FIRST Q U A D R A N T SPECTRAL SEQUENCES
IN A L G E B R A I C
K-THEORY
by
R.W. T h o m a s o n *
D e p a r t m e n t of M a t h e m a t i c s
M a s s a c h u s e t t s Institute of T e c h n o l o g y
Cambridge, M a s s a c h u s e t t s
02139
U.S.A.
By algebraic K - t h e o r y I u n d e r s t a n d the study of the f o l l o w i n g
process:
one takes a small c a t e g o r y
operation,
~;
"group completes"
on the c l a s s i f y i n g
space
the r e s u l t i n g space.
For
S
p r o v i d e d w i t h a "direct sum"
the m o n o i d structure
induced by
B=S; and then takes the h o m o t o p y groups of
R
a ring, this process applied to the
c a t e g o r y of f i n i t e l y g e n e r a t e d p r o j e c t i v e R-modules yields Q u i l l e n ' s
K,(R).
K a r o u b i ' s L - t h e o r y is also a special case of this g e n e r a l i z e d
a l g e b r a i c K-theory.
My aim in this paper is to show how K - t h e o r y may be a x i o m a t i z e d
as a g e n e r a l i z e d h o m o l o g y theory on the c a t e g o r y of such c a t e g o r i e s
S, and to give a c o n s t r u c t i o n that yields the K - t h e o r e t i c a n a l o g u e s
of m a p p i n g cones, m a p p i n g telescopes and the like.
interested only in the K - t h e o r y of rings,
K - t h e o r y should be useful t e c h n i c a l tools.
cone c o n s t r u c t i o n of
§6 may be useful
Even if one is
these results for g e n e r a l i z e d
In p a r t i c u l a r the m a p p i n g
in situations where appeal to
* A u t h o r p a r t i a l l y s u p p o r t e d by NSF Grant MCS77-04148.
333
Quillen's T h e o r e m B fails.
Two examples are given in §6 to illustrate
this point.
§i.
D e f i n i t i o n s of some algebraic
structures on categories.
K - t h e o r y assigns a graded abelian group to each small symmetric
m o n o i d a l category.
selected object
Recall this is a c a t e g o r y
0 e S,
a bifunctor
S
together with a
8 : S x S --~S, and natural
isomorphisms
e
:
(A
8
y:A~B
B)
•
C
~ P B
8
A
~pB@A
I:A
~,OeA
These n a t u r a l i s o m o r p h i s m s are subject to "coherence conditions;" we
require a certain five d i a g r a m s to commute,
and these imply all
(generic) d i a g r a m s made up of ~'s, y's, and l's commute.
II, §i, III,
The word
§i; or
[6], 3.3; or
"small" m e a n s only that
[9], VII,
S
§i, VII,
Check
[2],
§7; for details.
has a set of objects,
rather
than a proper class of them; this is a t e c h n i c a l i t y r e q u i r e d so the
c l a s s i f y i n g space
[14]
BS
exists.
The standard example of a symmetric m o n o i d a l c a t e g o r y is any
additive c a t e g o r y w i t h
~
given by d i r e c t sum.
The subcategory of
i s o m o r p h i s m s in an a d d i t i v e c a t e g o r y has an induced symmetric m o n o i d a l
structure.
A p e r m u t a t i v e c a t e g o r y is a symmetric monoidal one where the e's
and l's are r e q u i r e d to be i d e n t i t y natural transformations:
is s t r i c t l y a s s o c i a t i v e and unital.
thus
•
E v e r y symmetric m o n o i d a l c a t e g o r y
334
is e q u i v a l e n t
to a p e r m u t a t i v e
assume
are p e r m u t a t i v e
things
Let
2, ~
monoidal
be
functor
: S
([7],
whenever
symmetric
F
one
monoidal
~T
1.2;
[ii],
4.2),
so we m a y
convenient.
categories.
is a f u n c t o r
A lax
together
symmetric
with
natural
transformations
: 0T
such t h a t
the
following
(FA • FB)
FA •
~ FO S
8 FC
(FB • FC)
f
diagrams
~F(A
FB ~ FA
, F (B ~ A)
~el
lef
F
infinite
F
is a s t r o n g
loop
is a s t r i c t
identity
space
theory)
• FC
~F((A
FA • F(B
$ C)
P F(A
natural
functor
if in a d d i t i o n
transformation.
~
8 C)
(B ® C
)
FO S • FA
monoidal
syr~netric m o n o i d a l
• B)
• F ( O S • A)
~®i
symmetric
$ B)
F (A ~ B)
F~
FA
• F ( A ~ B)
commute
FA $ F B •
FA
OT
- FA $ FB
functor
F
(the u s u a l
f
if
is a lax
and
f
f
and
(strong,
notion
are
f
in
isomorphisms.
are the
strict)
335
p e r m u t a t i v e functor if its source and target are permutative.
If
F, G
are lax symmetric m o n o i d a l
m o n o i d a l natural t r a n s f o r m a t i o n
functors,
n : F ~ G
a symmetric
is a natural t r a n s f o r m a -
tion m a k i n g the following d i a g r a m s commute:
0
<
FA 8 FB
FO S
• F(A ~) B)
rl
GA q) GB
~
, G(A ~) B)
GOs
As an example,
any additive functor between two a d d i t i v e
categories is a strong symmetric m o n o i d a l
functor.
Any n a t u r a l
t r a n s f o r m a t i o n between two additive functors is a symmetric m o n o i d a l
natural transformation.
