work to compute the unitary K groups of finite fields. Six years later Evens and Friedlander computed the groups K(i)GL(Z/p2Z) for i = 3, 4 and for p greater-than-or-equal-to 5. K1 and K2 had been previously calculated by Bass and Milnor, respectively. In 1985, Aisbett found the groups K(i)(Z/p(n)Z) for 1 less-than-or-equal-to i less-than-or-equal-to 4, for all primes p and Snaith computed the groups K3(F2m[t]/(t2)) for m greater-than-or-equal-to 1. A year later Aisbett, Lluis-Puebla and Snaith obtained results on K3 of F(q)[t]/(t2) and F(q)[t]/(t3) for q a power of an odd prime. In this paper we compute some of the homology groups of the group SO(2n, Z/p2Z) and use these groups, along with some previous results on the homology of the groups SO(2n + 1, Z/p2Z), Sp(2n, Z/p2Z) and SL(n, Z/p2Z), to determine the groups K(i)G(Z/p2Z) for large primes p, where G = Sp or SO and 1 less-than-or-equal-to i less-than-or-equal-to 5. Unfortunately, there will be no exact definition of the meaning of large primes. This is due to the lack of explicit bounds of certain homology stability theorems. In addition, we are able to extend the work of Evens and Friedlander to determine bounds on the group K5GL(Z/p2Z) for primes p > 56.

[发布日期] 1991-03-29