Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 009 (April 12, 2011)
21
Some algebra: ltered and graded rings
A ring A is said to be ltered if the underlying
abelian group has a ltration
0 = F1(A) F0(A) i1 Fi(A) = A,
such that
Fp(A) Fq (A) Fp+q (A)
under the ring multiplication, and 1 F0(A). Th
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 007 (April 13, 2011)
16
Abelian category localization
Suppose that A is a small abelian category, and let
B be a full subcategory such that in every exact
sequence
0a aa 0
in A, a is an object of C if and only if b and a are
objects of B. Such a s
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 008 (March 9, 2011)
19
Ktheory with coecients
Suppose that n is some positive number. There is
a natural cobre sequence
p
n
E E E/n
in the category of spectra (or symmetric spectra),
where n is multiplication by n, meaning the map
in the stable
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 006 (March 14, 2008)
13
Homotopy bres
Here is another consequence of the Additivity Theorem:
Theorem 13.1. Suppose that f : M N is
an exact functor. Then the square
s(S(M) S(N) ES(N)
sS(M)
f
/
sES(N)
/
d
sS(N)
is homotopy cartesian.
Recall from Se
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 011 (April 12, 2011)
26
Ktheory of nite elds
We will sketch a proof of the following wellknown
result of Quillen [4]:
Theorem 26.1. Suppose that Fq is the eld
with q = pn elements. Then there are isomorphisms
Ki(Fq ) =
Z/(q j 1)
if i = 2j 1, j >
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 010 (April 12, 2011)
24
Rigidity
We will prove Suslins rst rigidity theorem [7]:
Theorem 24.1. Suppose that i : k L is an
inclusion of algebraically closed elds, and that
n is a number such that (n, char(k) = 1. Then
the induced map
i : K(k, Z/n)
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 003 (January 26, 2011)
5
Waldhausens s construction
The basic denitions and results of this section
rst appeared in Waldhausens seminal paper [3].
Many of the tricks in the proofs which are given
here appear in [1] and [2].
Suppose that C is some
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 002 (January 25, 2008)
2
Exact categories
Ill start with the canonical example.
Suppose that R is a unitary ring and let Mod(R)
of be the category of (left, right) Rmodules. Let
P(R) be the full subcategory of Mod(R) whose
objects are the nitely
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 005 (April 12, 2011)
10
Group completion
I say that a commutative diagram
/
X
Y
f
/
Z
W
of simplicial set maps is homology cartesian (for
some theory h) if there is a factorization
X BB
i /
V
BB
BB
p
f BB!
W
with i a trivial cobration and p a bra
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 001 (January 21, 2008)
1
Categorical homotopy theory
Much of the material in this section was invented
by Quillen to describe the Ktheory spaces, and
originally appeared in [3]. The theory has been
rened and extended over the years (see [1], for
Western University (Ontario)  Also known as University of Western Ontario
Algebraic KTheory
MATH 281

Spring 2011
Lecture 004 (March 9, 2011)
8
The Ktheory spectrum
Recall that there is a poset isomorphism
0 n n + 1,
=
and write
=0 :m+10m0nn+1
=
=
for each ordinal number map : m n.
Let X be a simplicial set, and write EX for the
simplicial set with
EXn = Xn+1 = X(0