Determinants In Triangulated Categories

Avishay Vaknin ^{∗} March 19, 2000

Abstract

In this paper we define the determinant function

det : {Automorphisms in τ }⟶K_{1}(τ),

whereτ is a small triangulated category, and K_{1} is Neeman’s first K– theory group for triangulated categories. This function is compatible with the classical determinant map

det : {Automorphisms in A}⟶K_{1}(A),

where A is a small abelian category. We prove that this determinant is a multiplicative additive function. As an application, we prove that the alternating sum determinant of bounded complex automorphisms over an abelian category A is an additive multiplicative homotopy invariant with values in K_{1}(A).

Contents

0 Introduction 2 1 The Loop Space 4 2 Double Distinguished Triangles 5 3 Nenashev’s Main Lemma 6 4 The 3′ × 3′ Relation 12 5 Determinant Elements 17

∗Department of Mathematics, The Hebrew University, Givat-Ram, Jerusalem 91904 Israel (vaaknin@math.huji.ac.il).

0 Introduction

One of the definitions of K_{1} of a ring R uses the formula

K_{1}(R) = GL(R)/[GL(R), GL(R)],

and we can view the map GL(R) → K_{1}(R) as a generalization of the determinant function on automorphisms of finite dimensional vector spaces. By definition

{Aut. of f.g. free mdules over R} = GL(R)⟶K_{1}(R)

is the universal homomorphism to an abelian group. The property det(ab) = det(a) + det(a) is called multiplicativity. This determinant function is also additive in the following sense. Given an automorphism of a short exact sequence of f.g. free modules

\xymatrix{0 \ar[r] & R^{n} \ar[r] \ar[d]^{a} & R^{n+m} \ar[r] \ar[d]^{b} & R^{m} \ar[r] \ar[d]^{c} & 0 \\ 0 \ar[r] & R^{n} \ar[r] & R^{n+m} \ar[r] & R^{m} \ar[r] & 0}

the equality det(b) = det(a) + det(c) holds. Moreover, the group K_{1} is the free group over the automorphisms of objects in the small exact category proj-R, divided by the multiplicativity and the additivity relations (additivity is the relation, which arise from an automorphism of a short exact sequence). Therefore,

{Automorphisms in proj-R}⟶K_{1}(R)

is the universal multiplicative additive function. For more details on K_{1} of a ring we refer to [14, Chapter 1 and 3].

Using Gersten’s ideas [1, section 4], Sherman [15, Section 3] describes the determinant of an automorphism α, as a π_{2} element in the nerve of the Q construction of proj-R. This description leads to the definition of the determinant for automorphisms in a general small exact category ε.

{Automorphisms in ε}⟶K_{1}(ε)

In [10, 11, 12], Nenashev computes explicitly Quillen’s K_{1} group [13] for a general small exact category ε. In the general case, the determinant function

{Automorphisms in ε}⟶K_{1}(ε)

is not universal, but it is still a multiplicative additive function. To describe the group K_{1}(ε) Nenashev uses Gillet-Grayson [2] simplicial model for the loop space of the Q–construction. The generators of K_{1}(ε) are double short exact sequences, and the relations are derived from double 3×3 commutative diagrams. The determinant function

{Automorphisms in ε}⟶K_{1}(ε)

is defined by attaching a double short exact sequence to an automorphism in ε, and the relations in K_{1}(ε) imply the multiplicativity and the additivity of the determinant function.

Neeman’s K–theory [4, 5, 6, 7, 8] for triangulated categories and the Gillet-Grayson model [9] for the K–theory loop space enable us to define the determinant function

{Automorphisms in τ }⟶K_{1}(τ),

where τ is a triangulated category. In this paper we suggest a translation of Nenashev’s work to the triangulated case. Double distinguished triangles replace the double short exact sequences, and we have a relation which is derived from coherent double 3′ × 3′ commutative diagrams. We define the determinant function

{Automorphisms in τ }⟶K_{1}(τ)

by attaching double distinguished triangles to automorphism in τ. We prove in Theorem 5.2 that this is an additive multiplicative function.

As an application, we compute explicitly in Corollary 5.4 the determinant function on automorphisms in the derived category of a small abelian category A. An automorphism of bounded complex over A represents an automorphism of an object in D^{b}(A), and hence, its determinant is well defined. We will show that the determinant of an automorphism of a bounded complex is the alternating sum of the classical determinant of its components. In other words

Theorem 0.1. Let A be a small abelian category, and let

c_{•} : Z_{•} → Z_{•}

be an automorphism of a bounded complex Z_{•} over A. The determinant of c_{•} is given by the formula

det(c_{•}) = ∑_{i}(−1)^{i} det(c_{i}).

From our previous statements, the formula above yields a multiplicative, additive, homotopy invariant function, with values in K_{1}(A). We should note that for the regular ring case, there is much simpler proof for the multiplicativity property, which I learned from Grayson.

