λ^{2} ON K_{1} IN TERMS OF DOUBLE SHORT EXACT SEQUENCES

Alexander Nenashev^{1}

Abstract. Given a commutative ring R and α ∈ GL(n, R), we have λ^{2}([α]) = [Λ^{2}α] − n[α] in K_{1}(R). We prove a generalization of this formula for K_{1} of an arbitrary exact category with operations.

1. Let M be an exact category with operations in the sense of [Gr2] and [Ne1]. In [Ne1], we define simplicial maps

(1.1)  Λ^{k} : Diag Sub_{k}G.M ⟶ G(k; M), k ≥ 1

which induce the λ-operations on the K-theory of M. The notation above has the following meaning:

G.M is a simplicial set, the so-called G-construction introduced by Gillet and Grayson in [GG];

G(k; M) is the ‘long G-construction’ which is a generalization of G.M introduced in [Gr1, Sect. 8] under the name of 0|SF_{k+1}; see also [Ne1] where the notation G(k; M) was introduced. For an explicit description of G(2; M) we refer to [Ne?] where this simplicial set is called ‘the triple G-construction’ and denoted by G.^{III}M.

Both Diag and Sub_{k} are a sort of subdivision functors, Diag Sub_{k}G.M is therefore a subdivision of the simplicial set G.M.

One can identify π_{m} of both simplicial sets in (1.1) with K_{m}(M), and the map Λ^{k} induces an operation λ^{k} : K_{m}(M) ⟶ K_{m}(M), m ≥ 0.

2. In this paper we will consider the case k = 2 and m = 1, and calculate the image of a generic element of K_{1}(M) under λ^{2}. Recall that every element of K_{1} comes from a double short exact sequence (d.s.e.s. for short) [Ne3], i.e., given x ∈ K_{1}(M), there exist data of the form

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

where both s_{i} = (A_{i} ⟶^{f_{i}} B_{i} ⟶^{g_{i}} C_{i}), i = 1, 2, are short exact sequences in M (s.e.s. for short), with the property:

Typeset by AMS-TEX

1This research has been possible thanks to a support of the Alexander von Humboldt Foundation (Germany)

If one associates to l the triangular loop μ(l) in G.M of the form

∗ ⟶^{e(A)} •^{(A,A)} ⟶^{e(l)} •^{(B,B)} ⟵^{e(B)} ∗

where e(A) = (0 ⟶ A ⟶^{1} A, 0 ⟶ A ⟶^{1} A) is the canonical edge from the base point to (A, A) (the same for B) and e(l) = (s_{1}, s_{2}) is the edge from (A, A) to (B, B) given by l, and denotes by m(l) the class of μ(l) in π_{1}G.M ≅ K_{1}(M), then x = m(l).

We now want to calculate the image of the loop μ(l) under the map Λ^{2}. First, note that an edge in G.M corresponds to (subdivides into) a pair of edges in Diag Sub_{2}G.M under the natural homeomorphism of their geometric realizations. Thus the loop Λ^{2}(μ(l)) consists of six edges. The two of them that are coincident to the base point of G(2; M) are in fact degenerate, hence Λ^{2}(μ(l)) looks like

(2.1)  ∗ ⟶ •^{Λ^{2}((A,A))} ⟶ • ⟶ •^{Λ^{2}((B,B))} ⟵ ∗

By the definition of Λ^{k} in [Ne1, Sect 4] applied to k = 2, we have

Λ^{2}((A, A)) = (A ∧ A, A ⊗ A, A ∘ A)
Λ^{2}((B, B)) = (B ∧ B, B ⊗ B, B ∘ B)

where A ∧ A = Λ^{2}A (resp. A ∘ A) denotes the exterior (resp. divided) square of A (the same for B), and the middle vertex in (2.1) is given by

(A ∧_{f_{1}} B, A ⊗ B, A ∘ A),

where A ∧_{f_{1}} B is the ‘exterior product’ of A and B with respect to the admissible mono f_{1} : A ↣ B (see [Gr2] or [Ne1] for definitions). The left side edge in (2.1) is given by

0 ⟶ A ∧ A ⟶^{1} A ∧ A
↑
0 ⟶ A ⊗ A ⟶^{1} A ⊗ A
↑
0 ⟶ A ∘ A ⟶^{1} A ∘ A

where the vertical s.e.s. is a Koszul complex, and the same, with A replaced by B, for the right side edge.

