ON THE K_{−i}-FUNCTORS

ERIK KJÆR PEDERSEN

0. Statement of results

The functors K_{0} and K_{1} have well-known descriptions in terms of projectives and automorphisms, respectively. The purpose of this paper is to give analogous descriptions of the K_{−i}-functors of Bass [1]. In fact given a ring R, we give two descriptions of K_{−i}(R), i ≥ 0; one as a Whitehead construction on a certain category of R-modules (elements represented by automorphisms), and one as a Grothendieck construction on a related category of Rmodules (elements represented by objects). The categories in question are associated with the category of ℤ^{i}-graded R-modules and bounded homomorphisms in the following sense:

0.1. Definition. Let R be a ring. A_{i}(R) denotes the category of ℤ^{i}-graded R-modules and bounded homomorphisms. This means an object A is a direct sum ⨁_{j_{1},...,j_{i}} A(j_{1}, …, j_{i}) of R-modules, and a morphism f : A → B is an R-module morphism, such that there exists k = k(f) satisfying

f(A(j_{1}, …, j_{i})) ⊆ ⨁_{\begin{matrix}h_{s}=−k \\ s=1,...,i \end{matrix}}^{k}(B(j_{1} + h_{1}, …, j_{i} + h_{i}).

Remark. A_{0}(R) is just the category of R-modules.

We shall be more interested in the full subcategory of A_{i}(R) with objects A satisfying A(j_{1}, …, j_{i}) are finitely generated free R-modules. This category we denote C_{i}(R). We shall some times write A_{i} and C_{i} instead of A_{i}(R) and C_{i}(R) when it is clear from the context which ring we are working with. Since C_{0}(R) is the category of finitely generated free Rmodules, it is helpful to think of C_{i}(R) as the category of ℤ^{i}-graded finitely generated free R-modules and bounded homomorphisms.

In A_{i} as well as C_{i} we have an obvious notion of direct sum (degree-wise). We define a sequence 0 → A → B → C to be exact if it is split-exact, i. e. , it is equivalent (in the category) to the sequence 0 → A → A⊕B → B → 0. We may now define K_{1} of the category C_{i+1}(R). That is the Abelian group generated by [A, α] where A is an object of C_{i+1}(R) and α an automorphism of A, subject to the relations

[A, αβ] = [A, α] + [A, β] and [B, β] = [A, α] + [C, γ]

when there is a commutative diagram

\xymatrix{0 \ar[r] & A \ar[d]^{α} \ar[r] & B \ar[d]^{β} \ar[r] & C \ar[d]^{γ} \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] & B \ar[r] & C \ar[r] & 0}

with exact rows. Note that this implies that [A, 1] = 0 and further that

A ⊕ B \underrightarrow{\{ \begin{matrix}1 & η \\ 0 & 1 \end{matrix} \}} A ⊕ B,

where η : B → A any bounded morphism, represents 0. An isomorphism of this type we shall call an elementary isomorphism.

0.2. Definition. K′_{−i}(R) = K_{1}(C_{i+1}(R)).

Given a category 𝔄 we define the category P 𝔄 as follows: An object is an idempotent in 𝔄, i. e. p : A → A with p^{2} = p, and a morphism ϕ : (A_{1}, p_{1}) → (A_{2}, p_{2}) is a morphism ϕ : A_{1} → A_{2} so that ϕp_{1} = p_{2}ϕ. In P C_{i}(R) we have an induced notion of direct sum, so we may form the Grothendieck group of P C_{i}(R).

0.3. Definition. We define K′′_{−i}(R) to be the Grothendieck group on the category P C_{i}(R) with the additional relation [A, 0] = 0 if i = 0.

We may now state our

Main Theorem. Let R be a ring. Then there are natural isomorphisms

K_{−i}(R) ≅ K′_{−i}(R) ≅ K′′_{−i}(R).

This result indicates that C_{i+1}(R) is some kind of nonconnective delooping of the category of finitely generated free R-modules. This is indeed the case, and it is the subject of a forthcoming joint work with C. Weibel.

It is a pleasure to acknowledge useful conversations with Hans J. Munkholm and Douglas R. Anderson in connection with this work.

