Download K-theory of Waldhausen categories

K-theory of Waldhausen categories
Riccardo Brasca
1
Categories with cofibrations
Definition 1.1 Let C be a category and let A ∈ ob(C) be an object of C.
If A is both initial and final, we call it a zero object and we indicate it with
the symbol 0.
Note that a zero object, if exists, it is uniquely determinated up to a
unique isomorphism. If a category C has a zero object all topological spaces
constructed from C become based topological spaces in a natural way. From
now on we suppose that all our categories have a zero object 0.
Definition 1.2 A category with cofibrations is a category C with a distinguished class of morphisms, closed under composition, called cofibrations, and
denoted with the arrow , such that the following properties are satisfied:
1. every isomorphism is a cofibration;
2. for all objects A of C, the morphism 0 A is a cofibration;
3. the pushout
A /
/
C /
/
B
B
`
A
C
exists provided that A B is a cofibration and the morphism C B qA C is a cofibration too.
Definition 1.3 An exact functor F : C → D between categories with cofibrations is a functor that preserves zero objects, cofibrations and pushout of
cofibrations.
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If C is a category with cofibrations the coproduct of A and B always
exists (it is A q0 B) and we can define the cokernel of a cofibration A B
as B/A := B qA 0. In this case we write
A B B/A
and we call such a sequence cofibration sequence or exact sequence. Note that
the cokernel is uniquely determinated up to a unique isomorphism.
Example 1.4 Clearly every abelian category becomes a category with cofibrations if we take for cofibrations the monomorphisms.
Example 1.5 Let C be an exact category, embedded in an abelian category
A, C becomes a category with cofibrations if we take for cofibrations the
admissible monomorphisms.
Proof. The only non trivial fact is that cofibrations are closed under pushout.
If A B is a cofibration then in A there is an exact sequence 0 → A → B →
C → 0 of objects of C. Let A → D be any morphism in C. The pushout
B qA D exists in A and the morphism D → B qA D is a monomorphism.
We have the exact sequence
a
0→D→B
D → C → 0,
A
which shows that, being C closed under extensions, B qA D is in C and that
D B qA D is a cofibration.
2
Waldhausen Categories and K0
Definition 2.1 A Waldhausen category is a category with cofibrations C
equipped with another class of morphisms closed under composition, called
∼
weak equivalences and denoted with A −
→ B, such that:
1. every isomorphism is a weak equivalence;
2. if we have the commutative diagram
Co
A /
∼
C0 o
∼
A0 /
∼
/
/
B
∼
B0
then B qA C −
→ B 0 qA0 C 0 is a weak equivalence.
2
Definition 2.2 Let C and D be Waldhausen categories. A functor F : C →
D is an exact functor if it is an exact functor between the corresponding
categories with cofibrations and preserves weak equivalences.
If C is an exact category it becomes a Waldhausen category if we take
the monomorphisms for cofibrations as above and the isomorphisms for weak
equivalences.
Definition 2.3 Let C be a small Waldhausen category. The group K0 (C)
is defined as the abelian group generated on symbols [A], one for each object
of C, with the following relations:
∼
1. A −
→ B ⇒ [A] = [B];
2. A B B/A ⇒ [B] = [A] + [B/A].
If there is an exact equivalence of categories F : C → D for some small Waldhausen category D we say that C is essentially small and we set K0 (C) =
K0 (D), clearly this is well defined. By definition an exact functor F : C → D
induces an homomorphism K0 (F ) : K0 (C) → K0 (D).
In this way K0 is a functor
K : Wcat → grp
C 7→ K0 (C)
from the category of small Waldhausen categories and exact functors to the
category of groups.
Clearly if C is an exact category this definition agrees with the classical
definition of K0 (C). From now on we suppose that all our categories are
essentially small.
Example 2.4 Let C be the category of based CW-complexes with only
finitely many cells and cellular maps. If we take for cofibrations the injective
maps and for weak equivalences the homotopy weak equivalences (maps that
induce isomorphism on homotopy groups) we obtain a Waldhausen category.
