ELEMENTS OF K_{1} VIA THE TRIPLE G-CONSTRUCTION

A. Nenashev

Abstract. We represent the elements of K_{1} of an exact category by means of twoand three-edge loops in the triple G-construction. The corresponding algebraic data are double short exact sequences and non-commutative 3 × 3-diagrams of short exact sequences. We state conjectural relations for the elements of K_{1} in terms of 3 × 3diagrams.

1. Review of the G- and the triple G-construction

In [GG] Gillet and Grayson attached a simplicial set G.A to any exact category A and proved that |G.A| is homotopy equivalent to Ω|Q A| ∼ Ω|S.A|, the equivalence being natural in A. Thus one can take the formula

K_{m}(A) = π_{m}(G.A) , m ≥ 0 ,

for a definition of the higher K-groups of A.

An n-simplex in G.A is a pair of triangular diagrams in A of the form

\begin{aligned}& & & & & & & & P_{n/n−1} & & & & & & & & & P_{n/n−1} \\ & & & & & & & & ↑ & & & & & & & & & ↑ \\ & & & & & & & & … & & & & & & & & & … \\ & & & & P_{2/1} & ⟶ & … & ⟶ & P_{n/1} & & & & & P_{2/1} & ⟶ & … & ⟶ & P_{n/1} \\ & & & & ↑ & & & & ↑ & & & & & ↑ & & & & ↑ \\ & & P_{1/0} & ⟶ & P_{2/0} & ⟶ & … & ⟶ & P_{n/0} & & & P_{1/0} & ⟶ & P_{2/0} & ⟶ & … & ⟶ & P_{n/0} \\ & & ↑ & & ↑ & & & & ↑ & & & ↑ & & ↑ & & & & ↑ \\ P_{0} & ⟶ & P_{1} & ⟶ & P_{2} & ⟶ & … & ⟶ & P_{n} & P′_{0} & ⟶ & P′_{1} & ⟶ & P′_{2} & ⟶ & … & ⟶ & P′_{n} \end{aligned} (1.1)

subject to the conditions:

(i) the quotient index subtriangles in both diagrams coincide; (ii) all the squares commute; (iii) all the sequences of the form P_{j} → P_{k} → P_{k/j}, P′_{j} → P′_{k} → P_{k/j}, and P_{j/i} → P_{k/i} → P_{k/j} with i ≤ j ≤ k are s.e.s’s in A. (For short, we will write s.e.s. for short exact sequence).

In particular, a vertex in G.A is a pair of objects (P, P′), and an edge connecting (P_{0}, P′_{0}) to (P_{1}, P′_{1}) is a pair of s.e.s’s (P_{0} → P_{1} → P_{1/0}, P′_{0} → P′_{1} → P_{1/0}) with equal cokernels.

Typeset by AMS-TEX

Key words and phrases. Exact category, K_{1}, G-construction, double short exact sequence.

The i-th face of (1.1) amounts to deleting all the objects whose indices contain i. For instance, the faces of a generic 2-simplex

t = [ \begin{matrix}& & & & P_{2/1} & & & & & P_{2/1} \\ & & & & ↑ & & & & & ↑ \\ & & P_{1/0} & ⟶ & P_{2/0} & & & P_{1/0} & ⟶ & P_{2/0} \\ & & ↑ & & ↑ & & & ↑ & & ↑ \\ P_{0} & ⟶ & P_{1} & ⟶ & P_{2} & P′_{0} & ⟶ & P′_{1} & ⟶ & P′_{2} \end{matrix} ]

are given by

d_{0}t = (P_{1} → P_{2} → P_{2/1}, P′_{1} → P′_{2} → P_{2/1})
d_{1}t = (P_{0} → P_{2} → P_{2/0}, P′_{0} → P′_{2} → P_{2/0})
d_{2}t = (P_{0} → P_{1} → P_{1/0}, P′_{0} → P′_{1} → P_{1/0}).