1. The Isomorphism K′_{−i}(R) ≃ K′′_{−i}(R)

In this section we define isomorphism ϕ^{s} : K′_{−i}(R) ≅ K′′_{−i}(R), where s = 1, …, i + 1. The construction we employ to define ϕ^{s} is based on a variation of a well-known construction due to Bass, Heller and Swan[2]. First we need some definitions. Let A be an object of C_{i}(R). We define

s^{+}(A)(j_{1}, …, j_{s}, …, j_{i}) = A(j_{1}, …, j_{s} − 1, … j_{i})  (1.1)

and

s^{−}(A)(j_{1}, …, j_{s}, …, j_{i}) = A(j_{1}, …, j_{s} + 1, … j_{i})  (1.2)

for 1 ≤ s ≤ i. There are obvious bounded isomorphism A ≅ s^{+}(A) and A ≅ s^{−}(A) induced by the identity, which we denote by s^{+} and s^{−} respectively. Also define

A^{s^{+}}(j_{1}, …, j_{i}) = \begin{cases}A(j_{1}, …, j_{i}) & if j_{s} ≥ 0 \\ 0 & if j_{s} < 0, \end{cases}  (1.3)

A^{s^{−}}(j_{1}, …, j_{i}) = \begin{cases}0 & if j_{s} ≥ 0 \\ A(j_{1}, …, j_{i}) & if j_{s} < 0, \end{cases}  (1.4)

for 1 ≤ s ≤ i. The following lemma was pointed out by the referee.

1.5. Lemma. Let A be an object in C_{i}(R). Then [A, 1] and [A, 0] represent 0 in K′′_{−i}(R) if i > 0.

Proof. Clearly [A, 1] = [A^{1^{+}}, 1] + [A^{1^{−}}, 1] but

A^{1^{+}} ⊕ ⨁_{k=1}^{∞}(1^{+})^{k}(A^{1^{+}}) = ⨁_{k=0}^{∞}(1^{+})^{k}(A^{1^{+}})

and 1^{+} is an isomorphism

⨁_{k=0}^{∞}(1^{+})^{k}(A^{1^{+}}) ≅ ⨁_{k=1}^{∞}(1^{+})^{k}(A^{1^{+}})

Note that even though the sum is infinite, it is finite in each degree since A^{1^{+}}(j_{1}, …, j_{i}) = 0 if j_{1} < 0. This proves [A^{1^{+}}, 1] = 0 in K′′_{−i}(R) and [A^{1−}, 1] is dealt with similarly. Clearly [A, 0] can be treated the same way. □

1.6. Remark. Lemma 1.5 proves that any two objects of C_{i}(R), i > 0, are stably isomorphic.

We now proceed to define the isomorphism ϕ^{s} : K′_{−i}(R) ≅ K′′_{−i}(R).

Given an object A of C_{i+1}(R), we have a direct sum decomposition A = A^{s^{−}} ⊕ A^{s^{+}} (by 1.3 and 1.4 above). We denote the projection on the first factor by

p_{−}^{s} : A → A  (1.7)

(the projection on the negative s-half space). Given an automorphism α : A → A in C_{i+1}(R), consider αp_{−}^{s}α^{−1}. Assuming α is bounded by k, this is a projection of A which is the identity on A(j_{1}, …, j_{s}, …, j_{i+1}) if j_{s} < −2k, and the 0-map if j_{s} > 2k. We define ϕ^{s}[A, α] by

\begin{matrix}\overline{A}(j_{1}, …, ĵ_{s}, …, j_{i+1}) = ⨁_{j_{s}=−2k}^{2k} A(j_{1}, …, j_{i+1}), \\ ϕ^{s}([A, α]) = [\overline{A}, αp_{−}^{s}α^{−1}] − [\overline{A}, p_{−}^{s}]. \end{matrix}  (1.8)

Several comments are in order here. The term [\overline{A}, p_{−}^{s}] represents 0 unless i = 0. A nice way to think of ϕ^{s}([A, α]) is to keep the ℤ^{i+1}-grading and notice we are looking at αp_{−}^{s}α^{−1} in a certain band around j_{s} = 0, the width of the band at least from −2k to 2k. Outside this band αp_{−}^{s}α^{−1} is equal to p_{−}^{s}; hence, when we subtract the restrictions of αp_{−}^{s}α^{−1} and p_{−}^{s}, the width of the band does not matter. Actually it is useful to notice that widening the band corresponds to stabilization.

1.9. Theorem. ϕ^{s} defines an isomorphism K′_{−i}(R) → K′′_{−i}(R)

To prove this theorem we first need to see that ϕ^{s} respects the relations in the definition of K′_{−i}(R).

1.10. Lemma. Let A and B be objects of C_{i+1}(R) and ψ : A ⊕ B → A ⊕ B a bounded projection satisfying

ψ|(A ⊕ B)(j_{1}, …, j_{i+1}) = \begin{cases}0 & if j_{s} > k \\ 1 & if j_{s} < −k \end{cases}

for some k. Let γ : A ⊕ B → A ⊕ B be an elementary isomorphism with matrix

γ = (\begin{matrix}1 & η \\ 0 & 1 \end{matrix}) , η : B → A

Then ψ and γψγ^{−1} restricted to a sufficiently big band around j_{s} = 0 represent the same element of K′′_{−i}(R).