We claim that K0 (C) ' Z. Indeed Dn /S n−1 ' S n , so we have S n−1 ∼
Dn S n , but 0 −
→ Dn so [S n ] = −[S n−1 ] for n ≥ 1 and by induction [S n ] =
(−1)n [S 0 ] for each n. Every CW-complex is obtainded with some glueing
of disks P
so K0 (C) is generated by [S 0 ]. The reduced Euler characteristic
χ̃(X) = n (−1)n+1 rk H̃n (C, ) gives a surjection K0 (C) → Z as required.
3
Definition 2.5 Let F , F 0 and F 00 be functors between Waldhausen categories C and D. A sequence F → F 0 → F 00 of functors and natural transformation is called an exact sequence of functors if for every object A of C we
have the exact sequence
F (A) F 0 (A) F 00 (A).
In this case we write F F 0 F 00 and we have K0 (F 0 ) = K0 (F ) + K0 (F 00 ).
Definition 2.6 Let C be a Waldhausen category. We say that C is saturated
if being f ◦ g a weak equivalence implies that f is a weak equivalence if and
only if g is.
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Chain complexes
Fix an exact category C, embedded in an abelian category A. The category
Chb (C) of bounded chain complexes with objects in C is a Waldhausen category, where the cofibrations are the degreewise monomorphisms and weak
equivalences are quasi-isomorphisms in A.
Proposition 3.1 If C is closed under kernel of surjections then we have
K0 (Chb (C)) ' K0 (C).
Proof. If C. is a chain complex, C[n]. is the chain complex with C[n]i = Ci+n .
If f : C. → D. is a map of complexes, cone(f ). is the complex with cone(f )n =
Cn−1 ⊕ Dn and differentials given by d(a, b) = (−d(a), d(b) − f (a)). If f is the
identity then cone(f ). is split exact, a splitting is given by s(b, c) = (−c, 0).
Given f : C. → D. we have a map α : D. → cone(f ). defined by α(b) = (0, c)
and a map β : cone(f ). → C[−1]. given by β(b, c) = −b. The sequence
0 → D. → cone(f ). → C[−1]. → 0
is exact. Taking for f the identity we have [cone(f ). ] = 0, so −[C. ] = [C[−1]. ]
for each complex C. . The natural functor F : C → Chb (C), that sends an
object of C to the complex concentrated in the 0-th piece, is exact, so it
induces a morphism K0 (F ) : K0 (C) → K0 (Chb (C)). Consider the Euler
characteristic χ : Chb (C) → C, it is an exact functor (it depends only on
the homology of the complex). We have K0 (χ) ◦ K0 (F ) = id. If C. is a
complex with only one non-zero term, at place, say, n, χ(C. ) = (−1)n Cn , but
(−1)n [F (Cn ). ] = [F (Cn )[−n]. ] = [C. ], so in this case K0 (F ) ◦ K0 (χ) = id. If
C. = . . . 0 → Cm → . . . → Cn → 0 → . . . is a complex consider C. /B. , where
B. has Cn at the n-th place and 0 elsewhere, and proceed by induction.
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4
Localization
Let C be a category with cofibrations equipped with two class of morphisms,
v and w, with v ⊆ w. Suppose that both v and w make C a Waldhausen
category when taken as weak equivalences, let us denote these categories with
Cv and Cw . Let Cw the full subcategory of Cv of all w-acylic objects (i.e.
objects A such that 0 A is in w). Note that the inclusion functors are
exact functors.
Proposition 4.1 With notations as above suppose that Cw is saturated and
that each morphism factors as a cofibration followed by a weak equivalence
of v. The inclusion functors induce the exact sequence
K0 (Cw ) → K0 (Cv ) → K0 (Cw ) → 0.