The reader is referred to [Ne1-3] for a better introduction to G.A. In the present paper we will also need another simplicial set G_{.}^{III}A with the properties:

(i) a vertex in it is a triple of objects (P′, P, P′′) ∈ A × A × A, the upper index III referring to this fact; (ii) an edge from (P′_{0}, P_{0}, P′′_{0}) to (P′_{1}, P_{1}, P′′_{1}) is a diagram of the form

P′_{0} ⟶ P′_{1} ⟶ P′_{1/0} ; ↓ ; P_{0} ⟶ P_{1} ⟶ P_{1/0} ; ↓ ; P′′_{0} ⟶ P′′_{1} ⟶ P′′_{1/0}  (1.2)

consisting of four s.e.s’s; (iii) an arbitrary n-simplex is a triple of admissible filtrations

P′_{0} ⟶ P′_{1} ⟶ … ⟶ P′_{n}
P_{0} ⟶ P_{1} ⟶ … ⟶ P_{n}
P′′_{0} ⟶ P′′_{1} ⟶ … ⟶ P′′_{n}

endowed with

a) a choice of quotients P′_{j/i}, P_{j/i}, P′′_{j/i}for all possible composed maps P′_{i} ⟶ P′_{j}, P_{i} ⟶ P_{j}, and P′′_{i} ⟶ P′′_{j};

b) short exact sequences

0 ⟶ P′_{j/i} ⟶ P_{j/i} ⟶ P′′_{j/i} ⟶ 0  (1.3)

for all 0 ≤ i ≤ j ≤ n such that the arrows in (1.3) commute with the obvious projections and inclusions of the quotient objects.

One defines the face and degeneracy maps for G^{III} in the obvious way, the same as for G. We refer to [G, sect. 8] for a more formal definition of G^{III}. In fact, an infinite series of “the long G’s” is introduced in [G], i.e., for any k ≥ 2 there is a G-construction whose vertices are k-tuples of objects, and all these G-constructions are naturally homotopy equivalent. In particular, we have an obvious embedding i : G ⟶ G^{III} given on vertices by (P, P′) ⟶ (P, P′, 0) which is a homotopy equivalence, and we can therefore identify

K_{1}(A) = π_{1}(G.A) = π_{1}(G_{.}^{III}A).

Given a s.e.s. s = (A′ ⟶ A ⟶ A′′) we denote by e(s) the edge from (0, 0, 0) to (A′, A, A′′) in G_{.}^{III}A given by

0 ⟶ A′ ⟶^{1} A′ ; ↓ ; 0 ⟶ A ⟶^{1} A ; ↓ ; 0 ⟶ A′′ ⟶^{1} A′′  (1.4)

(cf. (1.2)). Note that a generic 2-simplex in G_{.}^{III}A is a diagram of the form

\xymatrix{ &  &  &  &  &  & P′_{2/1} \ar@{-->}[dr] \\  &  &  &  &  &  &  & P_{2/1} \ar@{-->}[dr] \\  &  &  &  &  &  &  &  & P′′_{2/1} \\  &  &  & P′_{1/0} \ar[rrr] \ar@{-->}[dr] &  &  & P′_{2/0} \ar[uuu] \ar@{-->}[dr] \\  &  &  &  & P_{1/0} \ar@{-->}[dr] \ar[rrr] &  &  & P_{2/0} \ar[uuu] \ar@{-->}[dr] \\  &  &  &  &  & P′′_{1/0} \ar[rrr] &  &  & P′′_{2/0} \ar[uuu] \\ P′_{0} \ar[rrr] &  &  & P′_{1} \ar[rrr] \ar[uuu] &  &  & P′_{2} \ar[uuu] \\  & P_{0} \ar[rrr] &  &  & P_{1} \ar[rrr] \ar[uuu] &  &  & P_{2} \ar[uuu] \\  &  & P′′_{0} \ar[rrr] &  &  & P′′_{1} \ar[rrr] \ar[uuu] &  &  & P′′_{2} \ar[uuu]}

In particular, given a commutative diagram of s.e.s’s

d = \begin{matrix}A′ & ⟶ & B′ & ⟶ & C′ \\ ↓ & & ↓ & & ↓ \\ A & ⟶ & B & ⟶ & C \\ ↓ & & ↓ & & ↓ \\ A′′ & ⟶ & B′′ & ⟶ & C′′ \end{matrix}  (1.5)

we associate to it a 2-simplex t(d) given by

[figure: C′; C; C′′; A′; B′; A; B; A′′; B′′; 0; A′; 1; B′; 1; 0; A; 1; B; 1; 0; A′′; 1; B′′; 1]  (1.6)

Note that the edges of t(d) with the source (0, 0, 0) are e(s_{A}) and e(s_{B}), where s_{A} = (A′ ⟶ A ⟶ A′′) and s_{B} = (B′ ⟶ B ⟶ B′′) are s.e.s’s from (1.5).

