K–Theory for Triangulated Categories 3\frac{1}{2}(B): A Detailed Proof of the Theorem of Homological Functors

Amnon Neeman (amnon.neeman@anu.edu.au)

The Australian National University

Abstract. This article is a sequel to K–theory for triangulated categories 3 \frac{1}{2} (A). In this article, we complete the proof of the main theorem of the previous article.

AMS subject classification: 18F25, 18E30, 19D99

I would like to dedicate this article to Daniel Quillen and to Robert Thomason. This year we celebrate Quillen’s 60th birthday, and mourn the fifth aniversary of Thomason’s untimely death. Thomason helped me a tremendous amount, in the early days of this article. Quillen helped Thomason, when the étale descent theorem was still very new. Both Quillen and Thomason are outstanding mathematicians. They are also kind and decent human beings. Each went out of his way to help a young mathematician, whose work had been overlooked.

Table of Contents

7 Introduction 2 8 The proofs of the easy Steps 2 9 The two quasi–fibrations 17 10 Proof of Step 4.1 36

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.

7. Introduction

In the article K–theory for triangulated categories 3\frac{1}{2}(A), the proof of the main theorem was incomplete. We gave the proof, modulo seven technical lemmas. In the present article, we prove the seven technical lemmas. The reason for this organisation of the proof is that the seven technical lemmas have nearly identical proofs. It seems most illuminating to prove them all together.

We should very briefly remind the reader of the main theorem of K–theory for triangulated categories 3\frac{1}{2}(A).

Theorem. Let A and B be abelian categories. Any bounded δ–functor F : A ⟶ B induces a natural map in K–theory.

8. The proofs of the easy Steps

Step 2.1, Step 2.2, Step 3.1 and Step 6.2 have essentially identical proofs. For the purpose of exposition, it seems simplest to carefully go through the proof of Step 3.1, and then explain how each of the other Steps follows by an identical argument. We begin by recalling the statement of Step 3.1.

Step 3.1: The natural map

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

induces a homotopy equivalence.

gr85.tex; 3/04/2000; 15:34; p.2

K–Theory for Triangulated Categories 3 \frac{1}{2} (B) 3

Proof. The Segal fiber is the simplicial set [figure] 0  Gr[figure: [1,n]] X  . We will

prove it contractible. To do so, we will construct two simplicial homotopies, which combine to connect the identity to a contracting map. Recall that a simplicial homotopy takes an integer i with 0 ≤ i ≤ s, and an s–simplex, to an (s + 1)–simplex. The first homotopy we want to consider takes the integer i and the s–simplex

y_{s}  =  ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}] )

to the (s + 1)–simplex

gr85.tex; 3/04/2000; 15:34; p.3

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ii}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{s–i+1 \atop terms}}]

From this diagram, the reader can read off the form of the simplex, and the various objects that occur in different parts of the diagram. To define a simplex, however, one also need to know the maps (of degree 0) and the differentials (maps of degree 1). Here we go.

DEFINITION 8.1. The morphisms in the homotopy cell are given as follows. First, maps of objects of the same form are the obvious diago-

nals. For instance, the map Y ⊕X ⟶ Y′ ⊕X′ are matrices ( \begin{matrix}n_{1} & 0 \\ 0 & n_{2} \end{matrix} ) where n_{1} : Y ⟶ Y′ n_{2} : X ⟶ X′ are the natural maps, that is the ones from the original simplex. Furthermore, maps connecting objects from the original simplex are the original maps. We need only define the maps involving at least one object of the form Y ⊕ X or X ⊕ Z.

8.1.1. The vertical morphisms X ⟶ Y ⊕ X′ are taken to be ( \begin{matrix}0 \\ n \end{matrix} ),

where n : X ⟶ X′ is the “natural” map; that is, the one given in the original simplex.

8.1.2. The vertical morphisms Y ⊕X ⟶ Y′ are the matrices ( n_{1} n_{2} ) where n_{1} : Y ⟶ Y′ and n_{2} : X ⟶ Y′ are the natural maps.

8.1.3. The map X ⊕ Z ⟶ Z′ is the matrix ( 0 n ), where n : Z ⟶ Z′ is the natural map.

8.1.4. The map X⊕Z ⟶ Y is the matrix ( 0 n ), where n : Z ⟶ Y is the natural map.

8.1.5. The map Z ⟶ X ⊕Z′ are taken to be ( \begin{matrix}0 \\ n \end{matrix} ), where n : Z ⟶

Z′ is the natural map.

8.1.6. The map Z ⟶ Y ⊕X are taken to be ( \begin{matrix}n \\ 0 \end{matrix} ), where n : Z ⟶

Y is the natural map.

8.1.7. The maps X ⊕ Z ⟶ Y ⊕ X′ are matrices ( \begin{matrix}−n_{1} & n_{3} \\ n_{2} & 0 \end{matrix} ) where

n_{1} : X ⟶ Y, n_{2} : X ⟶ X′ and n_{3} : Z ⟶ Y are all the natural maps. Thus, up to one negative sign, the maps are all natural.

Now come the differentials:

8.1.8. The differential Y ⊕ X ⟶ Σ {Y′ ⊕ X′} is the zero map.

8.1.9. The differentials Y′ ⟶ Σ \{Y ⊕ X\} are matrices ( \begin{matrix}0 \\ n \end{matrix} ) where

n : Y′ ⟶ ΣX is natural.

8.1.10. The differential Y ⊕ X′ ⟶ ΣX is the map ( 0 n ) where n : X′ ⟶ ΣX is natural.

8.1.11. The differential Y ⊕ X′ ⟶ Σ \{X ⊕ Z\} is the map ( \begin{matrix}0 & 0 \\ n & 0 \end{matrix} )

where n : Y ⟶ ΣZ is natural.

8.1.12. The differential Y ⊕ X ⟶ ΣZ is the map ( n 0 ) where n : Y ⟶ ΣZ is natural.

8.1.13. The differential X ⊕ Z′ ⟶ Σ \{X ⊕ Z\} is the map ( \begin{matrix}0 & 0 \\ 0 & n \end{matrix} )

where n : Z′ ⟶ ΣZ is natural.