1 The Loop Space

In this paper we will use Gillet-Grayson model [2, 9] for the K–theory space. In Neeman’s notation, this simplicial model is described by the symbol

T [figure]  T [figure] [figure: 0]

We will denote this simplicial set by G(τ). An m–simplex is a pair of diagrams of the form

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0m}; Z_{mm}; ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0m}; Z_{mm}; ⋯; X_{0}; X_{m}; ⋯; X′_{0}; X′_{m}]

where the diagonal objects Z_{ii} are assumed to be the zero object. Each diagram is assumed to come with coherent differentials, i.e., the diagram comes with a differential for each square, and the differentials of large squares induces the differentials of smaller squares. The triangular parts of these two diagrams are assumed to be the same, including the differentials. The i^{th} face operation deletes simultaneously the i^{th} column in each of the two diagrams, and the i^{th} row in the triangular part of each diagram. The degeneracy operation inserts identities. And the diagrams must each split as a direct sum of diagrams which lift to models.

Theorem 1.1. The homotopy groups of the simplicial set G(τ) are the K–theory groups of the triangulated category τ.

Following Nenashev, we first study the simplices in low dimension. The 0-simplices are pairs of objects in τ. The 1-simplices are pairs of distinguished triangles having the same third vertex.

t : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} t′ : \xymatrix{X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z \ar[r]^{w′} & ΣX′}

We will denote this 1-simplex by (t, t′), and since the vertices of this simplex are (X, X′) and (Y, Y′), we will use the picture

\xymatrix{(X, X′) \ar[r]^{(t,t′)} & (Y, Y′)}

A 2–simplex diagram

\xymatrix{s_{1} : &  & 0 & s_{2} : &  & 0 \\  & 0 \ar[r] & Z \ar[u] \ar@{.>}[ddl]_{w_{1}} &  & 0 \ar[r] & Z \ar[u] \ar@{.>}[ddl]_{w_{2}} \\ 0 \ar[r] & X \ar[u] \ar[r]_{u} & Y \ar[u]_{v} \ar@{.>}[dll]^{d_{1}} & 0 \ar[r] & X \ar[u] \ar[r]_{u} & Y \ar[u]_{v} \ar@{.>}[dll]^{d_{2}} \\ A_{1} \ar[u] \ar[r]^{a_{1}} & B_{1} \ar[r]^{b_{1}} \ar[u]_{g_{1}} & C_{1} \ar[u]^{h_{1}} & A_{2} \ar[u] \ar[r]^{a_{2}} & B_{2} \ar[r]^{b_{2}} \ar[u]_{g_{2}} & C_{2} \ar[u]^{h_{2}}}

can be drawn in a double octahedron.

[figure: ΣA_{i}; ΣA_{i}; X; u; Y; v; d_{i}; Z; ΣX; e_{0}; B_{i}; b_{i}; g_{i}; C_{i}; h_{i}; Z; w_{i}; ΣB_{i}; Σg_{i}; A_{i}; a_{i}; A_{i}; e_{2}; e_{1}]

We will denote a 2–simplex which is this pair of diagrams by (s_{1}, s_{2}). The 0^{th} face is the pair of triangles in the row with the label e_{0}, the 1^{st} face is the pair of triangles in the column with the label e_{1}, and the 2^{nd} face is the pair of triangles in the column with the label e_{2}.

2 Double Distinguished Triangles

There is a special kind of edges, namely the edges associated to double distinguished triangles. A double distinguished triangle (d.d.t. for short) is a pair of distinguished triangles having the same objects.

t_{1} : \xymatrix{X \ar[r]^{u_{1}} & Y \ar[r]^{v_{1}} & Z \ar[r]^{w_{1}} & ΣX} t_{2} : \xymatrix{X \ar[r]^{u_{2}} & Y \ar[r]^{v_{2}} & Z \ar[r]^{w_{2}} & ΣX}

We will denote by e(l) the edge that the pair l = (t_{1}, t_{2}) defines. We will also denote this pair by

\xymatrix{X \ar[r]^{u_{1}} \ar[r]_{u_{2}} & Y \ar[r]^{v_{1}} \ar[r]_{v_{2}} & Z \ar[r]^{w_{1}} \ar[r]_{w_{2}} & ΣX.}  (2.1)

Given an object A, there is a standard edge from the base point (0, 0) to the point (A, A) given by the d.d.t.