Proof. The right map is onto because w adds more relations on K0 and by
definition the composite of the two maps is 0. Let L be the cokernel of
K0 (Cw ) → K0 (Cv ), it suffices to find an inverse of the induced map L →
∼
K0 (Cw ). Let f : C1 −
→ C2 be a morphism in w. We can factor f as C1 ∼
C −
→ C2 , where the second map is in v. It follows that in K0 (Cv ) we have
[C2 ] = [C] = [C1 ] + [C1 /C]. Being Cw saturated and v ⊆ w, C1 C is in
w. We have the diagram
0o
C1
C1
∼
0o
C1 /
/
C
it follows that 0 ' C1 qC1 0 C qC1 0 ' C/C1 is in w, so C/C1 is in Cw
and [C1 ] = [C2 ] in L. The map λ : K0 (Cv ) → L, λ(C) = [C], is therefore
well defined and it is the inverse required.
5
Higher K-theory
Definition 5.1 If C is any category we denote with ar(C) the corresponding
arrow category, i.e. the category whose objects are morphisms A → B in C,
and whose morphisms are commutative squares
/
A
/
A0
5
B
B0
Definition 5.2 Given a non negative integer n, we denote with [n] the category whose objects are the elements of the set {0, . . . , n}, with a morphism
i → j if and only if i ≤ j.
If we have i ≤ j ≤ k we have the diagram
/
i
/
i
j
k
and so we have a morphism (i → j) → (j → k). Similarly we have a
morphism (i → k) → (j → k).
Definition 5.3 Let ∆ be the simplicial category, i.e. the category whose
objects are the sets [n] = {0, . . . , n} and whose morphisms are non decreasing
functions.
Definition 5.4 Let C be any category. A simplicial object in C is a functor
A : ∆ → C, [n] 7→ An . Given a simplicial object the faces and degeneracies
maps are the maps corresponding to the injective non decreasing maps [n −
1] → [n] and surjective non decreasing maps [n] → [n − 1].
Definition 5.5 Given a simplicial `
set A : ∆ → set, [n] 7→ An its geometric
realization is the topological space (An × ∆n )/ ∼ (∆n is the standard nsimplex and to each An is given the discrete topology), where (x, s) ∼ (y, t)
if and only if f ∗ (y) = x and f∗ (s) = t, for some non decreasing f : [n] → [m],
where f ∗ and f∗ are the induced maps. It can be proved that it is a CWcomplex with an n-cell for each element of An that it is not in the image of
the degeneracies maps.
Definition 5.6 Let C be a category with cofibrations. With Sn C we denote
the category of functors A : ar[n] → C, (i → j) 7→ Aij such that:
1. Aii = 0 for all i;
2. for all i ≤ j ≤ k the induced morphism Aij Aik is a cofibration;
3. for all i ≤ j ≤ k we have the exact sequence Aij Aik Ajk .
Example 5.7 For example, an object of S4 C can be represented as the
6
following diagram
0 = A00 /
/
A01 /
0 = A11
/
/
/
A02 /
A12
0=
/
/
A22 /
A03 /
/
/
/
A23 /
/
/
A13
/
A04
0 = A33 /
A14
/
A24
A34
0 = A44
and two exact sequences are
A01 A02 A12 ,
A02 A04 A24 .
Given an n-filtration of an object A: 0 = A0 A1 . . . An = A
we can always find a diagram as above, with Aij = Ai /Aj , and the various
maps exist by universal properties. So we see that Sn C is equivalent to the
category of n-filtrations in C. The advantage of Sn C is that its definition
is clearly functorial with respect to any non decreasing map f : [n] → [m].
Given such a map we can send (i → j) to f ((i) → f (j)), so by composition
we get a functor Sm C → Sn C. In particular we have, for all i = 0, . . . , n
the face functor ∂i : Sn C → Sn−1 C corresponding to the unique injective non
decreasing map fi : [n−1] → [n] whose image misses i. Similarly we have, for
all i = 0, . . . , n, the degeneracy functor σi : Sn C → Sn+1 C corresponding to
the unique surjective non decreasing map gi : [n + 1] → [n] with two elements
mapping to i. In practice the map ∂0 erases the first row of the diagram,
while ∂i , i ≥ 1, erases the column that contains A01 and the i-th row, and reindexes each Aij . Note that in the category of n-filtrations the ∂i with i ≥ 1
corresponding to ignore the i-th step of the filtration, while ∂0 is not well
defined and requires the choice of each Ai /Aj . The functors σi in practice
add identities in the diagram.