Proof. Assume η is bounded by l > k. Define B′ and B′′ by

B′(j_{1}, …, j_{i+1}) = \begin{cases}B(j_{1}, …, j_{i+1} & if |j_{s}| ≤ 2l \\ 0 & if |j_{s}| > 2l \end{cases}

and B = B′ ⊕ B′′. Also define η′, η′′ : B → A as the composites B → B′ ⊕ 0 → B ⟶^{η} A and

B → 0 ⊕ B′′ → B ⟶^{η} A. Letting

γ′ = (\begin{matrix}1 & η′ \\ 0 & 1 \end{matrix}) and γ′′ = (\begin{matrix}1 & η′′ \\ 0 & 1 \end{matrix})

it is clear that γ = γ′ · γ′′ = γ′′ · γ′. But

γψγ^{−1} = γ′γ′′ψ(γ′′)^{−1}(γ′)^{−1} = γ′ψ(γ′)^{−1}.

This follows since ψ is only nontrivial in a small band around j_{s} = 0 and γ′′ is the identity in a bigger band around j_{s} = 0. But γ′ψ(γ′)^{−1} and ψ are equivalent projections in the band |j_{s}| ≤ 3l since γ′ restricts to an isomorphism of that band, □

This immediately leads to

1.11. Lemma. The construction 1.8 gives a well-defined homomorphism

ϕ^{s} : K′_{−i}(R) → K′′_{−i}(R).

Proof. If we have a diagram in C_{i+1}(R)

\xymatrix{0 \ar[r] & A \ar[d]^{α} \ar[r] & A ⊕ B \ar[d]^{γ} \ar[r] & B \ar[d]^{β} \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] & A ⊕ B \ar[r] & B \ar[r] & 0}

one easily sees that γ · (α^{−1} ⊕ β^{−1}) is an elementary isomorphism. Since ϕ^{s} commutes with direct sum, Lemma 1.10 shows

ϕ^{s}([A ⊕ B, γ]) = ϕ^{s}([A, α]) + ϕ^{s}([B, β]).  (1.12)

It follows directly from the definition that

ϕ^{s}([A, 1]) = 0  (1.13)

and now the standard identity

\begin{matrix}(\begin{matrix}β^{−1} & 0 \\ 0 & β \end{matrix}) = & (\begin{matrix}1 & β^{−1} \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ −β & 1 \end{matrix}) (\begin{matrix}1 & β^{−1} \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix}) \\ (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) & (\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix}) \end{matrix}  (1.14)

and repeated application of Lemma 1.10 shows that if α and β are isomorphisms of A in C_{i+1}(R), then

ϕ^{s}([A, αβ]) = ϕ^{s}([A ⊕ A, αβ ⊕ 1]) = ϕ^{s}([A ⊕ A, α ⊕ β])
= ϕ^{s}([A, α]) + ϕ^{s}([A, β]).
□

Here we used (1.13) in the first equality. It is easy to see

1.15. Lemma. The map ϕ^{s} is onto.

Proof. Let B be an object of C_{i}(R) and p : B → B a projection. Define A(j_{1}, …, j_{s}, …, j_{i+1}) = B(j_{1}, …, j_{s−1}, j_{s+1}, …, j_{i+1}) and α : A → A by

[figure: j_{s}; =; …; −2; −1; 0; 1; 2; …; A; =; α; …; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; …; A; B; B; B; B; B]  (1.16)

This is a standard construction of Swan [5]. We compute αp_{−}^{s}α^{−1} to be

[figure: j_{s}; =; …; −2; −1; 0; 1; 2; …; A; αp_{−}^{s}α^{−1}; =; …; B; 1; B; 1; B; p; B; 0; B; 0; …; A; B; B; B; B; B]

so ϕ^{s}([α]) = ([B, p]). □

The most difficult part of Theorem 1.9 is to prove ϕ^{s} is 1-1.

For this we need some condition to ensure elements represent 0 in K′_{−i}(R).

1.17. Definition. Let α be an automorphism of A in C_{i+1}(R). We say α is split at s-degree m if the following holds:

j_{s} ≥ m implies α(A(j_{1}, …, j_{s}, …, j_{i+1})) ⊂ ⨁_{\begin{matrix}k_{l};l≠s \\ k_{s}≥m \end{matrix}} A(k_{1}, …, k_{s}, …, k_{i+1})

and

j_{s} < m implies α(A(j_{1}, …, j_{s}, …, j_{i+1})) ⊂ ⨁_{\begin{matrix}k_{l};l≠s \\ k_{s}<m \end{matrix}} A(k_{1}, …, k_{s}, …, k_{i+1})

Heuristically the point of the definition is that α preserves the two halves of A given by j_{s} ≥ m and j_{s} < m, respectively.

