On K * Z/n) and K (Fq[t]/(t2)

210 Pages·1985·11.885 MB·English

by  Janet E. Aisbett, Emilio Lluis-Puebla, Victor Snaith

Memoirs of the American Mathematical Society
Number 329
Janet E. Aisbett, Emilio Lluis-Puebla
and Victor Snaith
(with an appendix by Christophe Soule)
2
OnK.(Z/n)andK,(F [t]/(t ))
q
Published by the
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September 1985 • Volume 57 • Number 329 (first of 6 numbers)
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TABLE OF CONTENTS
On K (Z/pn) and K (Z/pn)
3 4
Introduction *
§1.1: Some results from algebraic topology , 4
§1.2: Some explicit differentials 5
§1.3: Proofs 7
§1.4: Filtration of H* (M Z/p2;Z) 13
§11.1: Results of Evens-Friedlander, Lluis § Snaith 17
§11.2: K (Z/p2) for primes p > 3 20
§11.3: K (Z/4) = Z/12 24
3
§11.4: K (Z/9) = Z/8 @ Z/9 28
Appendix to §11.1 31
3
§111.1: Mod p cohomology of ker(SL Z/p > SL Z/p) 33
§111.2: Mod p cohomology of ker(SL Z/pk > SL Z/p), k > 3 39
1st Appendix to §111.1 (case p = 2) 44
2nd Appendix to §111.1 (commutator relations, and
the SL Z/p - action) 48
v
§IV.l: Integral cohomology of ker(SL Z/p > SL Z/p) 49
§1V.2: SL Z/p - invariants in H4(-;Z) of this kernel 54
Appendix to §IV. 1 62
Appendix to § IV. 2 , 64
§V. 1: K (Z/pk), K (Z/pk) for k an odd prime 68
3 4
§V.2: K (Z/2k) 73
3
§V1.1: Maps induced by reduction 79
SLZ > SLZ/pk
iii
TABLE OF CONTENTS
Notation 86
Bibliography 89
On K (F [t]/(t2) and K (Z/q),p an odd prime
0
£ 1: Introduction 91
§2: Proofs of §§1.1/1.2 92
§3: Group cohomology calculations 96
Bibliography 99
On K of dual numbers
Introduction - statement of results 101
§1: Computations of some k*-invariants 105
§2: Computation of H^T k;H*(M k)) for i = 0, 1 and 2 108
§3: R -invariants in H2(M k) 114
n n
§4: Estimates of H1 (T k;H2(M k)) 120
v n v n
§5: Vanishing of HX(GL k;H2(M k)) 132
§6: GL -invariants of H3(M k) 145
n n
§7: H (GLk;H*(MJk)} as a Hopf algebra 150
#
§8: Explicit generators for H(SLk;H(M k)) 154
§9: Determination of K (F [e]) 170
T
3 2
§10: On K (Z/4) and K of Witt vectors, W (F ) 177
3 3 2 m
§11: Some classes in H-(SLk;H.(M k)) and their
o 1 °°
d -differential 183
§12: List of notations and formulae for group actions 191
Appendix: Homological Stability of the Steinberg Group over the
integers, by C. Soule" 195
Bibliography , I99
iv
ABSTRACT
This collection of papers is unified by the theme of the calculation of the
low dimensional K-groups of the integers mod n and the dual numbers over a
finite field. Let GL IF act by conjugation on the m x m matrices, M F
m q m q
then H.(GL F : H.(M F ; A)) is computed when i + j < 3 and A = Z, Z
c
l m q j m q U)
0
or Z/p where q = p , From these results K.(Z/n) is computed for all n
2
when i < 3 (i = 4 in some cases), K.(F [t]/(t )) is computed for i < 3
when q = 2 and partial information is obtained when q is odd.
AMS (MPS) Subject Classification (1980): 18F25
Key words and phrases:
Algebraic K-theory, group cohomology, dual numbers,
Lyndon-Serre spectral sequence,
Library of Congress Cataloging-in-Publication Data
Aisbett, Janet E., 1951 —
OnK,(Z/n)andK,(F [t]/(t2)
q
(Memoirs of the American Mathematical Society, ISSN
0065-9266; no. 329)
On t.p. "*" and "q" are subscripts.
Bibliography: p.
1. K-theory. 2. Homology theory. 3. Spectral sequences
(Mathematics) I. Lluis, Emilio. II. Snaith, V. P. (Victor Percy),
1944- . III. Title. IV. Series.
QA3.A57 no. 329 [QA612.33] 510s [512'.55] 85-15802
ISBN 0-8218-2330-2
Introduction
The papers in this volume deal with certain low dimensional algebraic
K-theory calculations which can be performed once one has the low dimensional
cohomology of the extensions
M n F p - ^ GL Z/p2-> GL F
n n p
and
MnF *—* GL F [t]/(t2) — GL F
n £ £
P P P
In 1978 I made the necessary 2-primary cohomology calculations (which
appear here with minor - but significant - corrections). Emilio Lluis made
the analogous odd primary calculations in his thesis and they are summarised
here, too. His results confirmed those of Evens-Friedlander who did the case
2
when F , is the prime field and p > 5. They also computed K (Z/p ) and
P 3
2
K.(Z/p ) when p > 5, (while my calculations gave K (Z/4) only to a factor
of two). At this point, armed with the above cohomology calculations Janet
Aisbett, in her thesis (reproduced here in condensed form) managed to obtain
sufficient inductive control on the cohomology of SLZ/p (for all primes p)
that she could determine K (Z/n) for all n and sometimes K (Z/n), too. In
the course of my calculations I needed some information on H (SLZ) which
Christophe Soule kindly proved in an appendix. Actually, Janet Aisbett later
recovered Soule's result in the course of her work.
Since each paper has its own introduction I refer the reader to