In other words we have proved that S. C is a simplicial category, i.e. a
7
functor
S. C : ∆ → cat
[n] 7→ Sn C
Definition 5.8 Let C be any category. A bisimplicial object in C is a functor
A: ∆ × ∆ → C
([n], [m]) 7→ Anm
Definition 5.9 Let A : ∆ → Top be a topological simplicial space. The
geometric realization of A, denoted with |A|, is defined as in the case of
simplicial set, but the An ’s that appear in qn An × ∆n are given their natural
topology.
Definition 5.10 Given a bisimplicial set A : ∆ × ∆ → set we can construct
its geometric realization in three different ways:
1. for each m, Bm := |A.m | is a topological space, and B. is a topological
simplicial space, we can consider its geometric realization;
2. as above, starting with Cn = |An. |;
3. the diagonal Xn := Ann is a simplicial set, we can consider its geometric
realization.
It turns out that these three constructions give naturally homeomorphic results, denoted with |A|.
Each Sn C is a category, so we can consider its nerve N. Sn C that is a
simplicial set, as for any category. In this way we have a functor
N. S. C : ∆ × ∆ → set
([n], [m]) 7→ Nn Sm C
that is a bisimplicial set, in fact N. Sm C is a simplicial set for all m and Nn S. C
is a simplicial set for all n. Given any simplicial category A : ∆ → cat, we
denote the geometric realization of the bisimplicial set just defined with |A|.
Definition 5.11 Let C be a Waldhausen category. We denote with wC the
subcategory of C whose morphisms are weak equivalences in C. Clearly wC
is a category with cofibrations. By definition |S. wC| the geometric realization
of the bisimplicial set N. S. wC.
8
Proposition 5.12 If C is a Waldhausen category then |S. wC| is a connected
CW-complex.
Proof. The geometric realization of N. S. wC can be obtainded realizing the
diagonal, that is a simplicial set, so |S. wC| is a CW-complex. In this CWcomplex there is only one 0-cell, indeed, regarding |S. wC| as the realization of
the topological simplicial space [n] 7→ | N. Sn wC|, we have an n-cell for each
non-degenerate point of | N. Sn wC| and, being S0 wC the trivial category,
| N. S0 wC| has only one point. So all 1-cells are attached to the same point
and the CW-complex is connected.
Proposition 5.13 If C be a Waldhausen category then Σ| N. S1 wC| (the
reduced suspension) is the 1-skeleton of |S. wC|.
Proof. Again we regard |S. wC| as the realization of the topological simplicial
space [n] 7→ | N. Sn wC|. The degeneracy map σ0 : | N. S0 wC| → | N. S1 wC|
has as image the point corresponding to the diagram 0 0 0, that is 0, if
we identify S1 wC with wC. Therefore the only 1-cell of |S. wC| collapsed to
a point is that corresponding to the base point and the 1-skeleton is exactly
Σ| N. wC|.
The adjoint of the inclusion yelds a natural map | N. wC| ' | N. S1 wC| →
Ω|S. wC|, so a map ob(C) → π1 (|S. wC|).
Proposition 5.14 If C is a Waldhausen category then π1 (|S. wC|) ' K0 (C).