\xymatrix{0 \ar[r] \ar[r] & A \ar[r]^{1_{A}} \ar[r]_{1_{A}} & A \ar[r] \ar[r] & 0.}

We will denote this edge by e(A). We associate to the d.d.t. l in (2.1) a loop μ(l) of the form

\xymatrix{(X, X) \ar[rr]^{e(l)} &  & (Y, Y) \\  & (0, 0) \ar[ul]^{e(X)} \ar[ur]_{e(Y)}}

3 Nenashev’s Main Lemma

The first Lemma to prove is Nenashev’s main Lemma on admissible triples. A triangle contour in G(τ)

\xymatrix{ & (C_{1}, C_{2}) \\ (A_{1}, A_{2}) \ar[ur]^{e_{1}} \ar[rr]_{e_{2}} &  & (B_{1}, B_{2}) \ar[ul]_{e_{0}}}

is always given by three edges of the form

\xymatrix{ &  & ΣB_{i} \\  &  & L \ar[u]^{b′′_{i}} \\ e_{1} & A_{i} \ar[r]^{c_{i}} & C_{i} \ar[u]^{b′_{i}} \ar[r]^{c′_{i}} & M \ar[r]^{c′′_{i}} & ΣA_{i} \\ e_{2} & A_{i} \ar[r]^{a_{i}} & B_{i} \ar[u]^{b_{i}} \ar[r]^{a′_{i}} & N \ar[r]^{a′′_{i}} & ΣA_{i} \\  &  & e_{0}}

where i = 1, 2. We will say that the triple τ = (e_{0}, e_{1}, e_{2}) is admissible, if the diagram above can be completed to the a commutative diagram with the identity arrow A_{i} → A_{i}. In this case we can apply the octahedral axiom twice: once for i = 1, and one for i = 2, and get two octahedra.

[figure: ΣB_{i}; Σa′_{i}; ΣN; L; b′′_{i}; L; w_{i}; e_{1}; A_{i}; c_{i}; C_{i}; +; b′_{i}; c′_{i}; M; c′′_{i}; v_{i}; ΣA_{i}; e_{2}; A_{i}; a_{i}; B_{i}; b_{i}; a′_{i}; N; a′′_{i}; u_{i}; ΣA_{i}; e_{0}; e(l)]

In these diagrams the rows and the columns are distinguished triangles, and the tagged square is an MV –square with the differential

δ_{i} = Σa_{i}c′′_{i} = b′′_{i}v_{i}.

This means that the triangles

\xymatrix{B_{i} \ar[r]^{^{( \begin{matrix}a′_{i} \\ b_{i} \end{matrix} )}} & N ⊕ C_{i} \ar[r]^{^{(^{−u_{i} c′_{i}})}} & M \ar[r]^{δ_{i}} & ΣB_{i}}

are distinguished. The induced double distinguished triangle l in the second column is called a d.d.t. associated to the admissible triple (e_{0}, e_{1}, e_{2}). The next Lemma shows the relation between the loop (e_{0}, e_{1}, e_{2}) and the the loop μ(l). Since π_{1}(G(τ)) = K_{1}(τ) is abelian, we allow ourself to do all the computation in the first homology group H_{1}(G(τ)).

Lemma 3.1. If (e_{0}, e_{1}, e_{2}) is an admissible triple with an associated d.d.t. l, then there is an equality of cocycles

e_{0} − e_{1} + e_{2} = (C_{1}, C_{2}) ⊕ μ(l).

Remark 3.2. This Lemma is the translation of [11, Lemma 4.1] to the triangulated category case. The direct sum of a point with a simplex is defined as in [11, Remark 4.2]. The simplicial shift map (C_{1}, C_{2}) ⊕ (−) takes the simplex

[figure: ⋮; ⋯; ·; ·; ·; 0; Z_{0m}; 0; ⋮; ⋯; ·; ·; ·; 0; Z_{0m}; 0; ⋯; X_{0}; X_{m}; ⋯; X′_{0}; X′_{m}]

to the simplex

[figure: ⋮; ⋯; ·; ·; ·; 0; Z_{0m}; 0; ⋮; ⋯; ·; ·; ·; 0; Z_{0m}; 0; ⋯; C_{1} ⊕ X_{0}; C_{1} ⊕ X_{m}; ⋯; C_{2} ⊕ X′_{0}; C_{2} ⊕ X′_{m}]

Any edge connecting (C_{1}, C_{2}) to (C′_{1}, C′_{2}) yield a simplicial homotopy

(C_{1}, C_{2}) ⊕ (−)⟶(C′_{1}, C′_{2}) ⊕ (−).

In particular, if (C_{1}, C_{2}) is in the base point component, then (C_{1}, C_{2}) ⊕ (−) is homotopic to the identity map.

Corollary 3.3. If the triangle contour (e_{0}, e_{1}, e_{2}) lies in the base point component, then

e_{0} − e_{1} + e_{2} = μ(l).

Proof the Lemma. The proof uses the idea in [11, Page 33]. The following six double octahedra represent six 2-simplices in G(τ), which induce six relations in H_{1}(G(τ)). The relation in the Lemma is derived from these six relations. We will denote the vertex (C_{1}, C_{2}) by p. Here is the list of the double octahedra, each is followed by the induced relation in H_{1}(G(τ)).