8.1.14. The differential X ⊕ Z′ ⟶ ΣZ is the map ( 0 n ) where n : Z′ ⟶ ΣZ is natural.

8.1.15. The differential Z′ ⟶ Σ \{X ⊕ Z\} is the map ( \begin{matrix}0 \\ n \end{matrix} ) where

n : Z′ ⟶ ΣZ is natural.

8.1.16. The differential Y ⟶ Σ \{X ⊕ Z\} is the map ( \begin{matrix}n_{1} \\ n_{2} \end{matrix} ) where

n_{1} : Y ⟶ ΣX and n_{2} : Y ⟶ ΣZ are natural.

The remaining differentials are ones which do not involve Y_{i0} ⊕ X_{rj} or X_{r0} ⊕Z_{ij}. In other words, they are maps between objects in the original simplex; for these, we take always the “natural” map.

I have now defined a large diagram. It is an unpleasant computation, which I leave to the reader, that with definitions 8.1.1–8.1.7 the diagram commutes. Before we look at the differentials of 8.1.8–8.1.16 we have a commutative diagram. It also needs to be checked that the differentials are coherent. In other words, composing a differential with a map which is not a differential gives a differential. To repeat the same thing a third way, composing a map of type 8.1.8–8.1.16, either on the right or on the left, with a map obtained from 8.1.1–8.1.7, gives a map as in 8.1.8– 8.1.16. Finally, it needs to be checked that with the definitions of the differentials as in 8.1.8–8.1.16, any differential vanishes if its domain A and codomain ΣB are such that A and B are either in the same row or the same column of the simplex.

REMARK 8.2. There are non–computational ways to prove the above. However, one needs to introduce several other ideas. No one has ever been happy with my non–computational explanation of why this works; a written version may be found in Section III.1.

Next we should prove that all the squares in the diagram give rise to exact sequences in A. Here we appeal to Lemma 4.3. It suffices to check only the small squares, that is ones formed by taking two adjacent rows and two adjacent columns. There is a small list of cases to consider, and we leave this to the reader.

To show that we have a well defined cell, the morphisms must be shown to satisfy the restrictions that would qualify the diagram to belong to the simplicial set. In particular, we need to show that in the top square, the vertical maps are all epi. Why should the maps Y_{k0} ⊕ X_{rl} ⟶ Y_{k′0} ⊕ X_{rl} and Y_{k0} ⊕ X_{rl} ⟶ Y_{k′l} be epi?

For the map Y_{k0} ⊕ X_{rl} ⟶ Y_{k′0} ⊕ X_{rl} the epi–ness is obvious, since the map is obtained from the epi Y_{k0} ⟶ Y_{k′0} by adding X_{rl}. For the map Y_{k0} ⊕ X_{rl} ⟶ Y_{k′l}, decompose it as

Y_{k0} ⊕ X_{rl} ⟶ Y_{kl} ⟶ Y_{k′l}.

We know that Y_{kl} ⟶ Y_{k′l} is epi. It suffices therefore to prove that Y_{k0} ⊕ X_{rl} ⟶ Y_{kl} also is. But there is a square

\begin{matrix}Y_{k0} & ⟶ & Y_{kl} \\ ↑ & & ↑ \\ X_{r0} & ⟶ & X_{rl} \end{matrix}

and hence a long exact sequence,

H^{n} (Y_{k0} ⊕ X_{rl}) ⟶ H^{n} (Y_{kl}) ⟶ H^{n+1} (X_{r0}) .

But we know that X_{r0} ∈ T_{[1,n]}, and hence H^{n+1} (X_{r0}) = 0. From the exact sequence it then follow that Y_{k0} ⊕ X_{rl} ⟶ Y_{kl} is indeed epi.

Finally, it needs to be checked that the map sending an s–simplex and an integer i with 0 ≤ i ≤ s to an (s + 1)–simplex defined above gives a simplicial homotopy. In our case, this means concretely that the (s + 1)–simplex should not depend very much on the integer i. By its definition, it clearly does not. What happens to the k^{th} row depends on whether k is larger than i or smaller than it, but not on its precise value. Similarly, the maps between the rows depend only on where i is relative to the rows, not on the precise value of i. This immediately implies that we have a well defined simplicial homotopy.

The homotopy connects the identity to a map Θ, taking the s– simplex

y_{s} = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}] )

to the simplex

Θ(y_{s}) = ( [figure: ⋮; ..; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{s0}⊕X_{r0}; Y_{s0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}] )

gr85.tex; 3/04/2000; 15:34; p.8

K–Theory for Triangulated Categories 3 \frac{1}{2} (B) 9

and we would now like to reason that this map factor through the simplicial set [figure: 0; Gr; [1,n]]. Precisely, define maps

[figure] 0  Gr[figure: [1,n]] X  \overline{}ϕ \overline{}ψ  [figure: 0; Gr; [1,n]]

by letting

ϕ(y_{s})  =  ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{00}; Y_{s0}] )

Next we define ψ. If x_{s} is an s–simplex in [figure: 0; Gr; [1,n]], that is x_{s} is the diagram

x_{s}  =  ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}] )

then define ψ(x_{s}) to be the diagram

ψ(x_{s}) = ( [figure: ⋮; ..; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ..; ..; Y_{0}⊕X_{r0}; Y_{0}⊕X_{rt}; Y_{s}⊕X_{r0}; Y_{s}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}] )

Clearly, Θ = ψ ∘ ϕ, and hence ψ ∘ ϕ is homotopic to the identity.

The homotopy connecting Θ = ψ ∘ ϕ to the identity is really the key to the entire proof. We would like to introduce a shorthand for the homotopy. It will be denoted

[figure] 0  Gr[figure: [1,n]] ^{Gr[figure: [1,n]]}X

It connects the identity to a map we denote

gr85.tex; 3/04/2000; 15:34; p.10

[figure: 0; Gr; [1,n]; X]

The notation is meant to capture the fact that the homotopy forgets most of the square of Y ’s, leaving only the West face. In fact, it connects the identity to a map which replaces the square of Y ’s by the sums of the North face of the X’s and West face of the Y ’s. Also evident from the notation should be that this map Θ factors as ψ ∘ ϕ, through the

simplicial set [figure: 0; Gr; [1,n]].