Proof. We have to look at 1-cells, and, by what we have said above, there
is a 1-cell for each object of C (the one corresponding to 0 is collapsed to
the base point). If two points of | N. S1 wC| are connected by a path, the two
corresponding loop are homotopically equivalent, so we can consider only 1cells arising from elements of π0 (| N. S1 wC|), therefore 1-cells corresponding
to weak equivalence classes of objects of C. Therefore the generators of
π1 (|S. wC|) are the same as the generators of K0 (C). The only relations come
from attaching 2-cells. Again we can consider only elements of π0 (| N. S2 wC|)
and we have ∂1 (x) = ∂2 (x) + ∂0 (x) for each x ∈ π0 (| N. S2 wC|). But elements
of π0 (| N. S2 wC|) are weak equivalence classes of exact sequences, and ∂i
sends the exact sequence to its i-th term. In this way also the relations of
the two groups are equal as required.
9
Definition 5.15 Let C be a Waldhausen category. Its algebraic K-theory
space is K(C) := Ω|S. wC| and its algebraic K-theory groups are
Kn (C) := πn (K(C)) = πn+1 (|wS. C|).
By proposition 5.14 this definition is consistent with the one of K0 (C) given
above. Given two Waldhausen categories C and D, and given an exact
functor F : C → D, it is induced a continuous maps
K(F ) : K(C) → K(D),
and therefore homomorphisms
Kn (F ) : Kn (C) → Kn (D).
In other words K is a functor
K : Wcat → Top
C 7→ K(C)
from the category of Waldhausen categories and exact functors to the category of topological spaces.
Definition 5.16 An H-space X is a topological space X, toghether with a
map X ×X → X that satifies, up to homotopy, the axioms of the composition
law of a group.
Note that the coproduct functor q : C×C → C extends to each categories
Sn wC and it can be proved that it makes K(C) into an H-space. Furthermore
the map induced by an exact functor is an H-space map, so the functor K
has value in the category of H-spaces.
6
Relations with exact categories
We have already seen that every exact category C is a Waldhausen category
in a natural way. We want now to prove that the two possible definitions of
πn (C) agree.
Definition 6.1 If A. : ∆ → C, [n] 7→ An is a simplicial object in any category
C its Segal subdivsion Sub(A. ). is the simplicial object Sub(A. ). : ∆ → C,
[n] 7→ A2n+1 , the faces maps ∂i : A2n+1 → A2n+3 are defined as follows:
∂i0 = ∂i ◦ ∂2n+1−i , the definition of degeneracies maps is similar.
10
Example 6.2 Let C be any category. We denote by Sub(C) its Segal subdivision category, i.e. the category whose objects are morphisms A → B in
C and whose morphisms are commutative squares
/
AO
/
A0
B
B0
Note the diffence between this definition and that of ar(C). There are two
natural functor dom, cod : Sub(C) → C, domain and codomain: the first is
controvariant and the second covariant. If C is a category with an initial
object ∗ (in particular if C is a Waldhausen category) then dom and cod are
id
homotopy equivalences. Indeed the object of cod /x ∗ → x → x is initial,
so cod /x is contractible and by Quillen’s theorem A cod is an homotopy
equivalence. The proof for domain is similar. If we start with any category
C we have Sub(N. C). = N. Sub(C) so, if C has an initial object, | N. C| and
| Sub(N. C). | are homotopically equivalent.
Proposition 6.3 Let A : ∆ → set, [n] 7→ An , a simplicial set, then |A| and
| Sub(A). | are homeomorphic.
Proof. For the proof it is convenient to think to ∆n (the standard n-simplex)
as the set (t1 , . . . , tn ) ∈ Rn such that 0 ≤ t1 ≤ . . . ≤ tn ≤ 1. Let i : ∆n →
∆2n+1 be the map i(t1 , . . . , tn ) = 12 t1 , . . . , 12 tn , 12 , 1 − 21 t1 , . . . , 1 − 12 tn . The
maps A2n−1 × ∆n → A2n+1 × ∆2n+1 given by (a, x) 7→ (a, i(x)) induce a
continuous map | Sub(A)| → |A|. Note that ∆n can be subdivided in 2n
n-simplexes using the middle point of each edge. This ‘2n simplexes can
be indexed ∆nf , where f runs through a set of 2n maps [2n + 1] → [n]:
∆nf is the composition f∗ ◦ i (f∗ is the map induced by the structure of
cosimplicial object of [n] 7→ ∆n ). The maps An × ∆nf → A2n+1 × ∆n given by
(a, f∗ i(x)) 7→ (f ∗ (a), x) (f ∗ is the map induced by the structure of simplicial
object of A) induce a map |A| → | Sub(A)| that is the invese of the map
defined above.