For any of the above concepts,
we have a c o r r e s p o n d i n g n o n u n i t a l
v e r s i o n o b t a i n e d by d r o p p i n g any condition r e l a t i n g to the unit
§2.
O.
Passage to the a s s o c i a t e d spectrum and d e f i n i t i o n of a l g e b r a i c
K-theory.
Let
Sym Mon
be the c a t e g o r y of small symmetric m o n o i d a l
categories and lax symmetric monoidal
is a functor,
Spt
: Sym Mon
(connective)
spectra,
functors.
• Spectra,
By
[17], 4.2.1, there
into the category of
and a natural t r a n s f o r m a t i o n
which e x h i b i t s the zeroth space of the spectrum
B~
Spt(S)
c o m p l e t i o n of the c l a s s i f y i n g space of the category
S.
~ Spto(S),
as the group
This extends
the usual functors defined on the subcategory of strong symmetric
m o n o i d a l functors by
[II],
[15].
The proof of
[13] is easily adapted
336
to show
Spt
is u n i q u e l y d e t e r m i n e d up to natural e q u i v a l e n c e by
the above properties.
A symmetric m o n o i d a l n a t u r a l t r a n s f o r m a t i o n
a h o m o t o p y of maps of spectra from
Spt F
to
Spt G.
c o n s t r u c t a h o m o t o p y theory of symmetric monoidal
related via
Spt
space functor
induces
Thus one may
categories which is
to the h o m o t o p y theory of spectra,
a h o m o t o p y theory of categories
the c l a s s i f y i n g
~ : F ~ G
just as in
[14]
is c o n s t r u c t e d which is related by
B
to the h o m o t o p y theory of spaces.
The key thing to k e e p in m i n d is that a symmetric m o n o i d a l n a t u r a l
t r a n s f o r m is like a homotopy;
this motivates
the c o n s t r u c t i o n of §3
and its relation to the h o m o t o p y colimit c o n s t r u c t i o n s of §4.
I define K - t h e o r y as a functor from
groups by
K,(2)
the s p e c t r u m
Spt 2"
an a-spectrum,
Sym Mon to graded abelian
S
= ~,(Spt 2), the reduced stable h o m o t o p y groups of
As by spectrum I mean what used to be called
this is e q u i v a l e n t to
of the group completion of
BS.
~,(Spt 0 2),
Thus if
i s o m o r p h i s m s in an a d d i t i v e c a t e g o r y
S
~, my
the h o m o t o p y groups
is the s u b c a t e g o r y of
K,(2)
is
K,(~)
as
d e f i n e d by Q u i l l e n ' s plus c o n s t r u c t i o n or group c o m p l e t i o n method.
I p r e f e r to think of
h o m o t o p y groups,
§3.
K,
as
z~ Spt
as the former is more
rather than in terms of
"homological."
The fundamental c o n s t r u c t i o n and its first quadrant K - t h e o r y
spectral sequence.
I will n o w present a c o n s t r u c t i o n on diagrams in
it has a r e a s o n a b l e u n i v e r s a l m a p p i n g p r o p e r t y
ad hoc
construction),
Sym Mon,
show
(so it is not an
and then give a spectral sequence for its
K-groups.
Let
~
be a small category.
For simplicity,
consider a d i a g r a m
337
of permutative
category
categories;
of permutative
Let
Perm-hocolim
L, and
Xi
F : ~ --~Perm
be the permutative
where
is an object of
n[ (LI,X I) ..... (Ln,Xn)]
a functor
into the
categories.
F
n[(LI,X I) ..... (Ln,Xn)]
i.e.,
n = 0,1,2 ....
F(Li).
category with objects
L.
;
is an object of
l
A morphism
consists of data
--~k[ (LI,X I) ..... (Lk,Xk)]
(~;Zi;xi)
i)
a surjection
2)
maps
h i : L i ---~L~ (i)
3)
maps
xj
There
:
~ : {i ..... n }
in
is a straightforward
>> {I ..... k}
L
~iF(i i) (X i)
i£~
(j)
I will not explicitly
universal
of sets
,X:3
in
L:3
rule giving the composition
give it, but it is implicitly
of morphisms.
determined
by the
mapping property below.
The unit of P e r m - h o c o l i m
n[(LI,X I) ..... (Ln,Xn)]
F
is
O[ ], and
~
is given by
e m [ ( L n + l , X n + I) ..... (Ln+m, Xn+m)]
=
n+m[(Ll,X I) ..... (Ln+m,Xn+m)].
The universal mapping property
Lemma:
Strict permutative
correspond
G : Perm h o c o l i m F --~T
b i j e c t i v e l y with systems consisting of non-unital
permutative
functors
permutative
natural
i : L --* L
functors
is given by
in
G L : F(L)
--~
transformations
for each
G£
2; which must satisfy
lax
L e 2, and non-unital
: G L ~ G L-
• F(i)
the conditions
for each
G 1 = id
and
G Z • G£~ = G£~-.
Proof:
JL(X)
Let
JL
: F(L)
= I[(L,X)].
in an obvious way,
of
JR
: JL ~ JL"
--~ Perm h o c o l i m F
This
JL
and for
• F(Z).
be given by
is a non-unital
£ : L --~L"
lax permutative
there
functor
is an obvious choice
338
Given this system,
system
G L = G'JL,
the b i j e c t i v e c o r r e s p o n d e n c e
G£ = G.J£.
sends
G
to the
It is tedious but easy to see this
works.