To prove the contractibility of the Segal fiber [figure] 0  Gr[figure: [1,n]] X  ,

it suffices therefore to contract [figure: 0; Gr; [1,n]]. The contracting

homotopy is the “contraction to the terminal object”. It takes the s– simplex x_{s} above, and an integer i with 0 ≤ i ≤ s, to the (s + 1)– simplex

[figure: ⋮; ⋯; ·; ·; ·; 0; 0; 0; ⋮; 0; 0; ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0i}; Z_{ii}; ⋮; ⋮; ⋯; ⋯; Y_{0}; Y_{0}; Y_{i}; Y_{i}; ⋮; Y_{0}; Y_{i}]
□

Now I assert that Step 2.1, Step 2.2 and Step 6.2 are the same. Let us begin with Step 6.2.

Step 6.2: The projection

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]

induces a homotopy equivalence.

gr85.tex; 3/04/2000; 15:34; p.12

To see that the proof is identical with Step 3.1, it might be helpful to note that some of the hypotheses on the morphisms are redundant. Precisely, observe

LEMMA 8.3. Let

\begin{matrix}X′ & ⟶ & Y′ \\ ↑ & & ↑ \\ X & ⟶ & Y \end{matrix}

be a commutative square in Gr(A), with a differential ∂ : Y′ ⟶ ΣX, giving a long exact sequence. If X and X′ are in Gr_{[1,n]} and Y and Y′ in Gr_{[0,n]}, then the map Y ⟶ Y′ must be mono in Gr_{[0,n]}.

Proof. Consider the exact sequence

H^{0}(X) ⟶ H^{0}(X′ ⊕ Y) ⟶ H^{0}(Y′)

Because both X and X′ are in Gr_{[1,n]}, their H^{0} vanishes. The sequence therefore becomes

0 ⟶ H^{0}(Y) ⟶ H^{0}(Y′)

and its exactness is precisely the statement that Y ⟶ Y′ is mono in Gr_{[0,n]}. □

It follows that there is an equality of simplicial sets

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  [figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]

In other words, there is no need to impose the hypothesis that the vertical morphisms in the right column be mono in Gr_{[0,n]}; this follows from the existence of squares as in Lemma 8.3. Rewriting the map of Step 6.2 without explicitly insisting that the vertical maps be mono, we need to prove that the map

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]

induces a homotopy equivalence. We will show the contractibility of the Segal fiber

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] X  Y A

As in the proof of Step 3.1, the homotopy

Gr[figure: [1,n]]  Gr[figure: [0,n]] [figure] 0  ^{Gr[figure: [1,n]]}  X  Y A

is well–defined. The reader should note that to the top row of the X’s and Y ’s we are adding objects of Gr_{[1,n]}, and hence the sums lie in the category where they are claimed to lie. Furthermore, the horizontal maps on the right remain H^{0}–isomorphisms. The vertical epi–ness was checked in Step 3.1. The cell of the homotopy satisfies the hypotheses of membership in the simplicial set.

This homotopy allows one to factor the identity on the Segal fiber

through the simplicial set [figure: 0; Gr; [1,n]], which we already

know to be contractible.

Even easier is the proof of Step 2.1 and Step 2.2. Perhaps Step 2.1 is clearest. It asserts

Step 2.1: The natural map

Gr[figure: [m,n]]^{}^{}^{}^{}Gr[figure: [m,n]]  \overline{}^{[figure] [figure] Gr[figure: [m,n]]}

induces a homotopy equivalence.

There is no loss of generality in assuming m = 1 (we can always translate to that case). But then the statement becomes identical with Step 3.1, except for the triangle of Z’s which is now missing. The homotopies that contracted the Segal fiber in Step 3.1 will also contract the Segal fiber of Step 2.1, if we just forget the Z’s.

There remains Step 2.2. Recall the statement

Step 2.2: The natural map

Gr[figure: [m,n]]^{}^{}^{}^{}Gr[figure: [m,n]]  \overline{}^{[figure] [figure] Gr[figure: [m,n]]}

induces a homotopy equivalence.

It is almost identical with Step 2.1, except that the horizontal morphisms are now constrained to be mono. The reader should check that the two homotopies which combine to contract the Segal fiber in Step 2.1 respect horizontal monos; they carry the subset with horizontal monos to itself. Hence they contract the subset.

9. The two quasi–fibrations

In this Section we will prove Step 6.1 and Step 6.3. The proofs are not substantially more difficult than what we saw in Section 8, but since we are proving maps to be quasi–fibrations, the discussion of the Segal fiber is a little more delicate. We begin with Step 6.1.

Step 6.1: The natural projection

[figure] 0  Gr[figure: [0,n]] Gr[figure: [1,n]]  π  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

is a quasi–fibration.

Proof. We will apply Quillen’s Theorem B; that is, we will show that the face maps induce homotopy equivalences on the Segal fiber. The

Segal fiber is the simplicial set [figure] 0  Gr[figure: [0,n]] X  . On it, there

is a homotopy

[figure] 0  Gr[figure: [0,n]] ^{Gr[figure: [0,n]]}X

and we need a moment’s reflection to check that the cells of the homotopy are well defined; it needs to be checked that the objects and morphisms satisfy the restrictions placed on them. A simplex in the homotopy is a diagram

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ii}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{s–i+1 \atop terms}}]

and we need to check that (1) the horizontal maps Y_{k0} ⊕ X_{rl} ⟶ Y_{k0}⊕X_{rl′} are H^{0}–isomorphisms, and (2) the vertical maps Y_{k0}⊕X_{rl} ⟶ Y_{k′0} ⊕ X_{rl} and Y_{k0} ⊕ X_{rl} ⟶ Y_{k′l} are monoepi in T_{[0,n]}. Let us begin with the vertical maps. The epi–ness has already been checked, in the proof of Step 3.1. Why are these maps mono?