2. Two-edge loops in G^{III}

Recall that a double short exact sequence (a d.s.e.s. for short)

l = (A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C)

is a pair of short exact sequences s_{1} = (A ⟶^{f_{1}} B ⟶^{g_{1}} C) and s_{2} = (A ⟶^{f_{2}} B ⟶^{g_{2}} C) on the same objects. In [Ne1] we proved that every element of K_{1}(A) comes from a d.s.e.s. Namely, we associate to l a loop μ(l) in G.A of the form

\xymatrix{•^{(A,A)} \ar[rr]^{e(l)} &  & •^{(B,B)} \\  & •_{(0,0)} \ar[ul]^{e(A)} \ar[ur]_{e(B)}}

where e(X) denotes the edge (0 → X →^{1} X, 0 → X →^{1} X) for any X ∈ A and e(l) is the edge from (A, A) to (B, B) given by l, and prove that every loop in G.A is homotopic to μ(l) for some l. Since the map i : G → G^{III} is a homotopy equivalence, every loop in G_{.}^{III}A is homotopic to i(μ(l)) for some l. On the other hand, we can associate to l a simpler two-edge loop in G_{.}^{III}A,

κ(l) := (0, 0, 0)\xymatrix{ & e(s_{1}) \ar[dr] \\ • &  & • \\  & e(s_{2}) \ar[ur]} (A, B, C)  (2.1)

where for i = 1, 2, e(s_{i}) denotes the edge given by (1.4), and we always consider depicted loops as oriented clockwise.

Proposition 2.1. The loop i(μ(l)) is homotopic to κ(l)^{−1} for any d.s.e.s. l.

Proof. We construct two configurations of 2-simplices in G^{III}.A. The first configuration has the form

\xymatrix{•^{(A,B,C)} \ar[rrrr] &  &  &  & •^{(B,B⊕C,C)} \\  & •_{(A,A,0)} \ar[ul] \ar[urrr] \ar[rr] &  & •_{(B,B,0)} \ar[ur] \\  &  & •_{(0,0,0)} \ar[uull] \ar[ul] \ar[ur] \ar[uurr]}  (2.2)

It has i(μ(l)) as its inner loop and consists of four 2-simplices. The side 2-simplices are the simplices associated to the diagrams

A ⟶^{1} A ⟶ 0  B ⟶^{1} B ⟶ 0
1↓  ↓^{f_{2}}  ↓  ^{1}↓  ↓( \begin{matrix}1 \\ g_{2} \end{matrix} )  ↓

A ⟶^{f_{2}} B ⟶^{g_{2}} C  and  B \underrightarrow{( \begin{matrix}1 \\ g_{2} \end{matrix} )} B ⊕ C ⟶^{(g_{2},−1)} C
↓  ↓^{g_{2}}  ^{1}↓  ↓  ↓^{(g_{2},−1)}  ↓^{1}
0 ⟶ C ⟶^{1} C  0 ⟶ C ⟶^{1} C

as in (1.6). The upper simplices are given by

[figure: C; 1; C; 0; 0; C; 1; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; C; 1; _{( \begin{matrix}1 \\ −1 \end{matrix} )}; C⊕C; _{(_{1, 1} )}; _{(_{0, −1} )}; C; 1; C; A; 1; A; f_{1}; B; g_{1}; A; f_{2}; B; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; g_{2}; B⊕C; _{( \begin{matrix}g_{2} & 0 \\ −g_{2} & 1 \end{matrix} )}; 0; C; 1; 1; C; 1]

[figure: 0; C; 1; C; C; 1; 1; C; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; C; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; C⊕C; _{(_{0, −1} )}; _{(_{0, −1} )}; 0; C; 1; A; f_{1}; B; 1; g_{1}; B; g_{1}; A; f_{2}; B; _{( \begin{matrix}1 \\ g_{2} \end{matrix} )}; g_{2}; B⊕C; _{( \begin{matrix}g_{2} & 0 \\ −g_{2} & 1 \end{matrix} )}; 0; 0; C; 1]