The middle edges are given by

A ∧ A ⟶^{1∧f_{1}} A ∧_{f_{1}} B ⟶^{(1)} A ⊗ C
↑^{1}
A ⊗ A ⟶^{1⊗f_{2}} A ⊗ B ⟶^{1⊗g_{2}} A ⊗ C
↑
A ∘ A ⟶^{1} A ∘ A ⟶ 0

and

A ∧_{f_{1}} B ⟶^{f_{1}∧1} B ∧ B ⟶^{g_{1}∧g_{1}} C ∧ C
↑^{(2)}
A ⊗ B ⟶^{f_{1}⊗1} B ⊗ B ⟶^{g_{1}⊗1} C ⊗ B
↑^{(2)}
A ∘ A ⟶^{f_{2}∘f_{2}} B ∘ B ⟶^{(2)} X_{(2)}

Here and in the sequel, (i) over an arrow means that this arrow is induced by the maps f_{i} and g_{i} in the s.e.s. s_{i}. We do not have an appropriate notation for the term X_{(i)} which arises in fact while splitting the four-term exact sequence

0 ⟶ A ∘ A ⟶^{(i)} B ∘ B ⟶^{(i)} C ⊗ B ⟶^{(i)} C ∧ C ⟶ 0

into two short exact sequences.

3. Recall that for a commutative ring R and α ∈ GL(n, R), we have in K_{1}(R) [Su]

(3.1)  λ^{2}([α]) = [Λ^{2}α] − n[α].

If one thinks of l as an analogue of α and of m(l) as an analogue of [α] in K_{1}(R), then one faces the first problem at once: there is no reasonable definition of exterior powers on the class of double short exact sequences, i.e., given k ≥ 2, one cannot define Λ^{k}l as a d.s.e.s. or a collection of d.s.e.s’s with coefficients. What one can do is to consider the Grothendieck filtration (cf. [Gr2])

(3.2) Λ^{k}(s_{i}) = (Λ^{k}A ↣ A ∧ ⋯ ∧ A ∧_{f_{i}} B ↣ ⋯ ↣ A ∧_{f_{i}} B ∧ ⋯ ∧ B ↣ Λ^{k}B)

associated to each of the two s.e.s’s s_{i} = (A ⟶^{f_{i}} B ⟶^{g_{i}} C), i = 1, 2, which form l. All the terms in (3.2) except the side ones depend in general on i. On the contrary, the successive quotients have the form Λ^{r}A ⊗ Λ^{s}C, with r + s = k, and do not depend on i. The latter observation allows us to assign to this pair of filtrations a loop in G.M of the form

(3.3)  ∗ ⟶^{e(Λ^{k}A)} • ⟶^{e_{1}} • ⟶^{e_{2}} • ⋯ • ⟶^{e_{k}} • ⟵^{e(Λ^{k}B)} ∗

where

e_{r} = ( \underbrace{A ∧ ⋯ ∧ A}_{k−r+1} ∧_{f_{1}} \underbrace{B ∧ ⋯ ∧ B}_{r−1} ↣^{(1)} \underbrace{A ∧ ⋯ ∧ A}_{k−r} ∧_{f_{1}} \underbrace{B ∧ ⋯ ∧ B}_{r} ↠^{(1)} Λ^{k−r}A ⊗ Λ^{r}C,

\underbrace{A ∧ ⋯ ∧ A}_{k−r+1} ∧_{f_{2}} \underbrace{B ∧ ⋯ ∧ B}_{r−1} ↣^{(2)} \underbrace{A ∧ ⋯ ∧ A}_{k−r} ∧_{f_{2}} \underbrace{B ∧ ⋯ ∧ B}_{r} ↠^{(2)} Λ^{k−r}A ⊗ Λ^{r}C).

We call the pair of filtrations Λ^{k}(s_{1}), Λ^{k}(s_{2}), together with the successive quotients and projection maps, the k-th exterior power of the d.s.e.s. l and denote these data by Λ^{k}l. The associated loop (3.3) will be denoted by μ(Λ^{k}l). This suggests the form of the first term in the desired generalization of (3.1): [μ(Λ^{2}l)] instead of [Λ^{2}α].