1.18. Lemma. Let α be an automorphism of A in C_{i+1}(R), which is split at s-degree m. Then [A, α] represents the trivial element of K′_{−i}(R).

Proof. Define A′ and A′′ in C_{i+1}(R) by

A′(j_{1}, …, j_{s}, …, j_{i+1}) = \begin{cases}A(j_{1}, …, j_{s}, …, j_{i+1}), & j_{s} ≥ m \\ 0, & j_{s} < m \end{cases}

and A = A′ ⊕ A′′.

α restricts to isomorphisms α′ : A′ → A′ and α′′ : A′′ → A′′ and (A′ ⊕A′′, α′ ⊕α′′) = (A, α) so

[A, α] = [A′, α′] + [A′′, α′′].

We show these two terms are 0. Consider [A′, α′]. We define

B′ = ⨁_{l=0}^{∞}(s^{+})^{l}(A′) and B′′ = ⨁_{l=0}^{∞}(s^{+})^{2l}(A′)

and note that 1 ⊕ s^{+} is an isomorphism

1 ⊕ s^{+} : B′′ ⊕ B′′ ≅ B′  (1.19)

in C_{i+1}(R). As usual these infinite sums are finite in each degree, so they do make sense. Conjugating α′ : A′ → A′ by (s^{+})^{l} gives an automorphism (s^{+})^{l}(A′) → (s^{+})^{l}A′ which we will denote by (α′)_{l}. We now use a trick due to Farrell and Wagoner, really just the so-called “Eilenberg swindle”:

[A′, α′] = [B′, α′ ⊕ 1]

by stabilization and

α′ ⊕ 1 = (α′ ⊕ 1 ⊕ 1 ⊕ 1 ⊕ …)
= (α′ ⊕ (α′)_{1}^{−1} ⊕ (α′)_{2} ⊕ (α′)_{3}^{−1} ⊕ …)(1 ⊕ (α′)_{1} ⊕ (α′)_{2}^{−1} ⊕ …),

so we shall show these two automorphisms represent 0. But conjugating (α′ ⊕ (α′)_{1}^{−1} ⊕ ⋯) by the isomorphism 1.19 gives (B′′ ⊕ B′′, β ⊕ β^{−1}) where β = (α′ ⊕ (α′)_{2} ⊕ (α′)_{4} ⊕ ⋯).

We finish off using 1.14 and the fact that elementary automorphisms represent 0 in K′_{−i}(R). The other factor is dealt with similarly. □

Next we investigate what it means for some elements to be 0 in K′′_{−i}(R).

1.20. Lemma. Let A be an object of C_{i}(R) and p_{1},p_{2} projections on A. Then [A, p_{1}]−[A, p_{2}] = 0 ∈ K′′_{−i}(R) if and only if there are objects A′ and A′′ in C_{i}(R) and an automorphism ϕ of A ⊕ A′ ⊕ A′′ so that (p_{2} ⊕ 1 ⊕ 0) · ϕ = ϕ(p_{1} ⊕ 1 ⊕ 0).

Proof. The if part is trivial, so assume [A, p_{1}] = [A, p_{2}]. In case i > 0 we conclude there is a projection q : A′ → A′ of some object in C_{i}(R) so that (A ⊕ A′, p_{1} ⊕ q) is isomorphic to (A⊕A′, p_{2} ⊕q). But then (A⊕A′ ⊕A′, p_{1} ⊕q ⊕(1−q) is isomorphic to (A⊕A′ ⊕A′, p_{2}, ⊕q ⊕ (1 − q)). Conjugating (A′ ⊕ A′, q ⊕ (1 − q)) by \{ \begin{matrix}q & 1−q \\ 1−q & q \end{matrix} \} gives (A′ ⊕ A′, 1 ⊕ 0) so we obtain the desired result by letting A′′ = A′. In case i = 0 we only conclude (A ⊕ A′ ⊕ B′, p_{1} ⊕ q ⊕ 0) is isomorphic to (A ⊕ A′ ⊕ B′′, p_{2} ⊕ q ⊕ 0) since in this case we divide out by terms of the form (B, 0). But then B′ and B′′ are stably isomorphic and we are reduced to considerations as above. □