The second configuration

\xymatrix{•^{(A,B,C)} \ar[rr] &  & •^{(B,B⊕C,C)} \\  & •_{(0,0,0)} \ar[ur] \ar[ul]^{e(s_{2})} \ar[ul]_{e(s_{1})}}  (2.3)

consists of κ(l)^{−1} and a single 2-simplex

t (\begin{matrix}A & ⟶^{f_{1}} & B & ⟶^{g_{1}} & C \\ ↓^{f_{1}} & ↓( \begin{matrix}1 \\ g_{1} \end{matrix} ) & ^{1}↓ \\ B & \underrightarrow{( \begin{matrix}1 \\ 0 \end{matrix} )} & B ⊕ C & ⟶^{( 0, 1 )} & C \\ ↓^{g_{1}} & ↓^{(}^{g_{1},} ^{−1}^{)} & ↓ \\ C & ⟶^{1} & C & ⟶ & 0 \end{matrix}) .

The outer loops in (2.2) and (2.3) differ only in the edge from (0, 0, 0) to (B, B ⊕ C, C), hence the difference between i(μ(l)) and κ(l)^{−1} is homotopic to the loop

(0, 0, 0)\xymatrix{ & e(r_{1}) \ar[dr] \\ • &  & • \\  & e(r_{2}) \ar[ur]} (B, B ⊕ C, C)  (2.4)

where

r_{1} = (B \underrightarrow{( \begin{matrix}1 \\ g_{1} \end{matrix} )} B ⊕ C ⟶^{(g_{1},−1)} C), r_{2} = (B \underrightarrow{( \begin{matrix}1 \\ g_{2} \end{matrix} )} B ⊕ C ⟶^{(g_{2},−1)} C).

The 2-simplex

t (\begin{matrix}B & ⟶^{1} & B & ⟶ & 0 \\ ↓( \begin{matrix}1 \\ g_{1} \end{matrix} ) & ↓( \begin{matrix}1 \\ g_{2} \end{matrix} ) & ↓ \\ B ⊕ C & \underrightarrow{( \begin{matrix}1 & 0 \\ g_{2}−g_{1} & 1 \end{matrix} )} & B ⊕ C & ⟶ & 0 \\ ↓^{(}^{g_{1},} ^{−1}^{)} & ↓^{(}^{g_{2},} ^{−1}^{)} & ↓ \\ C & ⟶^{1} & C & ⟶ & 0 \end{matrix})

provides a homotopy between (2.4) and the one-edge loop

B  ⟶^{1}  B ⟶ 0
↓

B ⊕ C \underrightarrow{( \begin{matrix}1 & 0 \\ g_{2}−g_{1} & 1 \end{matrix} )} B ⊕ C ⟶ 0
↓
C  ⟶^{1}  C ⟶ 0

whose base point is (B, B ⊕ C, C). The 2-simplex

[figure: 0; 0; 0; B; 1; 1; B; 1; B; 1; B; 0; 0; 0; B; 1; 1; B; 1; C; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; B ⊕ C; _{( \begin{matrix}1 & 0 \\ g_{2}−g_{1} & 1 \end{matrix} )}; _{(_{1, 0} )}; B⊕C; _{(_{1, 0} )}; C; 1; C; 1; C]

shows that the latter loop is contractible □

Corollary 2.2. Every loop in G_{.}^{III}A is homotopic to a two-edge loop of the form (2.1).

Remark. Recall that the G-construction was defined in [GG] by applying the combinatorial loop space functor ω (cf. [GG]) to the S-construction: G.A := ωS.A. Gillet and Grayson proved that |ωS.A| ∼ Ω|S.A|, but one cannot go further with G, i.e., ωG.A does not reprsent the homotopy type of the loop space Ω|G.A|, because G.A does not satisfy the necessary condition:

If for a given pointed simplicial set X one has |ωX| ∼ Ω|X|, then every loop in X is homotopic to a two-edge loop.

The corollary above shows that G^{III} does satisfy this condition and one can therefore expect of ωG_{.}^{III}A to be at least a reasonable approximation to the loop space ΩK(A). We will discuss this question in a future paper.