Definition 6.4 Let A : ∆ → cat, [n] 7→ An , be a simplicial category. We
define sn (A) := ob(An ). We have that s. is a functor
s. : ∆ → set
[n] 7→ sn (A)
11
so a simplicial set. If C is a category with cofibrations we set sn C := sn (S. C).
Definition 6.5 Let C be an exact category. We denote with in QC the
category whose object are elements of Nn QC and whose morphisms are
commutative diagram with isomorphisms at each level.
We have that i. QC is a functor
i. QC : ∆ → cat
[n] 7→ in QC
so a simplicial category. Clearly sn (i. QC) = Nn QC.
Fix now an exact category C. We are going to show that the two definitions of πn (C) agree, proving that |S. wC| and | N. QC| are homotopically
equivalent. We prove this constructing a commutative diagram
|s. wC| o
r1
r3
| Sub(s. wC). |
|S. wC|
r2
/
r7
/ | N. QC|
r4
| Sub(S. wC). |
r5
/
r6
|i. QC|
in which all arrow will be homotopy equivalences.
The arrow r1 is defined and proved to be homotopy equivalences in proposition 6.3.
For the arrow r2 note that proposition 6.3 implies that | Nn S. wC| and
| Sub(Nn S. wC)| are homotopically equivalent for each n, so the same is
true for the geometric realization of the corresponding simplicial topological spaces.
If A : ∆ → C is a simplicial object we denote with A∗ : ∆/[1] → C the
composite with the furgetful functor.
Definition 6.6 Let A, B : ∆ → set be two simplicial sets and let f, g : A →
B be two simplicial maps. A simplicial homotopy from f to g is a natural
transformation α : A∗ → B ∗ such that αi0 = f and αi1 = g, where ij is the
object of ∆/[1] that on each [n] is the constant map to j ∈ [1].
It can be proved that a simplicial homotopy induces an homotopy between
the two continuous maps corresponding to f and g.
Proposition 6.7 Let F : C → D an exact functor between categories with
cofibrations. Then F induces a simplicial map s. F : s. C → s. D and a natural
transformation between two functors induces a simplicial homotopy.
12
Proof. The first part is clear, for the second let F, G : C → D be two exact
functors between categories with cofibrations and let η : C × [1] → D be a
natural transformation between F and G. To define a simplicial homotopy α
take any a : [n] → [1], we have to define αa : sn C → sn D. Given an element
of sn C, so a functor A : ar[n] → C let us define A0 : ar[n] → D as the
composite
ar[n]
(A,a∗ )
/ C × ar[1]id × cod/ C × [1]
η
/
D,
where a∗ (i → j) 7→ a(i) → a(j) and cod is the obvious codomain functor.
Now we are ready to define, and prove that it is an homotopy equivalence, the arrow r3 . Remember that |S. wC| is the geometric realization of the
bisimplicial set N. S. wC and that it can be obtainded realizing the simplicial
topological spaces [n] 7→ | Nn S. wC|. By definition we have s. wC = N0 S. wC.
Furthermore N1 S. wC = s. ar(wC) (where a cofibration in ar(C), for a category with cofibration C, is a cofibration at both levels). Therefore the first
faces and degeneracies maps arise from functors ar(wC) → wC that, being
each morphism in wC an isomorphism, are equivalences of categories and so
they are homotopy equivalences. The case n > 1 is the same, but instead
of the arrow category we have to consider the category on n-morphisms.
It follows that the geometric realization of the simplicial topological space
[n] 7→ | Nn S. wC| is homotopically equivalent to that of N0 S. wC = s. wC and
this allows us to define the homotopy equivalence r3 .