The proof of Theorem 1.9 is completed by

1.21. Lemma. The map ϕ^{s} is monic.

Proof. Assume ϕ^{s}([A, α]) = 0. In the terminology of 1.8 we have [\overline{A}, αp_{−}^{s}α^{−1}] − [\overline{A}, p_{−}^{s}] = 0 in K′′_{−i}(R). Thus we may use Lemma 1.20 to determine A′ and A′′ in C_{i}(R) so that (\overline{A} ⊕ A′ ⊕ A′′, αp_{−}^{s}α^{−1} ⊕ 1 ⊕ 0) is isomorphic to (\overline{A} ⊕ A′ ⊕ A′′, p_{−}^{s} ⊕ 1 ⊕ 0). However (\overline{A} ⊕ A′ ⊕ A′′, αp_{−}^{s}α^{−1} ⊕ 1 ⊕ 0) = (\overline{A} ⊕ A′ ⊕ A′′, (α ⊕ 1 ⊕ 1)(p_{−}^{s} ⊕ 1 ⊕ 0)(α ⊕ 1 ⊕ 1)^{−1}). Since we can replace (A, α) by (A ⊕ B, α ⊕ 1) where

B(j_{1}, …, j_{s}, …, j_{i+1}) = \begin{cases}A′(j_{1}, …, ĵ_{s}, …, j_{i+1}) & j_{s} = −1 \\ A′′(j_{1}, …, ĵ_{s}, …, j_{i+1}) & j_{s} = 0 \\ 0, & otherwise, \end{cases}

we may thus assume there is an isomorphism β : \overline{A} → \overline{A} so that βαp_{−}^{s}α^{−1} = p_{−}^{s}β. Extending β to all A by the identity, we get on all A that βαp_{−}^{s} = p_{−}^{s}βα. This means that βα is split at s-degree 0 so [A, βα] = 0 by Lemma 1.18. However β is the identity outside \overline{A}, so β is split at s-degree 2k + 1 where k is the bound for α, hence [A, β] = 0 and thus [A, α] = 0. □

This ends the proof of Theorem 1.9

2. The Bass-Heller-Swan homomorphisms

In this section we define homomorphisms

λ_{t}^{s} : K′_{−i}(R) → K′_{−i}(R[t, t^{−1}]), s = 1, 2, …, i + 1,

which will eventually be the Bass-Heller-Swan homomorphisms.

Let [A, α] represent an element of K′_{−i}(R). Consider the automorphism p_{t}^{s} : A[t, t^{−1}] → A[t, t^{−1}] given by

p_{t}^{s} = tp_{−}^{s} + (1 − p_{−}^{s})  (2.1)

(with inverse t^{−1}p_{−}^{+} + (1 − p_{−}^{s})). Consider the commutator between α (extended to a map of A[t, t^{−1}]) and p_{t}^{s}, [α, p_{t}^{s}]. Since α is bounded and commutes with multiplication by t, this is the identity on A(j_{1}, …, j_{i+1}) away from a band −k ≤ j_{s} ≤ k, where k is a bound for α. By restriction as in 1.8 we obtain that [α, p_{t}^{s}] is an i-graded bounded R[t, t^{−1}] automorphism of \overline{A}[t, t^{−1}] and we define

λ_{t}^{s}([A, α]) = [\overline{A}[t, t^{−1}], [α, p_{t}^{s}]].  (2.2)

2.3. Theorem. λ_{t}^{s} is a well-defined homomorphism of K′_{−i}(R) → K′_{−i+1}(R[t, t^{−1}]).

To prove λ_{t}^{s} respects the relations, we need to consider the following situation: Let A be an object of C_{i+2}(R) and γ an isomorphism of A which is 1 except for some band around j_{s} = 0. Then γ may be thought of as an isomorphism of a ℤ^{i+1}-graded object by restriction and thus defines an element of K′_{−i}(R). If β is a bounded isomorphism of A, then βγβ^{−1} is also the identity outside a sufficiently big band around j_{s} = 0 and we have

2.4. Lemma. [γ] = [βγβ^{−1}] ∈ K′_{−i}(R).

Proof. We stabilize βγβ^{−1} to βγβ^{−1} ⊕ 1 on A ⊕ A and note that βγβ^{−1} ⊕ 1 = (β ⊕ β^{−1})(γ ⊕ 1)(β^{−1} ⊕β). Now we proceed exactly as in Lemma 1.10 using 1.14 to complete the proof. □

Proof of Theorem 2.3. It is clear that λ_{t}^{s} sends exact sequences a to exact sequences. Now consider α and β, two automorphisms of an object A ∈ C_{i+1}(R).

λ_{t}^{s}[βα] = [βα, p_{t}^{s}] = βαp_{t}^{s}α^{−1}β^{−1}(p_{t}^{s})^{−1}
= β[α, p_{t}^{s}]β^{−1} · [β, p_{t}^{s}]

and, by Lemma 2.4, β[α, p_{t}^{s}]β^{−1} represents the same element as [α, p_{t}^{s}]. □

2.5. Theorem. If s_{1} < s_{2}, then the diagram

\xymatrix{ & K′_{−i+1}(R[t_{1}, t_{1}^{−1}]) \ar@{-->}[dr]^{λ_{t_{2}}^{s_{2}−1}} \\ K′_{−i}(R) \ar@{-->}[ur]^{λ_{t_{1}}^{s_{1}}} \ar[dr]_{λ_{t_{2}}^{s_{2}}} &  & K′_{−i+2}(R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}]) \\  & K′_{−i+1}(R[t_{2}, t_{2}^{−1}])) \ar@{-->}[ur]_{λ_{t_{1}}^{s_{1}}}}

is commutative.