3. Elements of K_{1} via non-commutative diagrams

Double short exact sequences do not look very natural as categorical data. In this section we will consider non-commutative, i.e. not necessarily commutative diagrams of the form

d = \begin{matrix}A′ & ⟶ & B′ & ⟶ & C′ \\ ↓ & & ↓ & & ↓ \\ A & ⟶ & B & ⟶ & C \\ ↓ & & ↓ & & ↓ \\ A′′ & ⟶ & B′′ & ⟶ & C′′ \end{matrix}  (3.1)

consisting of six short exact sequences. We will write for short n.c.d. for such a diagram. These data seem to us a bit more natural than a d.s.e.s., and one can regard any d.s.e.s. l = (A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C) as a particular case of n.c.d. by displaying it

in the form

d(l) = \begin{matrix}A & ⟶^{f_{1}} & B & ⟶^{g_{1}} & C \\ ^{1}↓ & & ^{1}↓ & & ^{1}↓ \\ A & ⟶_{f_{2}} & B & ⟶_{g_{2}} & C \\ ↓ & & ↓ & & ↓ \\ 0 & ⟶ & 0 & ⟶ & 0 \end{matrix}

The same as one assigns a loop μ(l) in G.A to a d.s.e.s. l, we now assign a loop μ(d) in G^{III}.A to a n.c.d. d:

μ(d) = \xymatrix{•^{(A′,A,A′′)} \ar[rr]^{e(d)} & & •^{(B′,B,B′′)} \\ & •_{(0,0,0)} \ar[ul]^{e(s_{A})} \ar[ur]_{e(s_{B})}}

Here e(s_{A}) and e(s_{B}) are the edges associated to the short exact sequences s_{A} = (A′ ⟶ A ⟶ A′′) and s_{B} = (B′ ⟶ B ⟶ B′′) (cf. (1.4)), and e(d) is the edge in G.^{III}A obtained by forgetting the arrows of s_{A} and s_{B} in d (cf. (1.2)). For any d.s.e.s. l, we obviously have

i(μ(l)) = μ(d(l)).  (3.2)

Let m(d) denote the homotopy class of μ(d) in K_{1}(A) = π_{1}(G.^{III}A). We know from [Ne1] that every element of K_{1}(A) can be represented as m(l) for some d.s.e.s. l. It follows from (3.2) that every element of K_{1}(A) can be represented as m(d) for some n.c.d. d and, moreover, as m(d(l)) for some l. A natural question now would be: what kind of relations in K_{1}(A) do we have for the elements m(d) in terms of diagrams? Below we state some relations which are either trivial or conjectural.

(i)′ If d commutes, then m(d) = 0. For in this case the loop μ(d) is the contour of the 2-simplex t(d) (cf. (1.6)).

(ii)′ (skew-symmetry) Let d^{t} denote the transposed diagram. Then m(d^{t}) = −m(d) for any n.c.d. d.

This relation is still conjectural in general, but for the diagrams of the form d(l) it follows from Proposition 2.1. For, the loop μ(d(l)^{t}) is essentially the same as κ(l) (cf. (2.1)) since the edge e(d(l)^{t}) is degenerate, and we are done by (3.2).

(iii)′ (Jacobi identity) Suppose we are given a 3 × 3 × 3 cubic diagram which consists of 27 objects and 27 short exact sequences and is not supposed to be subject to any commutativity condition, i.e. like (3.1) but three-dimensional. Numerate the directions 1,2,3. For any permutation i, j, k of 1,2,3 consider the three “faces” of the diagram (including the one in the middle of the cube) orthogonal to the i-th direction. Each of them is a 3 × 3-diagram of the form (3.1). Let us think of the j-th direction as horizontal and of the k-th direction as vertical. Denote by [i, j, k] the alternative sum of the elements m(d) corresponding to these three diagrams, the element corresponding to the middle face being taken with minus. Then

[1, 2, 3] + [2, 3, 1] + [3, 1, 2] = 0  (3.3)

in K_{1}(A).