The arrow r4 is defined by composition of homotopy equivalences, so it
is an homotopy equivalence too.
The arrow r5 is defined noting that an object of Sub(S. wC)n = S2n+1 wC
gives an object of in QC as in the following example, with n = 1: starting
from a diagram
0 = A00 /
/
A01 /
/ A02 /
/ A03
/ A12 /
/ A13
/ A23
0=
A11 /
0 = A22 /
0 = A33
13
We have the object A12 A02 A03 of i1 QC. Similarly, starting with an
object of Sub(S. wC)n = S2n+1 wC, we obtain an object of in QC. The associated functor is an equivalence of categories because in wC every morphism is
an isomorphism, so the geometric realizations of the two simplicial categories
are homotopically equivalent.
The definition of r6 and the proof that is an homotopy equivalent is quite
techical and we won’t do it, but is very similar to that of r3 , observing that
sn (i. QC) = Nn QC.
Now we have only to define r7 it will be automatically an homotopy
equivalence, being the composition of homotopy equivalence. Clearly an
element of Sub(s. wC). = s1 wC gives an element of ob(C) = N0 QC. As
for r5 , an element of Sub(s. wC)1 gives a morphism in QC, so an element of
N1 QC (see the diagram above). Let us see also the case n = 2, the cases
with n > 2 are very similar: by definition an element of Sub(s. wC)2 = s5 wC
is a diagram
0 = A00 /
/
A01 /
0 = A11 /
/A
/
/
A03 /
/ A12 /
/
A13 /
/
A23 /
02
0 = A22 /
A04 /
/
A14 /
/
A24 /
/
0=
A33 /
A05
/
A15
/
A25
/
/
0 = A44
/
/
A34 /
/
A35
A45
0 = A55
and we obtain an object of N2 QC, the composite of A23 A13 A14 with
A14 A04 A05 .
7
Applications
In this section we fix a Waldhausen category C.
14
Definition 7.1 Let C be a Waldhausen category. The category E(C) is the
category whose object are exact sequences A B C in C. We make
E(C) in to a Waldhausen category as follows: a morphism
A1 /
f1
B1 /
/
A2
//A
f2
/ B2
//
3
f3
B3
is a cofibration if f1 , f2 and the induced morphism B1 qA1 A2 → B2 are
cofibrations, and it is a weak equivalence if all fi ’s are weak equivalences.
There are three natural exact functors s, t, q : E(C) → C, which send the
exact sequence A B C in C to A, B and C.
Theorem 7.2 Let F F 0 F 00 be an exact sequence of functors between
Waldhausen categories C and D. Then K(F 0 ) = K(F ) + K(F 00 ) as H-space
maps and therefore Kn (F 0 ) = Kn (F ) + Kn (F 00 ).
Proof. Giving an exact sequence of functors F , F 0 and F 00 is, by definition,
the same thing as giving a functor G : C → E(D), so we may assume that
C = E(D) and prove that K(t) = K(s) + K(q). There is a natural functor
q : D × D → E(D) which sends (A, B) to A A q B B: it can be proved
that K(q) is an homotopy equivalence. Replacing E(D) with D × D it is
obvious that K(t) = K(s) + K(q).
Definition 7.3 A sequence 0 → A1 → . . . → An → 0 is called admissibly
exact if each map decomposes as Ai+1 Bi Ai and each Bi Ai Bi−1
is an exact sequence. A sequence 0 → F1 → . . . → Fn → 0 of functors between Waldhausen categories is called admissibly exact if every corresponding
sequence of objects is admissibly exact.
Proposition 7.4 Let 0 → F1 → . . . → Fn → 0 be an admissibly exact
sequence
of exact functors between Waldhausen categories. Then we have
P
i
i (−1) K(Fi ) = 0 as H-space maps.
Proof. It follows by induction on n, by theorem 7.2.