Proof. Let p_{i} represent p_{t_{i}} for i = 1, 2. Given [A, α] ∈ K′_{−i}(R) we have to compare

[[α, p_{1}], p_{2}] and [[α, p_{2}], p_{1}]

on A[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}] in a band around j_{s_{1}} = 0 and j_{s_{2}} = 0.

Using Lemma 2.4 we see that

[α, p_{2}] = αp_{2}α^{−1}p_{2}^{−1} ∼ α^{−1}p_{2}^{−1}αp_{2}

as an element of K′_{−i+1}(R[t_{2}, t_{2}^{−1}]), hence

[α, p_{2}] = [α^{−1}, p_{2}^{−1}]

. Now

[[α, p_{1}], p_{2}] = αp_{1}α^{−1}p_{2}αp_{1}^{−1}α^{−1}p_{2}^{−1}

since p_{1} and p_{2} commute, whereas [[α^{−1}, p_{2}^{−1}], p_{1}] = α^{−1}p_{2}^{−1}αp_{1}α^{−1}p_{2}αp_{1}^{−1}. This last expression however may be conjugated by p_{2}α to give αp_{1}α^{−1}p_{2}αp_{1}^{−1}α^{−1}p_{2}^{−1}. So applying Lemma 2.4 once again shows the two terms represent the same element, and we are done. □

2.6. Proposition. There is a standard identification of K′_{1} with K_{1} and of K′_{0} = K′′_{0} with K_{0} under which λ_{t} : K_{0}(R) → K_{1}(R[t, t^{−1}]) is the usual Bass-Heller-Swan homomorphism.

Proof. K′_{1}(R) is equal to K_{1}(R) by definition. We have seen K′_{0}(R) ≃ K′′_{0}(R) and, if we send a projection p : R^{n} → R^{n} to im(p) ⊂ R^{n}, this gives a direct summand (the other summand being im(1 − p)) and thus a finitely generated projective. On the other hand, if P is projective, we may find Q such that P ⊕ Q = R^{n} and 1_{P} ⊕ 0_{Q} will define the appropriate projection. If ϕ : P_{1} → P_{2} is an isomorphism of projectives, the diagram

\xymatrix{P_{1} ⊕ Q_{1} ⊕ P_{2} ⊕ Q_{2} \ar[d]^{1⊕0⊕0⊕0} \ar[r]^{ϕ⊕1⊕ϕ^{−1}⊕1} & P_{2} ⊕ Q_{1} ⊕ P_{1} ⊕ Q_{2} \ar[d]^{1⊕0⊕0⊕0} \\ P_{1} ⊕ Q_{1} ⊕ P_{2} ⊕ Q_{2} \ar[r]^{ϕ⊕1⊕ϕ^{−1}⊕1} & P_{2} ⊕ Q_{1} ⊕ P_{1} ⊕ Q_{2}}  (2.7)

shows that the corresponding projections are equivalent in K′′_{0}(R), so K′′_{0}(R) is isomorphic to K_{0}(R). The usual Bass-Heller-Swan homomorphism is described as follows: Let P be a finitely generated projective R-module. Choose Q so that P ⊕ Q = R^{n} and send [P] to [P [t, t^{−1}] ⊕ Q[t, t^{−1}], t ⊕ 1]. In terms of projections this means that p : R^{n} → R^{n} is sent to tp+1−p : R[t, t^{−1}]^{n} → R[t, t^{−1}]^{n}. If we are given [A, α] ∈ K′_{0}(R), the corresponding element in K′′_{0}(R) is given by [\overline{A}, αp_{−}α^{−1}] − [\overline{A}, p_{−}] where \overline{A} = ⊕_{l=−k}^{k}A(i).

The Bass-Heller-Swan construction thus gives tαp_{−}α^{−1} + 1 − αp_{−}α^{−1})(tp_{−} + 1 − p_{−})^{−1} = α(t · p_{−} + 1 − p_{−})α^{−1}(tp_{−} + 1 − p_{−})^{−1} = [α, p_{t}]. This completes the proof. □

We wish to show that the λ_{t}^{s}-homomorphisms we have constructed are split monomorphisms. The idea is to define a map K′_{−i+1}(R[t, t^{−1}]) → K′_{−i}(R) using the t-powers to give an extra grading. However given a ℤ^{i}-graded R[t, t^{−1}] isomorphism α : A[t, t^{−1}] → A[t, t^{−1}], we do not have a bound on the powers of t that may occur in expressing α. Hence we may not get a bounded isomorphism when we use the t-powers as gradings. This is the reason for the following slightly artificial step.