This relation is still a conjecture as well as (ii)′, and we will now try to justify both of them. Note first that (iii)′ implies (ii)′. For consider the cub

\xymatrix{A′ \ar[rrr] \ar@{-->}[dr] \ar[ddd] &  &  & A′ \ar[rrr] \ar@{-->}[dr] \ar[ddd] &  &  & 0 \ar[dr] \ar[ddd] \\  & A′ \ar[rrr] \ar@{-->}[dr] \ar[ddd] &  &  & B′ \ar[rrr] \ar@{-->}[dr] \ar[ddd] &  &  & C′ \ar@{-->}[dr] \ar[ddd] \\  &  & 0 \ar[rrr] \ar[ddd] &  &  & C′ \ar[rrr] \ar[ddd] &  &  & C′ \ar[ddd] \\ A \ar[rrr] \ar[dr] \ar[ddd] &  &  & A \ar[rrr] \ar[dr] \ar[ddd] &  &  & 0 \ar[dr] \ar[ddd] \\  & A \ar[rrr] \ar@{-->}[dr] \ar[ddd] &  &  & B \ar[rrr] \ar[dr] \ar[ddd] &  &  & C \ar[dr] \ar[ddd] \\  &  & 0 \ar[rrr] \ar[ddd] &  &  & C \ar[rrr] \ar[ddd] &  &  & C \ar[ddd] \\ A′′ \ar[rrr] \ar@{-->}[dr] &  &  & A′′ \ar[rrr] \ar@{-->}[dr] &  &  & 0 \ar[dr] \\  & A′′ \ar[rrr] \ar@{-->}[dr] &  &  & B′′ \ar[rrr] \ar@{-->}[dr] &  &  & C′′ \ar@{-->}[dr] \\  &  & 0 \ar[rrr] &  &  & C′′ \ar[rrr] &  &  & C′′}

Let us always numerate the directions in the following way:

1. the vertical direction;

2. the horizontal direction on the plane of the figure;

3. the direction “orthogonal” to the plane of the figure. The horizontal faces commute, hence [1, 2, 3] = 0, and we also have [2, 3, 1] = −m(d) and [3, 1, 2] = −m(d^{t}). It follows from (3.3) that m(d) + m(d^{t}) = 0.

Recall that we denote by D(A) the abelian group generated by the symbols ⟨l⟩ for all d.s.e.s’s l subject to relations (i) and (ii) of [Ne2, sect. 6]. Denote now E(A) the abelian group generated by the synbols ⟨d⟩ for all n.c.d’s d subject to relations (i)′ and (iii)′ posed on those symbols rather than on the elements m(d).

Lemma 3.1. There is a well-defined homomorphism f : D(A) ⟶ E(A) given on generators by f(⟨l⟩) = ⟨d(l)⟩.

Proof. If f_{1} = f_{2} and g_{1} = g_{2} in l (cf. (6.1) in [Ne2]), then d(l) commutes and ⟨d(l)⟩ = 0 in E(A) by virtue of (i)′. Thus f is well-defined with respect to relation (i).

Given a diagram

A′  ⇉_{a_{2}}^{a_{1}}  A  ⇉_{x_{2}}^{x_{1}}  A′′
f_{1} ⇊ f_{2}  g_{1} ⇊ g_{2}  h_{1} ⇊ h_{2}
B′  ⇉_{b_{2}}^{b_{1}}  B  ⇉_{y_{2}}^{y_{1}}  B′′
u_{1} ⇊ u_{2}  v_{1} ⇊ v_{2}  w_{1} ⇊ w_{2}
C′  ⇉_{c_{2}}^{c_{1}}  C  ⇉_{z_{2}}^{z_{1}}  C′′

in which the first arrows commute with the first ones and the second arrows commute with the second ones (cf. (5.1) and (6.2) in [Ne2]), consider the cube