Given a d i a g r a m
hocolim F
F : ~ --~ Sym Mon, there is an analogous Sym-Mon-
with the c o r r e s p o n d i n g universal m a p p i n g property.
say b e l o w about P e r m - h o c o l i m F
applies to it as well,
d e s c r i p t i o n of the objects and m o r p h i s m s of
All
I
The e x p l i c i t
Sym-Mon-hocolim F
differs
s l i g h t l y from that given above.
This c o n s t r u c t i o n turns out to have good p r o p e r t i e s with respect
to K-theory.
Theorem:
Let
F : ~ --~Perm
first quadrant
spectral
be a diagram.
sequence
E P,q
2
= Hp(L;KqF)
Here
H,(~;KqF)
coefficients
~ K p + q ( P e r m - h o c o l i m F)
is the h o m o l o g y of the c a t e g o r y
in the functor
i n f o r m a t i o n on this is
Then there is a natural
L ~-~ K F(L).
q
[5], IX §6 or
~
with
A c o n v e n i e n t source for
[14], §i.
I'll identify the E 2
t e r m with more familiar objects for the examples of §4.
This theorem is an immediate c o r o l l a r y of the theorem of §5
and the p r o p o s i t i o n of §4.
§4.
Facts about and e x a m p l e s of h o m o t o p y colimits.
To prepare the way for the s t a t e m e n t of the fundamental t h e o r e m
of §5, and to explain the s t r a n g e - l o o k i n g name
r e v i e w the h o m o t o p y colimit
B o u s f i e l d and Kan [i] .
Perm-hocolim,
(homotopy d i r e c t limit)
I will
c o n s t r u c t i o n of
Some version of this exists for every c a t e g o r y
339
a d m i t t i n g a r e a s o n a b l e h o m o t o p y theory, e.g.,
but
Sym Mon
and
[i] c o n c e n t r a t e s on the c a t e g o r y of simplicial sets.
Spectra;
I'll give
some of their results t r a n s l a t e d for the category of t o p o l o g i c a l
spaces, Top.
Let
a space
One can also read
F : ~ --~Top
Vogt
be a diagram.
c o r r e s p o n d e n c e b e t w e e n maps
gL
A s s o c i a t e d n a t u r a l l y to
T o p - h o c o l i m F, the h o m o t o p y colimit of
by a universal m a p p i n g p r o p e r t y
maps
[20] for this material.
: F(L) - - ~ X
([i], XII,
F.
F
is
It is c h a r a c t e r i z e d
2.3) e s t a b l i s h i n g a b i j e c t i v e
g : Top-hocolim F --~X
and h o m o t o p i e s r e l a t i n g them.
and a system of
With the
p h i l o s o p h y of §2 that symmetric m o n o i d a l natural t r a n s f o r m s are like
homotopies,
like that of
this u n i v e r s a l m a p p i n g p r o p e r t y of T o p - h o c o l i m F
Perm-hocolim F
given in the lenuna of §3
(cf.
is much
[18],
1.3.2).
For any g e n e r a l i z e d h o m o l o g y t h e o r y
first q u a d r a n t spectral sequence
[i], XII,
E 2p,q = Hp(~;EqF)
This c o n s t r u c t i o n
Example
I:
Let
E,
on
Top,
there is a
5.7
~ E p + q ( T O p - h o c o l i m F)
subsumes m a n y w e l l - k n o w n c o n s t r u c t i o n s .
F : ~ ---~Top
A
be the d i a g r a m
l. B
I
C
Then
Top-hocolim F
A --~ C.
is the double m a p p i n g c y c l i n d e r on
A ~
B
In this case the spectral sequence c o l l a p s e s to the long
exact M a y e r - V i e t o r i s
sequence
and
340
• • .---~ E q
(A)
•E
• E q (B) @ Eq(C)
(double m a p p i n g cylinder)
q
For
C
a point, the T o p - h o c o l i m is the m a p p i n g cone of
for
B
and
C
Example 2:
points,
Let
L
F(1) --~ F(2) --~F(3)
Then
....
F : L ----Top
, and
In the spectral
and
integers as a
is a diagram:
Top-hocolim F
sequence
A --~B;
A.
be the c a t e g o r y of the positive
p a r t i a l l y o r d e r e d set.
telescope.
it is the suspension of
8-- ...
is the m a p p i n g
Hp(L;EqF)
= 0
if
p > 0, and
H0(L;EqF)= = limn" EqF(n).
Example
3:
Let
object
*, and m o r p h i s m s being the elements of
F : L ---~Top
on
L
be a group
G
is a h o m o m o r p h i s m
of
G
F(*).
is
EG ×G F(*).
If
EG
c o n s i d e r e d as a category with one
G.
G --~Aut(F(*));
A functor
that is, an action
is a free acyclic G-complex,
Top-hocolim F
The spectral sequence is identified to the usual
one
Hp(G,EqF(*))
Example 4:
Let
~
s i m p l i c i a l space.
be
A °p.
~ E p + q ( E G ×G F(*))
Then
F : A°p ---.,'Top is just a
It follows from
[1], XII,
is the "thickened" g e o m e t r i c r e a l i z a t i o n
3.4 that
"ll li"
Top-hocolim
of Segal
is h o m o t o p y e q u i v a l e n t to the geometric r e a l i z a t i o n of
F.