2.8. Definition. Let R be a ring. We define C_{i}(R)[J, J^{−1}] where J = (t_{1}, …, t_{r}) as follows: We denote R[t_{1}, t_{1}^{−1}, …, t_{r}, t_{r}^{−1}] by R[J, J^{−1}], and given an R-module A, we denote the R[J, J^{−1}]-module A[t_{1}, t_{1}^{−1}, …, t_{r}, t_{r}^{−1}] by A[J, J^{−1}]. An object of C_{i}(R)[J, J^{−1}] is just an object of C_{i}(R), but a morphism A → B is an R[J, J^{−1}]-morphism A[J, J^{−1}] → B[J, J^{−1}] that can be written as a finite sum α = ∑ t_{1}^{n_{1}} ⋯ t_{r}^{n_{r}}α_{n_{a},··· ,n_{r}}, where α_{n_{1},··· ,n_{r}} are morphisms in C_{i}(R).

We also need the category P C_{i}(R)[J, J^{−1}] and the result analogous to Section 1:

2.9. Lemma. K_{1}(C_{i}(R)[J, J^{−1}]) ≅ K_{0}(P C_{i}(R)[J, J^{−1}]) for i > 0 and K_{1}(C_{i}(R)[J, J^{−1}]) is isomorphic to K_{0}(P C_{i}(R)[J, J^{−1}]) with the extra relations induced by [A, 0] = 0.

Proof. We define ϕ^{s} as in section 1, and note there are no infinite compositions, so everything we do in Section 1 (which corresponds to J = ∅) goes right through. □

We define

K_{−i}^{J}(R[J, J^{−1}]) = K_{1}(C_{i+1}(R)[J, J^{−1}]).  (2.10)

2.11. Remark. There is an obvious map K_{−i}^{J}(R[J, J^{−1}]) → K′_{−i}(R[J, J^{−1}]) induced by sending [A, α] to [A[J, J^{−1}], α]. By construction λ_{t}^{s} : K′_{−i}(R) → K′_{−i+1}(R[t, t^{−1}]) factors through K_{−i+1}^{t}(R[t, t^{−1}]) → K′_{−i+1}(R[t, t^{−1}]). It is also clear that λ_{t}^{s} generalizes to

λ_{t}^{s} : K_{−i}^{J}(R[J, J^{−1}]) → K_{−i}^{J,t}(R[J, J^{−1}, t, t^{−1}])

. Thinking of λ_{t}^{s} in this way as a homomorphism K′_{−i}(R) → K_{−i+1}^{t}(R[t, t^{−1}]) we will be able to define a left inverse, and we shall then eventually show K_{−i+1}^{t}(R[t, t^{−1}]) ≅ K′_{−i+1}(R[t, t^{−1}]).

Consider an element [B, β] of K_{−i+1}^{J,t}(R[J, J^{−1}, t, t^{−1}]) where J = (t_{1}, …, t_{r}) as above. We define

C(j_{1}, …, j_{s}, …, j_{i+1}) = B(j_{1}, …, ĵ_{s}, …, j_{i+1})[J, J^{−1}](t^{j_{s}}),

the R[J, J^{−1}]-submodule of B(j_{1}, …, ĵ_{s}, …, j_{i+1})[J, J^{−1}][t, t^{−1}] generated by t^{j_{s}}. We may clearly consider β an R[J, J^{−1}]-module isomorphism of C. Since the condition that β (and β^{−1}) may be written as a sum only involving finitely many t-powers will ensure that β is a bounded ℤ^{i+1}-graded automorphism of C, we may define μ_{t}^{s} : K_{−i+1}^{J,t}(R[J, J^{−1}, t, t^{−1}]) → K_{−i}^{J}(R[J, J^{−1}]) by

μ_{t}^{s}([B, β]) = [C, β].  (2.12)

Popularly speaking μ_{t}^{s} is the identity, only we consider the t-powers an extra grading placed at the sth coordinate.

2.13. Proposition. μ_{t}^{s} is a well-defined homomorphism and μ_{t}^{s} · λ_{t}^{s} = 1

Proof. We consider the case K′_{−i}(R) ⟶^{λ_{t}^{s}} K_{−i}^{t}(R[t, t^{−1}]) ⟶^{μ_{t}^{s}} K′_{−i}(R) and note the argument we give carries over to the general case as in the proof of Lemma 2.9.