[figure: ΣA_{i}; ΣA_{i}; M; c′′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; M ⊕ M; _{(_{c′′_{i} c′′_{i}})}; _{(_{1 0} )}; M; 0; ΣM; p⊕e(M); C_{i}; c′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; M ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ −1 & c′_{i} \end{matrix} )}; _{(_{1 0} )}; M; 0; ΣC_{i}; Σc′_{i}; A_{i}; c_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; e_{1}; ê]

⟹ p ⊕ e(M) − ê+ e_{1} = 0.  (3.1)

[figure: ΣA_{i}; ΣA_{i}; M; c′′_{i}; _{(\begin{matrix}1 \\ 0 \end{matrix})}; M ⊕ M; _{(_{c′′_{i} c′′_{i}})}; _{(_{0 1} )}; M; 0; ΣM; e′_{1}; C_{i}; c′_{i}; _{( \begin{matrix}c′_{i} \\ 1 \end{matrix} )}; M ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ −1 & c′_{i} \end{matrix} )}; _{(_{−1 c′_{i}})}; M; 0; ΣC_{i}; Σc′_{i}; A_{i}; c_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; e_{1}; ê]

⟹ e′_{1} − ê+ e_{1} = 0.  (3.2)

[figure: ΣA_{i}; ΣA_{i}; N; a′′_{i}; _{(\begin{matrix}1 \\ 0 \end{matrix})}; N ⊕ M; _{(_{a′′_{i} c′′_{i}})}; _{(_{0 1} )}; M; 0; ΣN; e′_{2}; B_{i}; a′_{i}; _{( \begin{matrix}a′_{i} \\ b_{i} \end{matrix} )}; N ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ −u_{i} & c′_{i} \end{matrix} )}; _{(_{−u_{i} c′_{i}})}; M; δ_{i}; ΣB_{i}; Σa′_{i}; A_{i}; a_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; e_{2}; e′]

⟹ e′_{2} − e′ + e_{2} = 0.  (3.3)

[figure: ΣA_{i}; ΣA_{i}; M; c′′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; N ⊕ M; _{(_{a′′_{i} c′′_{i}})}; _{(_{1 0} )}; N; 0; ΣM; p⊕e(N); C_{i}; c′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; N ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ −u_{i} & c′_{i} \end{matrix} )}; _{(_{1 0} )}; N; 0; ΣC_{i}; Σc′_{i}; A_{i}; c_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; e_{1}; e′]

⟹ p ⊕ e(N) − e′ + e_{1} = 0.  (3.4)

[figure: ΣB_{i}; ΣB_{i}; L; b′′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; M ⊕ L; _{(_{δ_{i} b′′_{i}})}; _{(_{1 0} )}; M; 0; ΣL; e′_{1}; C_{i}; b′_{i}; _{( \begin{matrix}c′_{i} \\ 1 \end{matrix} )}; M ⊕ C_{i}; _{( \begin{matrix}−1 & c′_{i} \\ v_{i} & 0 \end{matrix} )}; _{(_{−1 c′_{i}})}; M; 0; ΣC_{i}; Σb′_{i}; B_{i}; b_{i}; B_{i}; _{( \begin{matrix}c′_{i}b_{i} \\ b_{i} \end{matrix} )}; e_{0}; e′′]

⟹ e′_{1} − e′′ + e_{0} = 0.  (3.5)

[figure: ΣB_{i}; ΣB_{i}; M; δ_{i}; _{(\begin{matrix}1 \\ 0 \end{matrix})}; M ⊕ L; _{(_{δ_{i} b′′_{i}})}; _{(_{0 1} )}; L; 0; ΣM; p⊕e(l); N ⊕ C_{i}; _{(_{−u_{i} c′_{i}})}; _{( \begin{matrix}u_{i} & 0 \\ 0 & 1 \end{matrix} )}; M ⊕ C_{i}; _{( \begin{matrix}−1 & c′_{i} \\ v_{i} & 0 \end{matrix} )}; _{(_{v_{i} 0} )}; L; _{(\begin{matrix}w_{i} \\ 0 \end{matrix} )}; ΣN ⊕ ΣC_{i}; B_{i}; _{( \begin{matrix}a′_{i} \\ b_{i} \end{matrix} )}; B_{i}; _{( \begin{matrix}u_{i}a′_{i}=c′_{i}b_{i} \\ b_{i} \end{matrix} )}; e′_{2}; e′′]

⟹ p ⊕ e(l) − e′′ + e′_{2} = 0.  (3.6)

The double triangles ê, e′_{1}, e′ and e′′ are distinguished, since there exist the following triangle isomorphisms.