To interpret the
for any functor
i.e.,
for
E
E
E2
from
.
1
A °p
sequence,
[15], which
for "good"
recall that
into the c a t e g o r y of abelian groups,
a s i m p l i c i a l abelian group,
of the chain complex which
8 = [(-l)id
term of the spectral
F
F
in degree
This follows from
p
[i]
H,(A°P;E)
is
XII
'
Ep, and has d i f f e r e n t i a l
5.6 and
'
is the h o m o l o g y
[12]
22.1
'
"
One has analogous results in many c a t e g o r i e s a d m i t t i n g a h o m o t o p y
341
theory.
In particular,
may define
c o n s i d e r a functor
Spectra-hocolim F
F : ~
as follows.
Let
be the d i a g r a m of n t-~h spaces of the spectra.
As h o m o t o p y c o l i m i t s in
T o p - h o c o l i m Fn
induced by the maps
F n --~ ~Fn+l.
&
Tqp
, T o p - h o c o l i m IF n
IF n --~Fn+ 1
but o n l y a prespectrum.
F .
n
we get m a p s
- T o p - h o c o l i m Fn+ 1
These maps are not in general
spectrum
Top-hocolim Fn
is
not a
To this p r e s p e c t r u m one c a n o n i c a l l y
[10]; this s p e c t r u m is our Spectra-
As above, we have
Proposition:
F : ~
Top-hocolim
adjoint to the structure maps
so the sequence of spaces
a s s o c i a t e s an e q u i v a l e n t
Spectra,
Form
P a s s i n g to the a d j o i n t s again, we get maps
equivalences;
h o c o l i m F.
One
F n : =L - - ~ T o p
commute with suspensions,
T o p - h o c o l i m F n --~ ~ T o p - h o c o l i m Fn+ I.
spectrum,
~Spectra.
For any c o n n e c t i v e g e n e r a l i z e d h o m o l o g y theory
E,
on
there is a first q u a d r a n t spectral sequence n a t u r a l in
~Spectra
E2
P,q = Hp(~;EqF)
Proof:
~ E p + q ( S p e c t r a - h o c o l i m F).
Use the fact
E , ( S p e c t r a - h o c o l i m F) = l i ~ E k + n ( T O p - h o c o l i m F n)
n
and the spectral sequences for T o p - h o c o l i m F n.
Here one regards
E,
as a g e n e r a l i z e d h o m o l o g y t h e o r y on spaces
suspension
in the usual way, via the
s p e c t r u m functor.
For special diagrams, we may identify the
e x a m p l e s above.
In particular,
E2
term as in the
for a d i a g r a m of spectra
342
A
• B
1
the
Spectra-hocolim
is the m a p p i n g
A --~B,
and the s p e c t r a l
cofibre
sequence.
sequence
One m a y also c o n s i d e r
small categories.
This
that the c l a s s i f y i n g
homotopy
colimits
ingredient
§5.
Homotopy
Theorem:
Let
equivalence
B
(Spt F)
= Spt
mapping
Spectra-hocolim
This map w i l l be the equivalence.
of
resolution
diagra m s
T
category
of
F.
on
over
This
of p e r m u t a t i v e
degeneracy
the c a t e g o r y
It is shown
commutes
This
of
there
with
is an e s s e n t i a l
of the t h e o r e m
of §5.
Spt.
There
is a natural
(Perm-hocolim
property
of a h o m o t o p y
(Spt F) --~ Spt
The proof
uses
F)
colimit
(Perm-hocolim
F).
the r e s o l u t i o n
[19].
the m o n a d
permutative
[18].
be a functor.
The u n i v e r s a l
gives a n a t u r a l m a p
Use
Cat,
: Cat --~Top
by
of
of spectra
of proof:
technique
[17],
in the proof
: ~ --~Perm
in
equivalence.
are p r e s e r v e d
Spectra-hocolim
Sketch
in
spectrum
into the long exact
colimits
functor
points
colimits
F
homotopy
up to h o m o t o p y
at several
degenerates
is treated
space
cone or cofibre
operators
Cat
which
sends
it to c o n s t r u c t
is a simplicial
categories,
induced
a category
a Kliesli
object
of
standard
simplicial
in the c a t e g o r y
n ~-~ Tn+IF,
by the action
to the free
T
with
on
face and
F,
the
of
343
multiplication
augmented
to
of
F
Applying
T, and the unit of
via the action
Spectra-hocolim
a map of simplicial objects
realize"
such simplicial
T.
This simplicial
TF --~F.
(Spt ?) --~ Spt
in
Spectra.
spectra.
(Perm-hocolim
The augmentation
n ~-~ Spectra-hocolim
induces a map from
of
Spectra-hocolim
(Spt F), and one from the realization
(Perm-hocolim Tn+IF)
to
map is a h o m o t o p y e q u i v a l e n c e
calculation
given below.
Spectra-hocolim
equivalence,
equivalence
Spt
to
of
(Perm-hocolim F).
The first
the second,
by a
this, one is reduced to showing that
(Spt Tn+IF) --~Spt
using the usual
(Spt Tn+IF)
by general nonsense;
Granted
?), one gets
One may "geometrically
the realization
n ~--~ Spt
object is
(Perm-hocolim Tn+IF)
is an
fact that a simplicial map which
in each degree has a geometric realization
which
is an
is an
equivalence.
One next reduces to the theorem
colimits
in
Top
category on
and
~,
Cat.