Firstly, it suffices to consider the maps Y_{k0} ⊕ X_{rl} ⟶ Y_{k′l}. For if we know them all to be mono, then the composite

Y_{k0} ⊕ X_{rl} ⟶ Y_{k′0} ⊕ X_{rl} ⟶ Y_{k′l}

will be mono, forcing the first map, that is Y_{k0} ⊕ X_{rl} ⟶ Y_{k′0} ⊕ X_{rl}, to also be mono. Next, the map Y_{k0} ⊕ X_{rl} ⟶ Y_{k′l} can be decomposed as

Y_{k0} ⊕ X_{rl} ⟶ Y_{kl} ⟶ Y_{k′l}

where Y_{kl} ⟶ Y_{k′l} is known to be mono. It suffices to show

Y_{k0} ⊕ X_{rl} ⟶ Y_{kl}

to be mono. But recall that X_{rl} ∈ T_{[1,n]}, and hence its H^{0} vanishes. The map

H^{0} (Y_{k0} ⊕ X_{rl}) = H^{0} (Y_{k0}) ⟶ H^{0} (Y_{kl})

is induced by a horizontal map in the original simplex, and hence is not only a monomorphism, but also an isomorphism.

Next we need to check that the horizontal maps Y_{k0} ⊕ X_{rl} ⟶ Y_{k0} ⊕ X_{rl′} are H^{0}–isomorphisms. But H^{0}(X_{rl}) = H^{0}(X_{rl′}) = 0, and hence

H^{0} (Y_{k0} ⊕ X_{rl}) = H^{0} (Y_{k0}) = H^{0} (Y_{k0} ⊕ X_{rl′})

and the natural map induces an isomorphism.

Thus the homotopy is indeed well defined. It connects the identity to a map Θ, which factors as

[figure] 0  Gr[figure: [0,n]] X  \overline{}ϕ \overline{}ψ  [figure: 0; Gr; [0,n]]

It is very easy to compute what the composite ϕ ∘ ψ is. It is the map taking the simplex x_{s}

x_{s} = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}] )

to the simplex

ϕ(ψ(x_{s})) = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; X_{r0} ⊕ Y_{0}; X_{r0} ⊕ Y_{s}]  )

Now the simplicial set [figure: 0; Gr; [0,n]] is an H–space, with ad-

dition being given by direct sum. The map ϕ ∘ ψ is nothing other than translation in the H–space structure with respect to the 0–simplex

0 ⟶ X_{r0}

The H–space [figure: 0; Gr; [0,n]] is easily seen to be connected.

Actually, all that is relevant for us is that the 0–cell above can be connected to the trivial cell. Recalling that X_{r0} ∈ T_{[1,n]}, the following is a 1–cell connecting it to the trivial cell

0 ⟶ 0
↑  ↑
0 ⟶ X_{r0} ⟶ X_{r0}

The reader is asked to note that the map X_{r0} ⟶ 0 is a monoepi. It is clearly epi, and is mono because H^{0}(X_{r0}) = 0.

It follows that translation in the H–space with respect to some 0–cell

0 ⟶ X_{r0}

is homotopic to the identity. Thus ϕ ∘ ψ is homotopic to the identity, and ϕ and ψ are homotopy inverses of each other. The simplicial

sets [figure] 0  Gr[figure: [0,n]] X  and [figure: 0; Gr; [0,n]] are homo-

topy equivalent, and in particular the Segal fiber has a homotopy type independent of X. To prove that the face maps induce homotopy equivalences, consider the composite

[figure: 0; Gr; [0,n]]  ψ  \overline{}^{[figure] 0  Gr[figure: [0,n]] X} ∂ [figure] [figure] 0  Gr[figure: [0,n]] ∂X  ϕ  \overline{}[figure: 0; Gr; [0,n]]

Once again, all the maps are very explicit, and the composite can readily be computed to be translation in the H–space structure of

[figure: 0; Gr; [0,n]] by the 0–cell

0 ⟶ X_{rj}

This map is homotopic to the identity. The composite ϕ ∘ ∂ ∘ ψ is a homotopy equivalence, as are ϕ and ψ. It follows that ∂ is a homotopy equivalence. □

Step 6.3: The natural projection

Gr[figure: [1,n]]^{}^{}^{}^{}Gr[figure: [1,n]]  ^{}^{}^{}^{}Gr[figure: [0,n]]  \overline{}Gr[figure: [1,n]]^{}^{}^{}^{}[figure] [figure] ^{}^{}^{}^{}[figure] [figure]

induces a quasi–fibration.

Proof. We study the Segal fiber  Z Gr[figure: [1,n]]  Gr[figure: [0,n]] . More pre-

cisely, we will show that the face maps induce homotopy equivalences on the Segal fibers. There is a homotopy on this fiber whose symbol is

Z ^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]⊕Z_{SE} Gr[figure: [1,n]]  Gr[figure: [0,n]]

gr85.tex; 3/04/2000; 15:34; p.23

Let us write Z_{SE} for Z_{0t}, the South–East corner of the Z square. Then this homotopy has cells which are diagrams

[figure: ⋮; ⋮; ..; ..; Z_{00}; Z_{0t}; Z_{s0}; Z_{st}; ⋮; ⋮; ..; ..; Z_{00}⊕Y_{it}; Z_{0t}⊕Y_{it}; Z_{00}⊕Y_{rt}; Z_{0t}⊕Y_{rt}; ⋮; ⋮; ..; ..; X_{i0}⊕Z_{SE}; X_{iu}⊕Z_{SE}; X_{r0}⊕Z_{SE}; X_{ru}⊕Z_{SE}; ⋮; ⋮; ..; ..; Y_{00}; Y_{0t}; Y_{i0}; Y_{it}; ⋮; ⋮; ..; ..; X_{00}; X_{0u}; X_{i0}; X_{iu}; \}_{{r-i+1 \atop terms}}; \}_{{i+1 \atop terms}}; \{; \{]

To see that this homotopy is well–defined, the reader will note that in Definition 8.1, we defined a homotopy whose cells were

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ii}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{s–i+1 \atop terms}}]

If we choose any integer j with i ≤ j ≤ s, then we can look at the subset of the diagram above

gr85.tex; 3/04/2000; 15:34; p.25

[figure: ⋮; ⋮; ..; ..; Z_{ij}; Z_{is}; Z_{jj}; Z_{js}; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{j0}; Y_{jt}; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0j}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ij}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{j-i+1 \atop terms}}]

gr85.tex; 3/04/2000; 15:34; p.26

What we discover is precisely the dual of the homotopy cell of

Z ^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]⊕Z_{SE} Gr[figure: [1,n]]  Gr[figure: [0,n]]  .