[figure: ê; A_{i}; ^{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; M ⊕ C_{i}; ^{( \begin{matrix}1 & 0 \\ −1 & c′_{i} \end{matrix} )}; M ⊕ M; ^{(^{c′′_{i} c′′_{i}})}; _{(\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix})}; ΣA_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; M ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ 0 & c′_{i} \end{matrix} )}; M ⊕ M; _{(_{0 c′′_{i}})}; ΣA_{i}]

[figure: e′_{1}; C_{i}; ^{( \begin{matrix}c′_{i} \\ 1 \end{matrix} )}; M ⊕ C_{i}; ^{(^{−1 c′_{i}})}; _{( \begin{matrix}−1 & c′_{i} \\ 0 & 1 \end{matrix} )}; M; 0; ΣC_{i}; C_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; M ⊕ C_{i}; _{(_{1 0} )}; M; 0; ΣC_{i}]

[figure: e′; A_{i}; ^{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; N ⊕ C_{i}; ^{( \begin{matrix}1 & 0 \\ −u_{i} & c′_{i} \end{matrix} )}; N ⊕ M; ^{(^{a′′_{i} c′′_{i}})}; _{( \begin{matrix}1 & 0 \\ u_{i} & 1 \end{matrix} )}; ΣA_{i}; A_{i}; _{( \begin{matrix}0 \\ c_{i} \end{matrix} )}; N ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ 0 & c′_{i} \end{matrix} )}; N ⊕ M; _{(_{0 c′′_{i}})}; ΣA_{i}]

[figure: e′′; B_{i}; ^{( \begin{matrix}c′_{i}b_{i} \\ b_{i} \end{matrix} )}; M ⊕ C_{i}; ^{( \begin{matrix}−1 & c′_{i} \\ v_{i} & 0 \end{matrix} )}; _{( \begin{matrix}−1 & c′_{i} \\ 0 & 1 \end{matrix} )}; M ⊕ L; ^{(^{δ_{i} b′′_{i}})}; _{( \begin{matrix}1 & 0 \\ v_{i} & 1 \end{matrix} )}; ΣB_{i}; B_{i}; _{( \begin{matrix}0 \\ b_{i} \end{matrix} )}; M ⊕ C_{i}; _{( \begin{matrix}1 & 0 \\ 0 & b′_{i} \end{matrix} )}; M ⊕ L; _{(_{0 b′′_{i}})}; ΣB_{i}]

Relations (3.1) and (3.2) yields

p ⊕ e(M) − e′_{1} = 0,

the next two relations (3.3) and (3.4) imply the relation

p ⊕ e(N) − e′_{2} − (e_{2} − e_{1}) = 0,

and the relation

p ⊕ e(l) + e′_{2} − e′_{1} − e_{0} = 0

is derived from the last two relations (3.5) and (3.6). These three relations yields the identity

p ⊕ μ(l) = p ⊕ e(N) + p ⊕ e(l) − p ⊕ e(M) = e_{0} − e_{1} + e_{2}.

4 The 3′ × 3′ Relation

In [10, 11, 12] Nenashev proved that the K_{1} group of an exact category is the free abelian group on the double short exact sequences divided by the relations that arise from 3 × 3 commutative diagrams with double exact sequences in the columns and in the rows. Here we describe one way to translate the claim and the proof to the triangulated category case.

Consider the double (i = 1, 2) commutative diagram with a distinguished triangle in each row and column.

\xymatrix{ & ΣA′ \ar[r]^{Σf′_{i}} & ΣB′ \ar[r]^{Σg′_{i}} & ΣC′ \\ l′′ & A′′ \ar[r]^{f′′_{i}} \ar[u]^{a′′_{i}} & B′′ \ar[r]^{g′′_{i}} \ar[u]^{b′′_{i}} & C′′ \ar[r]^{h′′_{i}} \ar[u]^{c′′_{i}} & ΣA′′ \\ l & A \ar[r]^{f_{i}} \ar[u]^{a_{i}} & B \ar[r]^{g_{i}} \ar[u]^{b_{i}} & C \ar[r]^{h_{i}} \ar[u]^{c_{i}} & ΣA \ar[u]^{Σa_{i}} \\ l′ & A′ \ar[r]^{f′_{i}} \ar[u]^{a′_{i}} & B′ \ar[r]^{g′_{i}} \ar[u]^{b′_{i}} & C′ \ar[r]^{h′_{i}} \ar[u]^{c′_{i}} & ΣA′ \ar[u]^{Σa′_{i}} \\  & l_{A} & l_{B} & l_{C}}  (4.1)

We will call such a double diagram a double 3′×3′ commutative diagram, and we will denote it also by the following diagram.