Spt T~
,Cat,
Recall if
is equivalent
spectrum on the c l a s s i f y i n g
G : ~
[18], 1.2 relating h o m o t o p y
Z~BG = Z~ T o p - h o c o l i m
from
Cat
equivalence
transforms
has homotopy
on
TG
and
It remains
n ~-bSpt
BG = B(LIG).
T(~/G),
= Spt
(Spt Tn+IF)
Thus for
(Spt TG)
=
recall
given by the Grothendieck
One finds functors
and natural
between
so there are equivalences
(T(~/G)
= Spt
= Z (L/G) = Z (Top-hocollm BG).
we get
(Perm-hocolim Tn+IF)
as required.
only to indicate why the m a p of the realization
(Perm-hocolim Tn+IF)
equivalence.
C.
and that there is a natural
the two series of equivalences,
Spectra-hocolim
the suspension
On the other hand,
giving inverse h o m o t o p y equivalences
(Perm-hocolim TG)
Combining
BG.
colimits
G : ~ --~Cat,
Top-hocolim
Perm-hocolim
Spt
RIG
Z~B~,
Spectra-hocolim
Spectra-hocolim
construction
to
is the free permutative
space of the category
one has equivalences:
[18] that
T~
The functor
Spt
to
Spt
(Perm-hocolim F)
factors as the composite
is an
of an
of
344
infinite loop space machine and a functor which regards the c l a s s i f y ing space of a p e r m u t a t i v e c a t e g o r y as an E -space,
the m a c h i n e c o m m u t e s with geometric realization,
the r e a l i z a t i o n of the simplicial E~-space
is e q u i v a l e n t as a space to
paragraph,
and so by
of
as in
it suffices
B ( P e r m - h o c o l i m F).
to show
But by the p r e c e d i n g
n ~--bB(T(LITnF)),
[18] its r e a l i z a t i o n is e q u i v a l e n t to the c l a s s i f y i n g
or even
[A°Pf n --~ [T(~ITnF)] °p ]op.
a functor from this last c a t e g o r y to P e r m - h o c o l i m F
h o m o t o p y e q u i v a l e n c e of c l a s s i f y i n g
the proofs of
[17], VI,
2.3, VI,
To produce an honest proof,
For example,
As
n F P B ( P e r m - h o c o l i m Tn+IF)
this simplicial space is e q u i v a l e n t to
A°Pfn --~T(~ITnF),
applied.
[ii].
which
space
There is
induces a
spaces by an a r g u m e n t similar to
3.4,
This completes the sketch.
a few t e c h n i c a l tricks m u s t be
at various points there are p r o b l e m s w i t h
b a s e p o i n t s of spaces and with units of p e r m u t a t i v e categories.
One
deals with this by n o t i n g that if one adds a new d i s j o i n t unit
O
a p e r m u t a t i v e category,
homotopy.
F : L
Also,
~Perm
the a s s o c i a t e d s p e c t r u m doesn't change up to
the above proof works only in the case where
is such that for each m o r p h i s m
strict p e r m u t a t i v e
special case.
functor.
Finally,
£
in
L,
F(~)
is a
The general case is deduced from this
to avoid trouble with the u n i v e r s a l m a p p i n g
p r o p e r t y of Spectra-hocolim,
part of the a r g u m e n t must be done in
the c a t e g o r y of prespectra.
A fully d e t a i l e d honest proof will
appear elsewhere,
to
someday.
345
§6.
A s i m p l i f i e d m a p p i n g c o n e and h o w to use it.
The c a t e g o r y
Perm-hocolim F
a l t h o u g h not i m p o s s i b l y so.
is g e n e r a l l y somewhat complicated,
In certain situations of interest it may
be r e p l a c e d by a simpler h o m o t o p y e q u i v a l e n t construction.
indicate how to do this in the case of m a p p i n g cones.
tion should be useful
rings.
for p r o d u c i n g exact sequences in the K - t h e o r y of
([14],
§i)
It has the a d v a n t a g e that its h y p o t h e s e s are easier
to satisfy in p r a c t i c e than those of T h e o r e m B.
theorem,
This c o n s t r u c -
It can be e m p l o y e d in place of Q u i l l e n ' s T h e o r e m B
for this purpose.
I will
As w i t h Q u i l l e n ' s
it leaves one with the p r o b l e m of i d e n t i f y i n g what it gives
as the third terms in a long exact sequence of K-groups with what one
wants there.
This p r o b l e m is g e n e r a l l y that of showing some functor
induces a h o m o t o p y e q u i v a l e n c e of c l a s s i f y i n g
a t t a c k e d by the m e t h o d s of
of Q u i l l e n ' s T h e o r e m A.
spaces;
and may be
[14], e s s e n t i a l l y by c l e v e r n e s s and the use
These points should b e c o m e clear in the two
e x a m p l e s below.
To make the s i m p l i f i e d double m a p p i n g c y l i n d e r c o n s i d e r a
d i a g r a m of symmetric m o n o i d a l c a t e g o r i e s and strong symmetric m o n o i d a l
functors
A
V
~ S
U
C
Suppose e v e r y m o r p h i s m of
A
is an isomorphism.
category with objects
(C,A,B)
of
A m o r p h i s m in
A; and
B, of
B.
where
given by an e q u i v a l e n c e class of data:
C
P,
Let
P
is an o b j e c t of
(C,A,B)
be the
C;
--~ (C',A',B')
A,
is
346
i)
9 : A --~A 1 • A
• A2
an i s o m o r p h i s m in
2)
~i
: C e UA L ---~C ~
a m o r p h i s m in
3)
92
: VA 2 • B ---~B~
a m o r p h i s m in
B .