In other words, back when we defined all the morphisms of the cell in Definition 8.1 and checked the commutativity, coherence of differentials and that every square leads to a long exact sequence in A, we checked among other things that the smaller cell is well defined. Thus we have our homotopy.

We follow this by a homotopy whose shorthand is

Z ^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]⊕Z_{SE}  Gr[figure: [0,n]] \overline{}

This homotopy has cells

gr85.tex; 3/04/2000; 15:34; p.27

[figure: ⋮; ⋮; ..; ..; Z_{00}; Z_{0t}; Z_{s0}; Z_{st}; ⋮; ⋮; ..; ..; Z_{00}⊕Y_{0t}; Z_{0t}⊕Y_{0t}; Z_{00}⊕Y_{rt}; Z_{0t}⊕Y_{rt}; ⋮; ⋮; ..; ..; X_{00}⊕Z_{SE}; X_{0i}⊕Z_{SE}; X_{r0}⊕Z_{SE}; X_{ri}⊕Z_{SE}; ⋮; ⋮; ..; ..; X_{0i}; X_{0u}; X_{ri}; X_{ru}; _{(i+1)−terms}; _{(u−i+1)−terms}; ; ]

We have seen this homotopy cell before. Choose any integer j, with 0 ≤ j ≤ i. In the diagram

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ii}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{s–i+1 \atop terms}}]

consider the subset

gr85.tex; 3/04/2000; 15:34; p.29

[figure: ⋮; ⋮; ..; ..; Z_{0j}; Z_{0i}; Z_{jj}; Z_{ji}; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ji}; X_{r0}⊕Z_{js}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{j0}⊕X_{r0}; Y_{j0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}]

It is nothing more than the dual of the cell in our homotopy. Thus the morphisms are defined, and every square gives a long exact sequence. We need to check only that the morphisms satisfy the hypotheses for membership in our simplicial set. The only thing that needs checking is that the horizontal map X_{rj} ⊕ Z_{SE} ⟶ X_{rj′} is an H^{0}–isomorphism. But Z_{SE} ∈ Gr_{[1,n]}, and so at the level of H^{0} the map is the same as X_{rj} ⟶ X_{rj′}, which is an H^{0} isomorphism by hypothesis.

Taken together, these two homotopies connect the identity to some map. What is important about this map is that it factors through

Gr[figure: [1,n]] Gr[figure: [0,n]] . Precisely, we define two maps

K–Theory for Triangulated Categories 3 \frac{1}{2} (B) 31

Gr[figure: [1,n]]  Gr[figure: [0,n]]  \overline{}ψ \overline{}ϕ  ^{Z Gr[figure: [1,n]]  Gr[figure: [0,n]]}

where ϕ takes the simplex s_{(r,u)} = ( [figure: ⋮; ⋮; ⋯; ⋯; Z_{00}; Z_{0t}; Z_{s0}; Z_{st}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0u}; X_{r0}; X_{ru}] ) to the simplex

                                       gr85.tex; 3/04/2000; 15:34; p.31

32 Amnon Neeman

ϕ(s_{(r,u)}) = ( [figure: ⋮; Y_{0t}; Y_{rt}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0u}; X_{r0}; X_{ru}] ) ,

while the map ψ takes the simplex

t_{(r,u)} = ( [figure: ⋮; Y_{0}; Y_{r}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0u}; X_{r0}; X_{ru}] )

to the simplex

                                       gr85.tex; 3/04/2000; 15:34; p.32

ψ(t_{(r,u)}) = ( [figure: ⋮; ⋮; ..; ..; Z_{00}; Z_{0t}; Z_{s0}; Z_{st}; ⋮; ⋮; ..; ..; Z_{00}⊕Y_{0}; Z_{0t}⊕Y_{0}; Z_{00}⊕Y_{r}; Z_{0t}⊕Y_{r}; ⋮; ⋮; ..; ..; X_{00}; X_{0u}; X_{r0}; X_{ru}] ) .

We know that the composite ψ ∘ ϕ is homotopic to the identity on

Z Gr[figure: [1,n]] Gr[figure: [0,n]] . But the composite ϕ∘ψ is easy to compute. It takes the simplex

t_{(r,u)} = ( [figure: ⋮; Y_{0}; Y_{r}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0u}; X_{r0}; X_{ru}] )

to

[figure: ⋮; Y_{0} ⊕ Z_{SE}; Y_{r} ⊕ Z_{SE}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0u}; X_{r0}; X_{ru}]

This is nothing more nor less than translation in the H–space structure by the 0–cell

[figure: Z_{SE}; 0]

Since it is easy to see that ^{Gr[figure: [1,n]]} Gr[figure: [0,n]] is a connected H–

space, it follows that ϕ ∘ ψ is homotopic to the identity. Thus ϕ and

ψ are homotopy equivalences. Hence  Z Gr[figure: [1,n]]  Gr[figure: [0,n]] is ho-

motopy equivalent to ^{Gr[figure: [1,n]]} Gr[figure: [0,n]] , which is independent of

Z. But we need to prove further that the face maps induce homotopy equivalences.

To do this one considers the composite

Gr[figure: [1,n]]  Gr[figure: [0,n]]  ψ  \overline{}^{Z Gr[figure: [1,n]]  Gr[figure: [0,n]]} ∂ [figure]^{∂Z Gr[figure: [1,n]]  Gr[figure: [0,n]]} ϕ  \overline{}^{^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]}

Since all the maps are completely explicit, it is easily computed that the composite ϕ ∘ ∂ ∘ ψ is translation in the H–space structure of

Gr[figure: [1,n]] Gr[figure: [0,n]] by some 0–cell

[figure: Z_{0j}; 0]

for some Z_{0j} in  Z  . Therefore ϕ ∘ ∂ ∘ ψ is a homotopy equiva-

lence, and since ψ and ϕ also are, so is ∂. □

10. Proof of Step 4.1

The last thing that remains to be proved is

Step 4.1: Let γ be the map

[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]  γ  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [0,n]]

Up to homotopy, the identity on the Segal fiber of γ, that is the identity on the simplicial set

[figure] 0 Gr[figure: [0,n]] X,

factors through the simplicial set [figure: 0; Gr; [0,n]].