\xymatrix{ & ΣA′ \ar[r]^{Σf′_{1}} \ar[r]_{Σf′_{2}} & ΣB′ \ar[r]^{Σg′_{1}} \ar[r]_{Σg′_{2}} & ΣC′ \\ l′′ & A′′ \ar[r]^{f′′_{1}} \ar[r]_{f′′_{2}} \ar[u]^{a′′_{1}} \ar[u]_{a′′_{2}} & B′′ \ar[r]^{g′′_{1}} \ar[r]_{g′′_{2}} \ar[u]^{b′′_{1}} \ar[u]_{b′′_{2}} & C′′ \ar[r]^{h′′_{1}} \ar[r]_{h′′_{2}} \ar[u]^{c′′_{1}} \ar[u]_{c′′_{2}} & ΣA′′ \\ l & A \ar[r]^{f_{1}} \ar[r]_{f_{2}} \ar[u]^{a_{1}} \ar[u]_{a_{2}} & B \ar[r]^{g_{1}} \ar[r]_{g_{2}} \ar[u]^{b_{1}} \ar[u]_{b_{2}} & C \ar[r]^{h_{1}} \ar[r]_{h_{2}} \ar[u]^{c_{1}} \ar[u]_{c_{2}} & ΣA \ar[u]^{Σa_{1}} \ar[u]_{Σa_{2}} \\ l′ & A′ \ar[r]^{f′_{1}} \ar[r]_{f′_{2}} \ar[u]^{a′_{1}} \ar[u]_{a′_{2}} & B′ \ar[r]^{g′_{1}} \ar[r]_{g′_{2}} \ar[u]^{b′_{1}} \ar[u]_{b′_{2}} & C′ \ar[r]^{h′_{1}} \ar[r]_{h′_{2}} \ar[u]^{c′_{1}} \ar[u]_{c′_{2}} & ΣA′ \ar[u]^{Σa′_{1}} \ar[u]_{Σa′_{2}} \\  & l_{A} & l_{B} & l_{C}}

Each row and each column represents a double distinguished triangle, and the relation (sometimes) associated to the double commutative diagram (4.1) is

μ(l′) − μ(l) + μ(l′′) = μ(l_{A}) − μ(l_{B}) + μ(l_{C}).

Given a double commutative diagram (4.1) with distinguished triangles in the rows and in the columns, one can draw the following four double octahedra constructed by the octahedral axiom. The first one is the following octahedron.

[figure: ΣA′; ΣA′; A′′; a′′_{i}; _{(\begin{matrix}1 \\ 0 \end{matrix})}; A′′ ⊕ C′; _{(_{a′′_{i} h′_{i}})}; _{(_{0 1} )}; C′; 0; ΣA′′; e_{2}; A; a_{i}; D_{i}; C′; Σa′_{i}h′_{i}; ||; h_{i}c′_{i}; ΣA; Σa_{i}; A′; a′_{i}; A′; e(l_{A}); m]  (4.2)

In this octahedron we first complete the morphism

\xymatrix{A′′ ⊕ C′ \ar[r]^{^{(^{a′′_{i} h′_{i}})}} & ΣA′}

to a distinguished triangle, and get the column with the label m. Then, using the octahedral axiom, we choose a distinguished triangle to put in the row with the label e_{2}. Since the compositions

\xymatrix{A′′ ⊕ C′ \ar[r]^{^{(^{a′′_{i} h′_{i}})}} & ΣA′ \ar[r]^{Σa′_{i}} & ΣA & and \\ A′′ ⊕ C′ \ar[r]^{_{(_{0 1} )}} & C′ \ar[r] & ΣA}

coincide, the map C′⟶ΣA must be Σa′_{i}h′_{i}. In the next octahedron we already have the arrows in the column with the label m, and we choose a distinguished triangle to complete the row with the label e_{1} by the octahedral axiom. Again, the map A′′⟶ΣB′ must be Σf′_{i}a′′_{i}.

[figure: ΣA′; ΣA′; C′; h′_{i}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; A′′ ⊕ C′; _{(_{a′′_{i} h′_{i}})}; _{(_{1 0} )}; A′′; 0; ΣC′; e_{1}; B′; g′_{i}; D_{i}; A′′; Σf′_{i}a′′_{i}; ||; b′′_{i}f′′_{i}; ΣB′; Σg′_{i}; A′; f′_{i}; A′; e(l′); m]  (4.3)

The next step is to copy the above e_{1} and e_{2} rows to the following two octahedra, and then choose distinguished triangles to complete the columns with the labels ê and ẽ.

[figure: ΣD_{i}; ΣA′′; C′′; C′′; h′′_{i}; e(l_{B}); B′; b′_{i}; B; g′′_{i}b_{i}; =; c_{i}g_{i}; b_{i}; B′′; g′′_{i}; b′′_{i}; ΣB′; e_{1}; B′; D_{i}; A′′; f′′_{i}; b′′_{i}f′′_{i}; ΣB′; ê; e(l′′)]  (4.4)

[figure: ΣD_{i}; ΣC′; C′′; C′′; c′′_{i}; e(l); A; f_{i}; B; g′′_{i}b_{i}; =; c_{i}g_{i}; g_{i}; C; c_{i}; h_{i}; ΣA; e_{2}; A; D_{i}; C′; c′_{i}; h_{i}c′_{i}; ΣA; ẽ; e(l_{C})]  (4.5)

The first two double octahedra (4.2, 4.3) represent 2-simplices and yield the relations

e_{2} − m + e(l_{A}) = 0
e_{1} − m + e(l′) = 0.

By Nenashev’s basic Lemma 3.1, the other two double octahedra (4.4, 4.5) imply the relations

ê− e(l_{B}) + e_{1} = μ(l′′)
ẽ− e(l) + e_{2} = μ(l_{C}).