E q u i v a l e n t data are o b t a i n e d by c h a n g i n g
isomorphism;
thus if
the above data
(a • A ~ • b
This
(~,~i,92)
• 9,
~
a : A1
~ A~,
A2
up to
are isomorphisms,
,
92
• Vb -I 8 B)
•
is the s i m p l i f i e d double m a p p i n g c y l i n d e r of the diagram.
(C',A~,B ") =
structure w i t h
(C • C ", A ~ A', B ~ B').
functors
B --~P,
u n i v e r s a l m a p p i n g property.
on
and
is e q u i v a l e n t to
~i " C • Ua -I
symmetric monoidal
a point,
A1
b : A 2 ---~A~
It has the obvious syn~netric m o n o i d a l
(C,A,B) ~
A;
C --~P,
There are strong
and
~
has a simple
In the special case where
the c o n s t r u c t i o n of
P
C = O
is
yields the s i m p l i f i e d m a p p i n g cone
A --~ B.
One uses T h e o r e m A of
double m a p p i n g c y l i n d e r
[14] to show the canonical map from the
in the sense of §3 to the s i m p l i f i e d version
is a h o m o t o p y equivalence.
m a p p i n g cylinder"
Proposition:
for
P
This justifies the name
"double
and yields:
In the situation above,
there is a long exact M a y e r -
Vietoris sequence
--~ Ki+ I(P)
~ ~ K i(A)
* K i(B) • K i(C)
• K i(P) ~-~ ..-
As an example of how to use this construction,
I will give a
quick proof of Q u i l l e n ' s t h e o r e m that his two d e f i n i t i o n s of K - t h e o r y
agree
[3].
Let
~
be an a d d i t i v e category,
c a t e g o r y of isomorphisms
monoidal with
•
in
~.
and
A = Iso ~
As r e m a r k e d above,
given by direct sum.
finitely g e n e r a t e d p r o j e c t i v e R-modules,
For
~
A
the
is symmetric
the c a t e g o r y of
~GLn(R)
n
is cofinal in
A,
347
so
Ki(~)
K,(A)
= n~ Spt(A)
coincides
tion definition
Consider
= z i Spto(~)
with Quillen's
~ ~i(BGL(R) +)
"plus construction"
simplified double m a p p i n g cylinder
Thus
or group comple-
and
A,
ZA =
~
obtained
as the
from the diagram where both
By the Proposition
: Ki+I(Z ~) a Ki(A),
zi+ 1 BZA,
i > 0.
of K-theory.
now the suspension of
are points.
for
B
and
above we have an isomorphism
S
= ~i+l SPt(Z~)
Ki+I(Z~)
~ ~i+l Spto(Z~)
the last isomorphism being due to the fact
BZA
~
is connected,
hence group complete.
Now
ZA
morphism
has objects
A --~A"
which reduces
isomorphism
ZA
consists
of an equivalence
to giving an isomorphism
in
epimorphisms
in
(O,A,O), which I abbreviate
A1
and
A2 .
From
and m o n o m o r p h i s m s
A a A1 $ A" ~ A2 __~c~_~
in
• A 2 ~--~-~A'.
are those of
and whose morphisms
A ~--~gE ~-----~>A', where
mono- and epimorphisms,
as dotted arrows).
morphism
E & E"
induced by taking pullbacks
a functor
ZA ~ Q S ~ __~ Q ~
(Actually,
this
As
Ki(~)
a ~i+iBQS~
homotopy
QS~
EA
is easily seen
([16], §3) whose objects
A --~A"
are equivalence
the indicated arrows
E
classes
are splittable
(shown
and all arrows by the same isorelation.
as in Quillen's
is the opposite
by the above,
definition,
a diagram of
Q~
Composition
is
[14], and there is
that forgets the choice of splittings.
that the group completion
construction
In fact,
gives the equivalence
Q~
class of data,
together with a choice of splitting
Changing
A
~, with choices of splittings
to the category
of data
A.
& ~ A 1 • A" $ A 2, up to
~, one constructs
to be isomorphic
~;
~ : A
to
definition
category of the one in
to show
of
Ki(A)
K-theory
it remains only to show
= ~i+IBQ~;
[14].)
i.e.,
agrees with the QQS~ __~Q~
is a
equivalence.
To see this,
consider
but where the morphisms
the category
are classes of
Qse~
defined
like
A 4--<E ~ - - ~ A ' ,
of splitting made only for the epimorphism.
The functor
QSO~
with a choice
348
QS~ __~Q~
factors
is a h o m o t o p y
To show
p
p/A
Q S ~ __~ Q s e ~ __+Q~.
equivalence;
the proof
is a h o m o t o p y
the c a t e g o r i e s
But
as
p/A
contains
the
epimorphisms
j : B
j"
are e p i m o r p h i s m s
m A
splitting,
monoidal
(B
structure
proof of
(B"
in
to show this
[3];
down
~ A) = B ×A B
the virtue
This
A.