As for the second term in (3.1), we can regard α as an automorphism of R^{n} and rewrite n[α] as [R^{n}][α], where [R^{n}] is the class of R^{n} in K_{0}. Replacing the automorphisms by d.s.e.s’s, the right substitute for [R^{n}] will be the middle term of a d.s.e.s., and we can finally formulate the answer.

Proposition. For any d.s.e.s. l = (A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C), we have in K_{1}(M)

λ^{2}(m(l)) = [μ(Λ^{2}l)] − [B] · m(l)

4. First we describe a bi-simplicial set in which we will carry out our arguments. Recall that given an exact functor F : A ⟶ B, one can consider the combinatorial homotopy fiber of the map S.A ⟶ S.B which is known from [Gr1] under the notation 0|SF and which we will denote by C.[A ⟶ B] (we can also refer to [Ne5]). Both G.M and G.(2; M) are examples of this C.-construction: G.M = C.[M ⟶^{diag} M × M] and G.(2; M) = C.[SES(M) ⟶ M × M × M].

One can also consider the dual C.-construction defined by C^{op}.[A ⟶ B] = C.[A^{op} ⟶^{F^{op}} B^{op}]. An n-simplex in it is a pair of diagrams

B_{n−1\n}  A_{n−1\n}
↓  ↓
…  …
B_{1\2} ⟵ … ⟵ B_{1\n}  A_{1\2} ⟵ … ⟵ A_{1\n}
↓  ↓  ↓  ↓
B_{0\1} ⟵ B_{0\2} ⟵ … ⟵ B_{0\n}  A_{1} ⟵ A_{2} ⟵ … ⟵ A_{n}
↓  ↓  ↓
B_{0} ⟵ B_{1} ⟵ B_{2} ⟵ … ⟵ B_{n}

subject to the conditions:

(i) all the horizontal (resp. vertical) arrows are admissible epis (resp. monos); (ii) all the small squares commute; (iii) all the sequences of the form B_{j\k} ⟶ B_{k} ⟶ B_{j} and B_{j\k} ⟶ B_{i\k} ⟶ B_{i\j}

for i ≤ j ≤ k are s.e.s’s in B; the same for the A-diagram;

(iv) the image of the A-diagram under F coincides with the top part of the B-diagram.

Given a commutative square of exact categories

L = [\begin{matrix}B & ⟵ & A \\ ↓ & & ↓ \\ D & ⟵ & C \end{matrix}] ,

consider the mixed C./C^{op}.-construction C_{..}^{m}L associated to L. This is a bisimplicial set which is essentially the C.-construction with respect to the horizontal functors in L and the C^{op}.-construction with respect to the vertical functors in L (Warning: But it is not the composite of C. and C^{op}.). We leave it to the reader to give a formal definition and mention only how the low-dimensional bi-simplices in C_{..}^{m}L look like.

a) A (0, 0)-simplex is an object of D.

b) A (1, 0)-simplex (a horizontal edge) is an edge in C.[C ⟶ D], i.e. a s.e.s. D_{0} ⟶ D_{1} ⟶ D_{1/0} in D and an object C_{1/0} in C whose image equals to D_{1/0}.

c) A (0, 1)-simplex (a vertical edge) is an edge in C^{op}.[B ⟶ D], i.e. a s.e.s. D_{0} ⟵ D_{1} ⟵ D_{0\1} in D and an object B_{0\1} in B whose image equals to D_{0\1}.

d) A (1, 1)-simplex is the following data: d-1) a commutative diagram in D of the form

D_{0,0\1} ⟶ D_{1,0\1} ⟶ D_{1/0,0\1}
↓  ↓  ↓
D_{0,1} ⟶ D_{1,1} ⟶ D_{1/0,1}
↓  ↓  ↓
D_{0,0} ⟶ D_{1,0} ⟶ D_{1/0,0}

which consists of six s.e.s’s; d-2) a s.e.s. C_{1/0,0\1} ⟶ C_{1/0,1} ⟶ C_{1/0,0} in C whose image coincides with with the right vertical s.e.s. in the diagram; d-3) a s.e.s. B_{0,0\1} ⟶ B_{1,0\1} ⟶ B_{1/0,0\1} in B whose image coincides with with the upper horizontal s.e.s. in the diagram; d-4) an object A_{1/0,0\1} in A whose images under A ⟶ B and A ⟶ C coincide with B_{1/0,0\1} and C_{1/0,0\1} respectively.