\xymatrix{A′ \ar[rrr]^{a_{1}} \ar@{-->}[dr]^{f_{1}} \ar[ddd]_{1} &  &  & A \ar[rrr]^{x_{1}} \ar[dr]^{g_{1}} \ar[ddd]_{1} &  &  & A′′ \ar@{-->}[dr]^{h_{1}} \ar[ddd]_{1} \\  & B′ \ar[rrr]^{b_{1}} \ar@{-->}[dr]^{u_{1}} \ar[ddd]_{1} &  &  & B \ar[rrr]^{y_{1}} \ar[dr]^{v_{1}} \ar[ddd]_{1} &  &  & B′′ \ar@{-->}[dr]^{w_{1}} \ar[ddd]_{1} \\  &  & C′ \ar[rrr]^{c_{1}} \ar[ddd]_{1} &  &  & C \ar[rrr]^{z_{1}} \ar[ddd]_{1} &  &  & C′′ \ar[ddd]_{1} \\ A′ \ar[rrr]^{a_{2}} \ar@{-->}[dr]^{f_{2}} \ar[ddd] &  &  & A \ar[rrr]^{x_{2}} \ar[dr]^{g_{2}} \ar[ddd] &  &  & A′′ \ar@{-->}[dr]^{h_{2}} \ar[ddd] \\  & B′ \ar[rrr]^{b_{2}} \ar@{-->}[dr]^{u_{2}} \ar[ddd] &  &  & B \ar[rrr]^{y_{2}} \ar[dr]^{v_{2}} \ar[ddd] &  &  & B′′ \ar@{-->}[dr]^{w_{2}} \ar[ddd] \\  &  & C′ \ar[rrr]^{c_{2}} \ar[ddd] &  &  & C \ar[rrr]^{z_{2}} \ar[ddd] &  &  & C′′ \ar[ddd] \\ 0 \ar[rrr] \ar[dr] &  &  & 0 \ar[rrr] \ar[dr] &  &  & 0 \ar[dr] \\  & 0 \ar[rrr] \ar[dr] &  &  & 0 \ar[rrr] \ar[dr] &  &  & 0 \ar[dr] \\  &  & 0 \ar[rrr] &  &  & 0 \ar[rrr] &  &  & 0}

The horizontal faces commute, hence [1, 2, 3] = 0. We have

[2, 3, 1] = ⟨d(l′)⟩ − ⟨d(l)⟩ + ⟨d(l′′)⟩
[3, 1, 2] = ⟨d(l_{A})^{t}⟩ − ⟨d(l_{B})^{t}⟩ + ⟨d(l_{C})^{t}⟩
= −⟨d(l_{A})⟩ + ⟨d(l_{B})⟩ − ⟨d(l_{C})⟩

i.e., (iii)′ implies ⟨d(l′)⟩ − ⟨d(l)⟩ + ⟨d(l′′)⟩ = ⟨d(l_{A})⟩ − ⟨d(l_{B})⟩ + ⟨d(l_{C})⟩. Thus f is well-defined with respect to relation (ii)■

If the conjectural relations (ii)′ and (iii)′ are true for the elements m(d), then we have a well-defined map

E(A) ⟶ K_{1}(A), ⟨d⟩ ↦ m(d)

and, as a consequence, the map f : D(A) ⟶ E(A) is injective since the composite map D(A) ⟶ K_{1}(A) is an isomorphism [Ne3].

Conjecture. The map f : D(A) ⟶ E(A) is an isomorphism for any exact category A.

Remark. We do not know how to produce a non-trivial d.s.e.s. out of a n.c.d. in order to define an inverse map E(A) ⟶ D(A).

Acknowledgement. I would like to express my distinguished gratitude to the University of Nice (France) in which a part of the work was done. I am equally grateful to the Alexander von Humboldt Foundation for its support and to the University of Bielefeld (Germany) for its hospitality during the preparation of the paper. I thank Tony Bak for his interest to my work and stimulating discussions.

References

[GG] H. Gillet and D. Grayson, On the loop space of the Q-construction, Illinois J. Math. 31 (1987), no. 4, 574-597.

[G] D. Grayson, Long exact sequences in algebraic K-theory, Illinois J. Math. 31 (1987), no. 4, 598-617.

[Ne1] A. Nenashev, Double short exact sequences produce all elements of Quillen’s K_{1}, Algebraic K-Theory (Banaszak et al., Editors), Contemp. Math., AMS, vol. 199 (1996), pp. 151-160.

[Ne2] A. Nenashev, Double short exact sequences and K_{1} of an exact category, K-Theory 14 (1998), no. 1, 23-41.

[Ne3] A. Nenashev, K_{1} by generators and relations, J. Pure Appl. Algebra 131 (1998), no. 2, 195-212.