"For any
is the
is,
Using
in
j : B
with
and so
here
to
of
has a symmetric
structure,
the
is c o n t r a c t i b l e "
is contractible,
clever part of Q u i l l e n ' s
of m y m a c h i n e r y
>> A
a choice
<S,Ec>
p/A,
[14].
so
this
P,
Q~
show
are s p l i t t a b l e
from
A,
only
so as a d e f o r m a -
subcategory
over
C
in
is similar.
is m e r e l y
that
proof
as
given
it gets one
to this crux very quickly.
As a second
Lichtenbaum
a certain
example,
conjecture.
functor
to c o m p l e t e
closed
field
of the prime
I will
by showing
i- K, ( G )
By the work
is so.
states
that
p, and
~
• K, ([)
Hiller
of Quillen:
~ x q, then
this
but
if
• K.(~)
to u n d e r s t a n d i n g
of Howard
to the c o n j e c t u r e
the p r o b l e m
to showing
I have been
~
map
x I
BGL(~q) +
i-~ q
: BGL(~) + --~BGL(~) +.
[4],
Let
~q
® @
K-theory
,~
is the h o m o t o p y
Equivalently,
there
that
unable
closure
~O
in c h a r a c t e r i s t i c
this c o n j e c t u r e
: ~
the
is an a l g e b r a i c a l l y
is the a l g e b r a i c
P
is a short exact sequence
there
is crucial
how one can a p p r o a c h
equivalence,
in c h a r a c t e r i s t i c
field,
indicate
reduce
the c o n j e c t u r e
O
Proof of this
I will
is a h o m o t o p y
the proof
Recall that
p.
~
subcategory,
This a r g u m e n t
objects
g : B ---Z~-~
B~
induced by p u l l b a c k
A
and
morphisms
j'g = j.
[3], p. 227 that
required.
for each
whose
>) A, and whose
p : Q s e ~ ---bQ~
QS~ __,Qse~
subcategory,
subcategory
and such that
~ A) ®
applies
full
show
by T h e o r e m A I need
are c o n t r a c t i b l e
tion retract,
: B~
that
equivalence,
as a r e f l e x i v e
I'll
is e q u i v a l e n t
be the F r o b e n i u s
fibre of
s h o u l d be a fibre
349
sequence
of infinite
I'll produce
conjecture
the cofibre
of
one then wants
Let
~ =
structure
Spt0(B)
loop spaces,
~GL
n
n(Fq),
x BGL(~) +.
the functor
~ OF
sequence
BGL(Fq) + --~BGL(~) + ;
to show it's
~ =
induced by direct
~ ~
or cofibre
~
n
sum.
: A - - ~ B.
GL n(~),
with
Spt0(~)
P
symmetric
= ~
the simplified
This
to prove the
BGL(~) +
Thus
Consider
of spectra.
monoidal
× BGL(~q)
mapping
has objects
+
cone
,
=P
(A,B), with
on
A
q
a vector
(A,B)
42
space over
--p (A',B')
: B ~
in
B-IB
[3].
group,
Then
of
There
be Quillen's
Spt0(B-IB)=
=
= ~
Then
p(A,B)
with the T-module
morphism
a vector
=
~.
A morphism
4 : A & A 1 e A" • A 2,
x BGL(~) +,
Let
and as
(B-IB) 0
monoidal
(B,~qB) ,
(A,B) --~ (A',B')
in
category,
as described
~0(B-IB)_
_
is a
be the connected
B(B-IB) 0 = BGL(~) +
structure
given by the morphism
space over
group completed
= B(B-IB)._
_
(0,0).
by
B
.
is a symmetric
on objects
B
& ~ B"
Spt0(B-IB)
component
and
q
is a class of data:
(~ ® A2)
Let
F
functor
where
changed
of
~
(B-IB)
0 ==
p : ~ --* (B-IB) 0
~qB
by
is the
~
~q : ~ - - ~ .
vector
space
On a
given by data as above,
determined
given
p
is
([3]) by the object
0 A2, and the pair of isomorphisms
B •
(~ ~I~
42
A2)
q
~qB •
(~ O F
A 2) ~ ~qB • ~q(~ O F
q
where
one uses
the canonical
~q(42 )
A 2)
isomorphism
~ @F
A2 ~ ~q~ O F q
q
which
is the identity on the subgroup
Fq
®F
A2"
q
The diagram
, ~qB"
q
A2
350
× BGL(]Fq) +
• ~
Spt 0 (A)
•
× BGL(~) +
i-~ q
Spt 0 (B)
II
Spt 0 (P )
~ S p t 0 (__B)
Spt 0 (P)
and the b o t t o m row is a fibre sequence,
zeroth spaces of a cofihre sequence of spectra.
As both
group complete,
and
(B-IB) 0
are
as the sequence of
Thus Q u i l l e n ' s
c o n j e c t u r e is e q u i v a l e n t to the s t a t e m e n t that
equivalence.
• Spt 0 (B-IB)=
_ 0)
u
S p t 0 (A)
commutes,
D BGL (k) +
Spt0(P)
is an
c o n n e c t e d and so
this is in fact e q u i v a l e n t to the functor
p : P --~ (B-IB) 0
b e i n g a h o m o t o p y equivalence.
So far, I h a v e been unable to show this, but there are signs
that it is true.
One could try to appeal to Q u i l l e n ' s T h e o r e m A.
I k n o w the fibre
(0,0)/p
that all
torsors
is c o n t r a c t i b l e by Lang's T h e o r e m
for the F r o b e n i u s action on
w i t h t r i v i a l i z a t i o n unique up to
is in general d i s c o n n e c t e d ,
contractible.