Proof. The problem here is that, for the first time, our favorite homotopy is not defined. We would like to take the homotopy whose cells are

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; X_{r0}⊕Z_{0i}; X_{r0}⊕Z_{0s}; X_{r0}⊕Z_{ii}; X_{r0}⊕Z_{is}; ⋮; ⋮; ..; ..; Y_{00}⊕X_{r0}; Y_{00}⊕X_{rt}; Y_{i0}⊕X_{r0}; Y_{i0}⊕X_{rt}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}]

This large diagram tries hard to be a cell of a homotopy. It is commutative, the differentials are coherent and all squares give long exact sequences. The problem with this diagram is that it is not a cell in our simplicial set. The problem has in fact been highlighted by a dashed box, for the reader’s convenience. The vertical morphisms Y_{k0}⊕X_{rj} ⟶ Y_{k′j} are not in general monoepi. They are epi, but not necessarily mono. This diagram fails to satisfy the restrictions that would entitle it to be a simplex in our simplicial set.

But there is a natural map X_{r0} ⟶ X_{r0} ⊕ Z_{kj}, given by the matrix

( \begin{matrix}1 \\ 0 \end{matrix} ). This continues to define a map of X_{r0} to everything to the right

and above X_{r0} ⊕ Z_{0i}. On the other hand, Definition 8.1 gives us very explicitly all the maps in the large diagram. It is easily computed that the map from X_{r0} to an object of the form Y_{k0} ⊕ X_{rj} is

( \begin{matrix}−n_{1} \\ n_{2} \end{matrix} ) : X_{r0} ⟶ Y_{k0} ⊕ X_{rj}

where n_{1} : X_{r0} ⟶ Y_{k0} and n_{2} : X_{r0} ⟶ X_{rj} are the natural maps. The map from X_{r0} to the top of the simplex, that is to objects of the form Z_{kj} or Y_{kj}, is easily seen to be zero, for instance because it factors through Z_{ii} = 0.

We have defined a map from X_{r0} to the top right part of the diagram. Let X_{r0}^{<1} be the “truncation” of X_{r0}. That is,

H^{q}(X_{r0}^{<1}) = \{ \begin{matrix}H^{q}(X_{r0}) & if q ≤ 0 \\ 0 & if q > 0 \end{matrix}

It is clear that the composites

X_{r0}^{<1} ⟶ X_{r0} ⟶ X_{r0} ⊕ Z_{kj}  and  X_{r0}^{<1} ⟶ X_{r0} ⟶ Y_{k0} ⊕ X_{rj}

induce morphisms

X_{r0}^{<1} = H^{0} (X_{r0}^{<1}) ⟶ H^{0} (X_{r0} ⊕ Z_{kj})

and

X_{r0}^{<1} = H^{0} (X_{r0}^{<1}) ⟶ H^{0} (Y_{k0} ⊕ X_{rj})

I assert that these are monomorphisms. This is clear for the map

X_{r0}^{<1} = H^{0} (X_{r0}^{<1}) ⟶ H^{0} (X_{r0} ⊕ Z_{kj})

but not so immediate for the second map. To prove it, consider the square

\begin{matrix}Y_{k0} & ⟶ & Y_{kj} \\ ↑ & & ↑ \\ X_{r0} & ⟶ & X_{rj} \end{matrix}

There is an induced exact sequence

0 ⟶ H^{0} (X_{r0}) ⟶ H^{0} (Y_{k0} ⊕ X_{rj}) ⟶ H^{0} (Y_{kj}) ⟶

and from this we deduce that, indeed, H^{0} (X_{r0}) ⟶ H^{0} (Y_{k0} ⊕ X_{rj}) is mono. Not only that, but the quotient map

\frac{H^{0} (Y_{k0} ⊕ X_{rj})}{H^{0} (X_{r0})} ⟶ H^{0} (Y_{kj})

is mono. We can therefore form a diagram

[figure: ⋮; ..; ·; ·; ·; 0; Z_{is}; 0; ⋮; ⋮; ..; ..; Y_{i0}; Y_{it}; Y_{s0}; Y_{st}; ⋮; ..; ·; ·; ·; 0; Z_{0i}; 0; ⋮; ⋮; ..; ..; \frac{X_{r0}⊕Z_{0i}}{X_{r0}^{<1}}; \frac{X_{r0}⊕Z_{0s}}{X_{r0}^{<1}}; \frac{X_{r0}⊕Z_{ii}}{X_{r0}^{<1}}; \frac{X_{r0}⊕Z_{is}}{X_{r0}^{<1}}; ⋮; ⋮; ..; ..; \frac{Y_{00}⊕X_{r0}}{X_{r0}^{<1}}; \frac{Y_{00}⊕X_{rt}}{X_{r0}^{<1}}; \frac{Y_{i0}⊕X_{r0}}{X_{r0}^{<1}}; \frac{Y_{i0}⊕X_{rt}}{X_{r0}^{<1}}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}; \}_{{i+1 \atop terms}}; \}_{{s-i+1 \atop terms}}]

and this diagram does define a cell in the Segal fiber. We have just checked that the maps inside the dashbox above are mono. The fact that they are epi we know from the proof of Step 3.1. What we still need to check is that the vertical maps X_{rj} ⟶ \frac{Y_{00}⊕X_{rj}}{X_{r0}^{<1}} are mono. But the composite

X_{rj} ⟶ \frac{Y_{00} ⊕ X_{rj}}{X_{r0}^{<1}} ⟶ Y_{ij}

is mono, and hence so is the first map. Thus the homotopy really is well defined.