these for further details.
Victor Snaith
April 1982
Received by the editors April 12, 1982 and, in revised form April 12, 1983
and March 2k 1985.
9
Research by the third author partially supported by a Natural Sciences
Engineering and Research Council of Canada Grant.
VI
ON K (Z/p") AND K (Z/pu)
3 4
Janet E. Aisbett
INTRODUCTION
The algebraic K-groups of the finite fields and their algebraic closures
were computed by Qui11en in [Q2]. Since then, there has been only a handful
of complete calculations of any of the higher K-groups (K. for i > 3). Lee
and Szczarba [L-S] showed that the Karoubi subgroup Z/48 of K„(Z) was the full
group. Evens and Friedlander [E-F] computed K„(Z/p ) and K.(F [t]/(t )) for
i < 4 and prime p greater than 3. Snaith [Sn] has obtained the groups
K (F [t]/(t2)) for m > 1.
3 m
This paper contains computations of the groups K-(Z/n), and K.(Z/p ) for
prime p > 3. These complete the partial results on K (Z/4) by Snaith
and on K„(Z/9) by Lluis [L12], and extend the work of Evens and Friedlander.
Our main results are contained in the following theorem announced in [Al].
Theorem. Take k > 1 and 0 < i < 2.
(a) K _ (Z/2k) = Z/21 © Z/2l(k"2) © Z/C21-!). If p is an odd prime,
2i 1
K2i-l(Z/pk) = Z/Pi(k*13 « Z/Cp1-!). The map K^^CZ/p**1) -> K^^CZ/p*}
induced by reduction is the reduction epimorphism at all primes p.
(b) For prime p > 3, K (Z/pk) = 0. K (Z/2k) = Z/2. K (Z/3k) = 0.
2i 2 2
The results on K are due to Bass, K to Milnor and Dennis and Stein. Tech
2
niques used are simple. The method is unsuitable for higher dimensional com
putations.
Our results are consistent with the Karoubi conjecture that for odd
k+
primes, BGLZ/p is the homotopy fibre of the difference of Adams operations,
pK -pK-1
V - ¥F . However, Priddy [P] has disproved the conjecture in the cases
p > 3 and k = 2.
This work is organized as follows. There are 6 chapters. Each chapter
begins with a short summary of its contents, as does each of the sections.
Lengthy and mechanical proofs are relegated to an appendix following the rele
vant chapter. Internal references to sections, etc. are explained, along with
other symbols and usages, in the notation pages at the end of the paper.
The first chapter brings together sundry results used later.
1
2 JANET E. AISBETT
2
The second chapter is a summary of the computations of the groups K.Z/p
for prime p. The first section lists some results of Snaith, Lluis and Evens
and Friedlander; the second, third and fourth sections complete the calcul-
2
ation of K Z/p for all primes p.
The third chapter starts on the real computational work of the paper by
deriving inductively the GL Z/p-modules H*(G ; Z/p) for k > 3, where if r is
induced by the reduction,
Gk = ker(r : SL Z/pk -> SL Z/p) .
n n r n VJ
Throughout n is large with n^O mod p, and (n,p-l) = 1. Partial results only
are obtained when p = 2.
The fourth chapter computes some integral cohomology groups of G , in
terms of quotients etc. of the GL Z/p-modules H*(M Z/p; Z/p). The results
quoted in Chapter II are then used to determine low-dimensional E**-terms in
the Serre spectral sequence
H*(SL Z/p; *(Gk; Z)) => H*(SL Z/pk; ).
n H n Z
The fifth chapter computes groups H*(SL Z/p ; Z), * < 5, from this spec
tral sequence. Only partial results for p = 2 or 3 are available. From these
come the K^. groups.
The last chapter uses Quillen's description of the homology of GLZ/p to
derive simply the well-known result that the 3-stem injects into K„Z and
passes withkemel of order 2 into K~Z/p (or maps onto K_Z/p if p = 2 or 3).
k
It then considers the maps K„Z -> K~Z/p induced by the reduction when k > 1.
These maps are described by maps onto what are effectively the direct summands
K~Z/p when p is odd, or K-Z/4 when p = 2. This last statement uses a theorem
of Stein.
A Note on the Truncated Polynomial Rings F [t]/(t ) .
2
When k = 2, computation of the cohomology of GL IF [t]/(t ) is closely
related to that of GL Z/p , since these are respectively the total spaces in
the split and non-split extensions M Z/p >—• E—•-*• GL Z/p, n > 3. (See
Snaith and Lluis, this volume.) Analagous methods to those used in this
paper can also be applied to the sequences
ker r >-* SL R[t]/(tk) - ^ SL R
to determine low-dimensional K-groups of the truncated polynomial rings over
ON K (Z/pn) AND K (Z/pn) 3
3 4
the commutative ring R; again, knowledge of low-dimensional groups of the form
HX(GL R; H-'CM R; Z)) would be central to the proof. The work of Snaith, Lluis
and others has been applied in this context to computing K, when R is a finite
field, or the rational integers [A2],
I am glad to record my debt to Vic Snaith for his enthusiasm and advice.