Hence,

μ(l′′) − μ(l_{C}) = [e(l) + e(l_{A}) − e(l_{B}) − e(l′)]
+ [ê− ẽ]
= [μ(l) + μ(l_{A}) − μ(l_{B}) − μ(l′)]
+ [ê− ẽ].

This is the desired relation with an additional two-edge cycle.

The problem is that in general ê− ẽ may not vanish. In the degenerate case, where the objects C′, C and C′′ are zero, we can first take D_{i} = B, the arrow D_{i} → B to be the identity morphism, and then make the natural choice for the rest of the morphisms.

Definition 4.1. A coherent double 3′ ×3′ commutative diagram is a a pair of two commutative diagrams as in (4.1) with distinguished triangles in the rows and in the columns, such that there exist four octahedra (4.2,4.3,4.4,4.5) with

ê = ẽ.

Proposition 4.2. A coherent double 3′ × 3′ commutative diagram (4.1) yields the relation

μ(l′) − μ(l) + μ(l′′) = μ(l_{A}) − μ(l_{B}) + μ(l_{C}).

As we said above, a double 3′ × 3′ commutative diagram (4.1) with C′ = C = C′′ = 0 is coherent.

5 Determinant Elements

Definition 5.1. The d.d.t. associated to and automorphism a : A → A is

\xymatrix{l(a) : & A \ar[r]^{a} \ar[r]_{1_{A}} & A \ar[r] \ar[r] & 0 \ar[r] \ar[r] & ΣA}

and the determinant element in π_{1}(G(τ)) = K_{1}(τ) associated with a is

det(a) = μ(l(a)).

Theorem 5.2. The determinant function

det : {Automorphisms in τ }⟶K_{1}(τ)

is a multiplicative additive function, which is compatible with the classical determinant map

det : {Automorphisms in A}⟶K_{1}(A)

where A is a small abelian category. More precisely:

Multiplicativity det(ab) = det(a) + det(b), where a and b are two automorphisms of the same object in τ;

Additivity det(b) = det(a) + det(c), where a, b and c are the components of a distinguished triangle automorphism in τ;

\xymatrix{A \ar[r]^{u} \ar[d]^{a} & B \ar[r]^{v} \ar[d]^{b} & C \ar[r]^{w} \ar[d]^{c} & ΣA \ar[d]^{Σa} \\ A \ar[r]^{u} & B \ar[r]^{v} & C \ar[r]^{w} & ΣA}

Compatibility if τ has a bounded t-structure and the heart is the abelian category A, then the diagram

\xymatrix{\{Automorphisms in A\} \ar@{^{(}->}[r] \ar[d]^{det} & \{Automorphisms in τ \} \ar[d]^{det} \\ K_{1}(A) \ar[r]^{∼} & K_{1}(τ)}

commutes.

Proof. Multiplicativity
\xymatrix{ & ΣX \ar[r]^{Σa} \ar[r]_{1_{ΣX}} & ΣX \ar[r] & 0 \\ l′′ & 0 \ar[r] \ar[u] & 0 \ar[r] \ar[u] & 0 \ar[r] \ar[u] & 0 \\ l & X \ar[r]^{ba} \ar[r]_{1_{X}} \ar[u] & X \ar[r] \ar[u] & 0 \ar[r] \ar[u] & ΣX \ar[u] \\ l′ & X \ar[r]^{a} \ar[r]_{1_{X}} \ar[u]^{1_{X}} & X \ar[r] \ar[u]^{b} \ar[u]_{1_{X}} & 0 \ar[r] \ar[u] & ΣX \ar[u]^{1_{ΣX}} \\  & l_{A} & l_{B} & l_{C}}

Additivity For a distinguished triangle automorphism in τ

\xymatrix{A′ \ar[r]^{a′} \ar[d]^{f′} & A \ar[r]^{a} \ar[d]^{f} & A′′ \ar[r]^{a′′} \ar[d]^{f′′} & ΣA′ \ar[d]^{Σf′} \\ A′ \ar[r]^{a′} & A \ar[r]^{a} & A′′ \ar[r]^{a′′} & ΣA′}

one has the coherent double 3′ × 3′ commutative diagram

\xymatrix{ & ΣA′ \ar[r]^{Σf′} \ar[r]_{1} & ΣA′ \ar[r] \ar[r] & 0 \\ l′′ & A′′ \ar[r]^{f′′} \ar[r]_{1} \ar[u]^{a′′} \ar[u]_{a′′} & A′′ \ar[r] \ar[r] \ar[u]^{a′′} \ar[u]_{a′′} & 0 \ar[r] \ar[r] \ar[u] \ar[u] & ΣA′′ \\ l & A \ar[r]^{f} \ar[r]_{1} \ar[u]^{a} \ar[u]_{a} & A \ar[r] \ar[r] \ar[u]^{a} \ar[u]_{a} & 0 \ar[r] \ar[r] \ar[u] \ar[u] & ΣA \ar[u]^{Σa} \ar[u]_{Σa} \\ l′ & A′ \ar[r]^{f′} \ar[r]_{1} \ar[u]^{a′} \ar[u]_{a′} & A′ \ar[r] \ar[r] \ar[u]^{a′} \ar[u]_{a′} & 0 \ar[r] \ar[r] \ar[u] \ar[u] & ΣA′ \ar[u]^{Σa′} \ar[u]_{Σa′} \\  & l_{A} & l_{A} & l_{0}}

Here μ(l′) = det(f′), μ(l) = det(f), μ(l′′) = det(f′′), and the identity follows.