We put

W..(M) = C_{..}^{m} [\begin{matrix}SES(M) & ⟵ & SES(M) \\ ↓ & & ↓ \\ M × M × M & ⟵ & SES(M) \end{matrix}] .

Note that one can identify W_{·0}(M) with C.[SES(M) ⟶ M × M × M] = G.(2; M). Applying the technique of [Ne2] or calculating a spectral sequence as in [GG, p. 596], one proves that the embedding G.(2; M) ↪ W..(M) is a homotopy equivalence.

Proof of the proposition. In Figure 1 of Appendix we have:

a) The inner four-edge loop is the image of μ(Λ^{2}l) in G.(2; M) under the embedding i : G.M ↪ G.(2; M);

b) The outer loop is Λ^{2}(μ(l)) (cf. Sect. 2);

c) I, II, S_{A}, and S_{B} are (1, 1)-simplices in W..(M) given by diagrams in Appendix.

This shows that μ(Λ^{2}l) is homotopic to a loop which which differs from Λ^{2}(μ(l)) in one edge; namely, we replace f_{1} and g_{1} by f_{2} and g_{2} in the middle horizontal s.e.s. in the diagram

A ∧_{f_{1}} B ⟶^{f_{1}∧1} B ∧ B ⟶^{g_{1}∧g_{1}} C ∧ C
↑^{(2)}
A ⊗ B ⟶^{f_{1}⊗1} B ⊗ B ⟶^{g_{1}⊗1} C ⊗ B
↑^{(2)}
A ∘ A ⟶^{f_{1}∘f_{1}} B ∘ B ⟶^{(2)} X_{(2)}

which defines the right horizontal edge of Λ^{2}(μ(l)). Denote this edge by e_{(1)} and the other one (with g_{2} and f_{2}) by e_{(2)}. Then the difference λ^{2}(m(l)) − [μ(Λ^{2}l)] is equal to the homotopy class of the loop τ formed by e_{(1)} and e_{(2)} (clockwise on Figure 1).

Lemma. τ is freely homotopic to the loop μ(l ⊗ B)^{−1}.

Proof. For i = 1, 2, consider the non-commutative diagram

d_{i} = (\begin{matrix}A ∧_{f_{1}} B & ⟶^{f_{1}∧1} & B ∧ B & ⟶^{g_{1}∧g_{1}} & C ∧ C \\ ^{(1)}↑ & & ↑ & & ↑^{(2)} \\ A ⊗ B & ⟶^{(i)} & B ⊗ B & ⟶^{(i)} & C ⊗ B \\ ^{(1)}↑ & & ↑ & & ↑^{(2)} \\ A ∘ A & ⟶^{(2)} & B ∘ B & ⟶^{(2)} & X_{(2)} \end{matrix})

Then τ is homotopic to μ(d_{1})μ(d_{2})^{−1} in the notation of [Ne4]. By virtue of the conjectural skew-symmetry relation for the symbol m(d) [Ne4, p. 9], this is homotopic to μ(d_{1}^{t})^{−1}μ(d_{2}^{t}), which is homotopic to ν(l^{t} ⊗ B). The latter is homotopic to i(μ(l^{t}⊗B)) ∼ i(μ(l⊗B))^{−1}, which completes the proof of Lemma and Proposition.■

References

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[Gr2] D. Grayson, Exterior power operations on higher K-theory, K-Theory 3 (1989), no. 3, 247-260.

[Ne1] A. Nenashev, Simplicial definition of λ-operations in higher K-theory, Adv. Soviet Math., vol. 4, AMS, Providence, RI, 1991, pp. 9-20.

[Ne2] A. Nenashev, Comparison theorem for lambda operations in higher algebraic K-theory, Astérisque 226 (1994), 335-369.

[Ne3] 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.

[Ne4] A. Nenashev, Elements of K_{1} via the triple G-construction, Preprint 98-098, Universität Bielefeld (Germany).

[Ne5] A. Nenashev, On the combinatorial loop space of the triple G-construction, Preprint, Bielefeld (1999).

[Su] A. A. Suslin, K-theory and K-cohomology of certain group varieties, Adv. Soviet Math., vol. 4, AMS, Providence, RI, 1991.