Fix now a quasi compact scheme X and a vector bundle E on X of
π
rank r. Let P = P(E) → X the associated projective space bundle. For
i = 0, . . . , r − 1 we have a functor
ui : VB(X) → VB(P)
F 7→ π ∗ (F)(−i)
15
We are going to generalize the fact that K0 (P) ' K0 (X)r proving that K(P)
and K(X)r are homotopically equivalent. First of all we want to replace
K(P) with K(MR(P)).
Proposition 7.5 The inclusion MR(P) ⊆ VB(P) induces an homotopy
equivalence K(MR(P)) ' K(P).
Proof. Being K(P) ' limn K(MR(P)(n)) it suffices to prove that each inclu−→
sion in : MR(P)(n) ⊆ MR(P)(n − 1) induces an homotopy equivalence. We
have exact functors
λi : MR(P)(n − 1) → MR(P)(n)
F 7→ F(i) ⊗OX
i
^
E
and an exact sequence
0 → F → λ1 (F) → . . . → λr (F) → 0.
P
It follows by proposition 7.4 that ri=1 (−1)i K(λi ) is an homotopy inverse of
K(in ).
Theorem 7.6 There is an homotopy equivalence K(P) ' K(X)r .
Proof. It suffices to prove that K(MR(P)) ' K(X). Remember that we are
in an exact category, so product and coproduct are naturally isomorphic.
The functors ui give a functor u : VB(X)r → VB(P) → MR(P), where the
last arrow is an homotopy equivalence. We have seen that there are exact
functors Ti : MR(P) → VB(X),
assemble to form an exact functor
L which
r
i
t : MR(P) → VB(X) , t = (−1) Ti . By Quillen’s canonical resolution of
a Mumford-regular bundle we have an exact sequence
0 → ur−1 ◦ Tr−1 (F) → . . . → u0 ◦ T0 (F) → F → 0,
P
i+1
with F Mumford-regular. This shows that r−1
K(ui ◦ Ti ) = id, so
i=1 (−1)
K(t) is, up to homotopy, a right inverse for K(u). Now define the exact
functors vi : MR(P) → VB(X) by vi (F) = π∗ (F(i)), this assemble to form
an exact functor v : MR(P) → K(X)r . We have seen that the matrix corresponding to K(v ◦ u) is triangular, and that its diagonal entry are homotopy
to the identity, so K(u) has, up to homotopy, a left inverse, as required.
16
Definition 7.7 Let C be a Waldhausen category. A cylinder functor on C is
a functor T : ar(C) → C with three natural transformations j1 : dom → T ,
j2 : cod → T and p : T → cod such that, for every morphism f : A → B in
C the following diagram commutes.
j1
A DD
/
T (f ) o
j2
zzz
DD f
zzzzz
DD
z
p
z
z
DD
D! zzzzzzz
B
B
Furthermore the following axioms must be satisfied:
1. T (0 A) = A for every object A;
2. given f : A → B, the morphism j1 q j2 : A q B T (f ) is a cofibration;
∼
3. given (a, b) : f → f 0 , if a and b are weak equivalences then T (f ) −
→
0
T (f ) is a weak equivalence;
4. given (a, b) : f → f 0 , if a and b are cofibrations then T (f ) T (f 0 )
is a cofibration and the induced map A0 qA T (f ) qB B 0 T (f 0 ) is a
cofibration, where f : A → B and f 0 : A0 → B 0 .
We always assume the cylinder axiom: all morphisms p : T (f ) → B are weak
equivalences.
Definition 7.8 Let C be a Waldhausen category. We say that C satifies the
extension axiom if, given any map f : E → E 0 between two exact sequences
E and E 0 , such that the maps between the first and last terms are weak
equivalences the the other map is a weak equivalence too.
Theorem 7.9 We use notations as in proposition 4.1. Suppose that C is
saturated, satifies the extension axiom and has a cylinder functor. Then
K(Cw ) → K(Cv ) → K(Cw )
is an homotopy fibration sequence.
This theorem is the fundamental step in the proof of the following theorem
Theorem 7.10 Let C an exact category, then K(Chb (C)) and K(C) are
homotopically equivalent.
17