G L n ( ~ q) .
n
(~)
are t r i v i a l
Unfortunately,
(B,B)/p
although I s u s p e c t each c o m p o n e n t is
One could hope to show
c o n s i d e r i n g the G r o t h e n d i e c k
GL
[8]
spectral
H*(p)
is an i s o m o r p h i s m by
sequence
HP((~-IB) 0,
Hq(/p))
~ HP+q(P)
and p r o v i n g it c o l l a p s e s by
a n a l y s i s of
H*(/p)
and the action of the
GL
n
(~)
thing like Tits b u i l d i n g s seems to play a role here.
reader is invited to try to make sense of this.
on it.
Some-
The interested
351
§7.
Axiomitization
of K-theory
as a generalized h o m o l o g y
theory on
SymMon.
I will give an axiomatiziation
The axioms are reminiscent
characterizing
of the usual axioms
homology theory on Top if one accepts
Sym-Mon-hocolim
reassuring
is the analogue
the idea that the appropriate
cone.
picture of K-theory as a g e n e r a l i z e d
This gives a
homology
theory on
Sym Mon,
and suggests
homology
theories on S~ectra ought to be true for K-theory.
of the usual theorems
I do not see how to characterize
or exact categories
Consider
in any similar
the following
I.
(Homotopy axiom).
If
with
~01H,(F)
~01H,(B);
Here
or
H,(A)
[14].
By
with respect
known,
Spt0(F)
II.
space.
By
then
to the statement
~01H,(A),
that
from Sym Mon
is an isomorphism.
A, as in
[5], IX,
~0A C H0(A).
H.(Spt 0 A),
K,(F)
localized
As is well
so this axiom is
is an isomorphism
if
is a h o m o t o p y equivalence.
(Cofibre
sequence
axiom).
For
§6,
H,(B__A), the h o m o l o g y of
I mean the homology
subset
to
to
an
after group completion,
K,(F)
is isomorphic
is isomorphic
K,
is a m o r p h i s m that induces
~-coefficients
to the m u l t i p l i c a t i v e
~01H,(A)
equivalent
H,(A)
to rings
graded abelian groups.
is h o m o l o g y of the category
[14],
the classifying
~
restricted
about
fashion.
F : A--~ B
i s o m o r p h i s m on homology
: z01H,(A)
K-theory
four axioms on a functor
to the category of n o n - n e g a t i v e l y
K..
for a generalized
of the m a p p i n g
that analogues
the functor
F : A --~B
a morphism,
let
352
--~P
be the m a p p i n g cone on
F.
Then there is a long exact
sequence
--~ Ki+ 1 (P)
~
~ K i (A)
Here the m a p p i n g cone
as in §4, Example
III
zi (P) ~ . . . .
• K i (B)
is the
Sym M o n - h o c o l i m of a diagram
i.
(Continuity axiom).
If
A.,
E 1
i e I
is a d i r e c t e d system of
symmetric m o n o i d a l categories,
li~ K,(Ai ) & K , ( l i ~ ~i ) .
IV
2Z
(Normalization axiom).
integers
Let
n, and whose m o r p h i s m s
be the c a t e g o r y whose objects are
are all identity morphisms.
have the symmetric m o n o i d a l structure
K 0 (2Z) = ~ ,
Theorem:
K i(zZ) = 0
If
K.
for
n • m = n+m.
is any functor from
S[m Mon
F : ~
bB
the h o m o t o p y c a t e g o r y of
Sym Mon
to n o n - n e g a t i v e l y
such that
SPectra.
then
K,
is
Spt.
K.
induces a functor
(obtained by f o r m a l l y
Spt(F)
is an equivalence)
into
suppose first one knew the induced
map of h o m o t o p y c a t e g o r i e s was an equivalenoe.
homotopy.
~
B e c a u s e of the h o m o t o p y axiom,
out of the h o m o t o p y c a t e g o r y of
i n v e r t i n g all
2Z
i > 0.
isomorphic to algebraic K-theory,
Idea of Proof:
Let
Then
graded abelian groups s a t i s f y i n g the above four axioms,
naturally
z o (~) ---, o
Then one shows
K.
is stable
Using a x i o m II and the tower of h i g h e r c o n n e c t e d coverings
of a system, w h i c h is a sort of u p s i d e - d o w n P o s t n i k o v tower,
reduce to c h e c k i n g
K.
is ~
on E i l e n b e r g - M a c L a n e spectra.
one can
Using
a x i m m II again, one can shift d i m e n s i o n s until one is d e a l i n g w ~ t h
353
K(z,0)-spectra.
generated;
By a x i o m III, reduce to the case
by axiom II, to the case
again to the case
~ = ~;
~
is cyclic;
z
is f i n i t e l y
and by axiom II
which holds by axiom IV.
While I do not k n o w the two h o m o t o p y c a t e g o r i e s are the same,
I can show the h o m o t o p y c a t e g o r y of
Spectra,
Spt 0.
Sym Mon
is a retract of that of
and that the r e t r a c t i o n does not change the h o m o t o p y type of
Proof of this involves 2 - c a t e g o r y theory, and a g e n e r a l i z a t i o n
of the t h e o r e m of
§5, so I'll say no more about it.
b e t w e e n the h o m o t o p y c a t e g o r i e s
This relation
is strong enough to make possible an
a r g u m e n t a l o n g the lines of the first paragraph.
354
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