Research announcements: On K3and Kaof the integers mod n (original) (raw)

BY JANET AISBETT Quillen [7] defines an algebraic A^-functor from the category of associative rings to that of positively graded abelian groups, with K f (R) = ÏI^BGLR 4 ") for i > 1. K x and K 2 correspond respectively to the 'classical' Bass and Milnor definitions. The /^-images of finite fields and their algebraic closures were computed by Quillen in [8]. Since then, there has been only a handful of complete calculations of any of the higher A^-groups (K t for i > 2). Lee and Szczarba [4] showed that the Karoubi subgroup Z/48 of K 3 (Z) was the full group. Evens and Friedlander [3] computed KfZ/p 2) and K ((F p [t] lit 2)) for i < 5 and prime p greater than 3. Snaith in [1] and, with Lluis, in [5], fully determined K 3 (¥ m [t]/(t 2)) for m > 1 and prime p other than 3. This note summarizes computations of the groups K 3 (Z/n), and K^(Z/p k) for k > 1 and prime p > 3. These complete the recent partial results on K 3 (Z/4) by Snaith and on K 3 (Z/9) by Lluis, and extend the work of Evens and Friedlander. The theorem stated below is consistent with the Karoubi conjecture that for odd primes, BGLZ/p k+ is the homotopy fibre of the difference of Adams operations, ty p-ty p _1. However, Priddy [6] has disproved the conjecture in the cases p > 3 and k-2. I am most grateful to Victor Snaith for his supervision of the thesis in which these results originally appeared. Details of the proofs can also be found in [H-THEOREM. Take k > 1 and 0 < i < 2. (a) K 2{ _ l (Z/2 k) = Z/2< eZ/2'<*-2 > 0Z/(2''-1). K 2i _^Zjp^ = 2/p'(fc-1) 0 z/(p'-l)ifp is an odd prime. For all primes, the map K^^iZ/p^^-^K^^Z/p") induced by reduction is the obvious surjection. (b) For prime p > 3, K 2i (Z/p k) = 0. K 2 (Z/3 k) = 0. K 2 (Z/2 k) = Z/2. K t is due to Bass, K 2 to Milnor, Dennis Stein.