The homotopy connects the identity on the Segal fiber to a map, which can be written in the form ψ ∘ ϕ. Let us denote the object X_{r0}^{<1} by the letter A. Consider the simplicial set

[figure: 0; Gr; [0,n]; A]

A simplex in [figure: 0; Gr; [0,n]; A] is a commutative diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}; A]

together with coherent differentials. Define maps

K–Theory for Triangulated Categories 3 \frac{1}{2} (B) 41

[figure] 0  Gr[figure: [0,n]] X  \overline{}ϕ \overline{}ψ  [figure: 0; Gr; [0,n]; A]

Concretely, the map ϕ takes the simplex s_{s} = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}] ) to the simplex

gr85.tex; 3/04/2000; 15:34; p.41

42 Amnon Neeman

ϕ(s_{s}) = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{00}; Y_{s0}; A] ) , while the map ψ takes the simplex

t_{s} = ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}; A] )

to the simplex

                                       gr85.tex; 3/04/2000; 15:34; p.42

ψ(t_{s}) = ( [figure: ⋮; ..; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ..; ..; \frac{Y_{0}⊕X_{r0}}{A}; \frac{Y_{0}⊕X_{rt}}{A}; \frac{Y_{s}⊕X_{r0}}{A}; \frac{Y_{s}⊕X_{rt}}{A}; ⋮; ⋮; ..; ..; X_{00}; X_{0t}; X_{r0}; X_{rt}] ) .

Perhaps we should give a little thought to why the map ψ is well– defined. It is pehaps not immediately clear how to show that the vertical maps X_{rj} ⟶ \frac{Y_{0}⊕X_{rj}}{A} are mono.

But we know that that the map α : A ⟶ Y_{0} is mono. Hence the map

( \begin{matrix}−α & 0 \\ β & 1 \end{matrix} ) : A ⊕ X_{rj} ⟶ Y_{0} ⊕ X_{rj}

ia also mono. Dividing by the inclusion of A ∈ T_{[0,0]}, we obtain a mono

X_{rj} ⟶ \frac{Y_{0} ⊕ X_{rj}}{A}

as required. Thus we do have a well–defined simplicial map.

The composite ψ ∘ ϕ is nothing other than the map we have shown

to be homotopic to the identity on the Segal fiber [figure] 0  Gr[figure: [0,n]] X  .

We have so far factored the identity through [figure: 0; Gr; [0,n]; A]. To

complete the proof of Step 4.1, it therefore suffices to show

LEMMA 10.1. The simplicial sets [figure: 0; Gr; [0,n]; A] and

[figure: 0; Gr; [0,n]] are homotopy equivalent.

Proof. We begin by defining an auxiliary bisimplicial set, whose symbol is

[figure: Gr; [0,0]; 0; Gr; [0,n]; A] .

An (s, t)–simplex is a diagram

[figure: ⋮; W_{0}; W_{t}; ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}; A]

The curly arrow connecting A with the W ’s in the symbol for the simplicial set is meant to indicate that the morphisms A ⟶ W_{j} are all mono. There is only one morphism A ⟶ W_{j}, namely the composite A ⟶ Y_{i} ⟶ W_{j}. But without writing explicitly the strange looking curly arrow, there seem to be no restrictions on that map; after all, the maps Y_{i} ⟶ W_{j} are unrestricted. The curly arrow is not some new map, only a new restriction. Note also that W_{i} is an object of Gr_{[0,0]} = A.

It is trivial that the projection

[figure: Gr; [0,0]; 0; Gr; [0,n]; A; π; 0; Gr; [0,n]; A]

induces a homotopy equivalence. The Segal fiber is the nerve of a category having an initial object. A t–simplex in the Segal fiber is a diagram

[figure: ⋮; W_{0}; W_{t}; ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}; A]

and Y_{s}^{<1} is initial for the W ’s. There is a map

                     gr85.tex; 3/04/2000; 15:34; p.47

[figure: (; Gr; [0,0]; A; ); ×; (; 0; Gr; [0,n]; ); R; Gr; [0,0]; 0; Gr; [0,n]; A]

sending the simplex

s_{(s,t)} = ( [figure: ⋮; W_{0}; W_{t}; A] ) × ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}] )

to the simplex

                                       gr85.tex; 3/04/2000; 15:34; p.48

K–Theory for Triangulated Categories 3 \frac{1}{2} (B) 49

R(s_{(s,t)}) = ( [figure: ⋮; W_{0}; W_{t}; ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0} ⊕ A; Y_{s} ⊕ A; A] ) .

There is a natural candidate for the inverse of R. It is the product of the two natural projections

gr85.tex; 3/04/2000; 15:34; p.49

[figure: Gr; [0,0]; 0; Gr; [0,n]; A; g; f; 0; Gr; [0,n]; A; Gr; [0,0]; A]

We will prove that the composites ( \begin{matrix}f \\ g \end{matrix} ) ∘ R and R ∘ ( \begin{matrix}f \\ g \end{matrix} ) are both homotopy equivalences. Assume for a second we have proved this. It would then follow that the map

[figure: (; Gr; [0,0]; A; ); ×; (; 0; Gr; [0,n]; ); R; Gr; [0,0]; 0; Gr; [0,n]; A]

has both a right and a left inverse in the homotopy category, and hence is a homotopy equivalence. On the other hand the target of

the map has the homotopy type of [figure: 0; Gr; [0,n]; A], while the

source is the product of [figure: 0; Gr; [0,n]] with the contractible

space [figure: Gr; [0,0]; A]. Thus the Lemma.