Compatibility Letτ be a triangulated category with a bounded t–structure and a heart A, and let a : X → X be an automorphism of an object in the heart. The determinant element det(a) in [12] is the loop associated to the double short exact sequence

\xymatrix{0 \ar[r] \ar[r] & X \ar[r]^{1} \ar[r]_{a} & X} ,

which is equivalent to the loop associated to the double short exact sequence

\xymatrix{X \ar[r]^{a} \ar[r]_{1} & X \ar[r] \ar[r] & 0} .

See Definition 1.1 in [12], and the discussion in the second paragraph in page 38 there. The edge associated to the last double short exact sequence and the corresponding loop in G(A) maps to e(l(a)) and μ(l(a)), respectively, by the homotopy equivalence inclusion

G(A) ↪ G(τ).

As an immediate conclusion, we compute the affect of the suspension functor on automorphisms.

Lemma 5.3.

det(Σa) = − det(a).

Proof. Let a : X⟶X be an automorphism of an object X in a triangulated category τ. There is a distinguished triangle automorphism

[figure: X; a; 0; ΣX; Σa; ΣX; Σa; X; 0; ΣX; ΣX]

and by additivity, 0 = det(Σa) + det(a).

Corollary 5.4. Let A be a small abelian category, and let

c_{•} : Z_{•} → Z_{•}

be an automorphism of a bounded complex Z_{•} over A. The determinant of c_{•} is given by the formula

det(c_{•}) = ∑_{i}(−1)^{i} det(c_{i}).

We conclude that the sum above depends only on the homotopy class of c_{•}, and that it satisfies the multiplicative and additive identities of Theorem 5.2.

Proof. Let c_{•} : Z_{•} → Z_{•} be an automorphism of a bounded complex Z_{•} with top degree n.

\xymatrix{Z_{•} \ar[d]^{c_{•}} & … \ar[r] & 0 \ar[r] \ar[d] & Z_{m} \ar[r] \ar[d]^{c_{m}} & … \ar[r] & Z_{n−1} \ar[r] \ar[d]^{c_{n−1}} & Z_{n} \ar[r] \ar[d]^{c_{n}} & 0 \ar[r] \ar[d] & … \\ Z_{•} & … \ar[r] & 0 \ar[r] & Z_{m} \ar[r] & … \ar[r] & Z_{n−1} \ar[r] & Z_{n} \ar[r] & 0 \ar[r] & …}

The proof is by induction on the size of the complex. By Lemma 5.3, if Z_{•} is concentrated in degree n, then det(c_{•}) = (−1)^{n} det(c_{n}), and the induction starts.

For the general case, let Y_{•} be the complex concentrated in degree n with Y_{n} = Z_{n}, and let X_{•} be the de-suspension of the brutal truncation of Z_{•} under degree n, i.e., X_{i} = Z_{i−1} for i ≠ n + 1 and X_{n+1} = 0. Let a_{•} be the restriction of c_{•} to X_{•}, and let b_{•} be the restriction of c_{•} to Y_{•}. The map Z_{n−1} → Z_{n} defines a morphism X_{•} → Y_{•}, with mapping cone Z_{•}. Now we have an automorphism (a_{•}, b_{•}, c_{•}) of the distinguished triangle

X_{•}⟶Y_{•}⟶Z_{•}⟶ΣX_{•},

and by additivity

det(c_{•}) = det(b_{•}) − det(a_{•}).

By the induction hypothesis

det(a_{•}) = ∑_{i}(−1)^{i} det(a_{i}) = ∑_{i<n+1}(−1)^{i} det(c_{i−1}),

and by Lemma 5.3,

det(b_{•}) = det(Σ^{−n}c_{n}) = (−1)^{n} det(c_{n}).

Therefore,

det(c_{•}) = det(b_{•}) − det(a_{•})

= (−1)^{n} det(c_{n}) − ∑_{i<n+1}(−1)^{i} det(c_{i−1})

= (−1)^{n} det(c_{n}) + ∑_{i<n}(−1)^{i} det(c_{i})

= ∑_{i}(−1)^{i} det(c_{i}).

Acknowledgments

Most of this work was done while the author was visiting the Australian National University, Canberra. The author would like to express his gratitude to Amnon Neeman for his interest, guidance, and many helpful discussions.

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