It therefore remains only to show that both ( \begin{matrix}f \\ g \end{matrix} )∘R and R∘( \begin{matrix}f \\ g \end{matrix} ) are homotopy equivalences. We compute easily what the composites are. The composite ( \begin{matrix}f \\ g \end{matrix} ) ∘ R takes the simplex

s_{(s,t)} = ( [figure: ⋮; W_{0}; W_{t}; A] ) × ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}] )

to the simplex \{( \begin{matrix}f \\ g \end{matrix} ) ∘ R\} (s_{(s,t)}) below

( [figure: ⋮; W_{0}; W_{t}; A] ) × ( [figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0} ⊕ A; Y_{s} ⊕ A] )

There is a commutative square

[figure: (; Gr; [0,0]; A; ); ×; (; 0; Gr; [0,n]; ); p; 0; Gr; [0,n]; ( \begin{matrix}f \\ g \end{matrix} ) ∘ R; α; (; Gr; [0,0]; A; ); ×; (; 0; Gr; [0,n]; ); p; 0; Gr; [0,n]]

where the map p is the second projection from the product, and the map α is translation in the H–space structure of [figure: 0; Gr; [0,n]] with respect to the 0–cell

0 ⟶ A.

Since the H–space is connected, it follows that α is homotopic to the identity. The map p, being a projection with contractible fiber, is a homotopy equivalence. It follows that ( \begin{matrix}f \\ g \end{matrix} ) ∘ R is homotopic to the identity, hence certainly a homotopy equivalence.

Next we must show that the map R ∘ ( \begin{matrix}f \\ g \end{matrix} ) is a homotopy equivalence. By Segal’s theorem, it suffices to show it a homotopy equivalence after realising just one simplicial structure. We realise only the structure which varies the integer s. We then have two homotopies, which combine to connect R ∘ ( \begin{matrix}f \\ g \end{matrix} ) to the identity map; they have for their cells the two diagrams [figure: ⋮; W_{0}; W_{t}; ⋮; ..; ·; ·; ·; Z_{ii}; Z_{is}; Z_{ss}; ⋮; Y_{i}⊕W_{0}; Y_{s}⊕W_{0}; ⋮; ..; ·; ·; ·; Z_{00}; Z_{0i}; Z_{ii}; ⋮; ⋮; ..; ..; Z_{0i}⊕Σ^{−1}W_{0}; Z_{0s}⊕Σ^{−1}W_{0}; Z_{ii}⊕Σ^{−1}W_{0}; Z_{is}⊕Σ^{−1}W_{0}; ⋮; Y_{0}; Y_{i}; A] and

56 Amnon Neeman

[figure: ⋮; W_{0}; W_{t}; ⋮; ..; ·; ·; ·; Z_{ii}; Z_{is}; Z_{ss}; ⋮; Y_{i}⊕W_{0}; Y_{s}⊕W_{0}; ⋮; ..; ·; ·; ·; Z_{00}; Z_{0i}; Z_{ii}; ⋮; ⋮; ..; ..; Z_{0i}⊕Σ^{−1}\{ \frac{W_{0}}{A} \}; Z_{0s}⊕Σ^{−1}\{ \frac{W_{0}}{A} \}; Z_{ii}⊕Σ^{−1}\{ \frac{W_{0}}{A} \}; Z_{is}⊕Σ^{−1}\{ \frac{W_{0}}{A} \}; ⋮; Y_{0}⊕A; Y_{i}⊕A; A] □

                                       gr85.tex; 3/04/2000; 15:34; p.56

References

1. A. A. Beilinson, J. Bernstein and P. Deligne. Analyse et topologie sur les éspaces singuliers. Astérisque 100, Soc. Math. France (1982).

2. A. Borel and J. P. Serre. Le théorème de Riemann–Roch, Bull. Soc. Math. France 86 (1958), 97-136.

3. C. Giffen, and A. Neeman. K–theory for triangulated categories. Preprint.

4. R. Hartshorne. Residues and duality, SLN 20 (1966).

5. V. A. Hinich and V. V. Schechtman. Geometry of a category of complexes and algebraic K–theory. Duke Math. J. 52 (1985), 399-430.

6. J. F. Jardine. The multiple Q–constructions. Canad. J. Math. 39 (1987), 1174-1209.

7. A. Neeman. Some new axioms for triangulated categories. J. of Algebra 139 (1991) 221–255.

8. A. Neeman. The Brown Representability Theorem and phantomless triangulated categories. J. of Algebra 151 (1992) 118–155.

9. A. Neeman. K–theory for triangulated categories I(A): homological functors. Asian Journal of Mathematics 1 (1997) 330–417.

10. A. Neeman. K–theory for triangulated categories I(B): homological functors. Asian Journal of Mathematics 1 (1997) 435–519.

11. A. Neeman. K–theory for triangulated categories II: the subtlety of the theory and potential pitfalls. Asian Journal of Mathematics 2 (1998) 1–125.

12. A. Neeman. K–theory for triangulated categories III(A): the theorem of the heart. Asian Journal of Mathematics 2 (1998) 495–589.

13. A. Neeman. K–theory for triangulated categories III(B): the theorem of the heart. Asian Journal of Mathematics 3 (1999) 557–608.

14. D. Quillen. Higher algebraic K–theory I, SLN 341 (1973), 85-147.

15. G. Segal. Categories and cohomology theories. Topology 13 (1974), 293–312.

16. R. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In: The Grothendieck Festschrift ( a collection of papers to honor Grothendieck’s 60’th birthday) Volume 3 pp. 247–435, Birkhäuser 1990.

17. J. L. Verdier. Catégories dérivées, état 0. SGA 4 \frac{1}{2} , 262-308 (SLN 569, 1977).

18. F. Waldhausen. Algebraic K–theory of generalized free products I, II. Ann. of Math. 108 (1978), 135-256.

19. F. Waldhausen. Algebraic K-theory of spaces. SLN 1126 (1985), 318-

419.

Address for Offprints: Center for Mathematics and its Applications School of Mathematical Sciences John Dedman Building The Australian National University Canberra, ACT 0200 AUSTRALIA

gr85.tex; 3/04/2000; 15:34; p.58