The fact that β may be written as a finite sum ∑ t^{j}β_{j} ensures that β becomes a bounded automorphism of C. Since μ_{t}^{s} is essentially the identity, it will respect all relations. To prove that μ_{t}^{s} · λ_{t}^{s} is the identity, consider an element of K′_{−i}(R). Using Theorem 1.9 (see in particular Lemma 1.15) we may assume α is of the form

[figure: j_{s}; =; …; −2; −1; 0; 1; 2; …; =; α; …; p; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; …; B; B; B; B; B]

where B is an object of C_{i}(R) and p : B → B a projection. It is easy to compute the commutator [α, p_{t}^{s}], and we get

[figure: j_{s}; =; …; −2; −1; 0; 1; 2; …; [α, p_{t}^{s}]; =; …; B; 1; B; 1; B; tp+1−p; B; 1; B; 1; …; B; B; B; B; B]

so λ_{t}^{s}([α]) = [B[t, t^{−1}], tp + 1 − p].

When we turn the t-powers into gradings we get back α on the nose, so we are done. □

As mentioned above (Remark 2.11) we get a map K_{−i}^{J}(R[J, J^{−1}]) → K′_{−i}(R[J, J^{−1}]) sending [A, α] to [A[J, J^{−1}], α]. This map induces an isomorphism.

2.14. Proposition. K_{−i}^{J}(R[J, J^{−1}]) ≅ K′_{−i}(R[J, J^{−1}]).

Proof. The proof is by induction on i, the induction starting with i = −1. In this case the R[J, J^{−1}]-module is finitely generated so it is no restriction to require a bound on the powers of t_{i} (remember J = (t_{1}, …, t_{r})). The slight difference between A[J, J^{−1}] where A is a finitely generated free R-module, and a finitely generated free R[J, J^{−1}]-module causes no trouble. Assume inductively that K_{−i+1}^{J}(R[J, J^{−1}]) → K′_{−i+1}(R[J, J^{−1}]) is an isomorphism for all rings R. In the commutative diagram

\xymatrix{K_{−i}^{J}(R[J, J^{−1}]) \ar[rr] \ar[d]^{λ_{t}^{1}} &  & K′_{−i}(R[J, J^{−1}]) \ar[d]_{λ_{1}^{t}} \\ K_{−i+1}^{J,t}(R[J, J^{−1}, t, t^{−1}]) \ar[d]^{μ_{t}^{1}} \ar[r] & K′_{−i+1}(R[J, J^{−1}, t, t^{−1}]) & K_{−i+1}^{t}(R[J, J^{−1}][t, t^{−1}]) \ar[l] \ar[d]^{μ_{t}^{1}} \\ K_{−i}^{J}(R[J, J^{−1}]) \ar[rr] &  & K′_{−i}(R[J, J^{−1}])} (2.15)

the middle horizontal maps are isomorphisms by induction hypothesis. Since μ_{t}^{1} · λ_{t}^{1} = 1, it follows that K_{−i}^{J}(R[J, J^{−1}]) → K′_{−i}(R[J, J^{−1}]) is an isomorphism. □

In view of Proposition 2.13 and 2.14 we have proved the following:

2.16. Theorem. λ_{t}^{s} : K′_{−i}(R) → K′_{−i}(R[t, t^{−1}]) is a split monomorphism with left inverse

given by K′_{−i}(R[t, t^{−1}]) ≅ K_{−i}^{t}(R[t, t^{−1}]) ⟶^{μ_{t}^{s}} K′_{−i}(R).

We have not discussed how λ_{t}^{s} and μ_{t}^{s} depend on s. Note, that if g ∈ Gl(i + 1, ℤ) is used to regrade an object A of C_{i+1}(R) by A^{g}(j_{1}, …, j_{i+1}) = A(g(j_{1}, …, j_{i+1})), the identity 1_{g} : A^{g} → A is not a bounded automorphism of A^{g}. But if α is a bounded automorphism of A, 1_{g}α1_{g}^{−1} is a bounded automorphism of A^{g}. This defines an action of Gl(i + 1, ℤ) on K_{−i}(R) which is given by

2.17. Lemma. g ∈ Gl(i + 1, ℤ) acts on K′_{−i}(R) by multiplication by det(g).

Proof. First we show that if g is elementary, g = E_{rs}(a), the action is trivial. If [A, α]

is regraded by g we get the composite A^{g} ⟶^{1_{g}} A ⟶^{α} A ⟶^{1_{g}^{−1}} A^{g}. But λ_{t}^{s}([A^{g}, 1_{g}^{−1}α1_{g}]) = [A[t, t^{−1}]^{g}, [1_{g}^{−1}α1_{g}, p_{t}^{s}]] and since 1_{g} and p_{t}^{s} commute, we get [A[t, t^{−1}]^{g}, 1_{g}^{−1}[α, p_{t}^{s}]1_{g}] and restrict to a band around j_{s} = 0. But g = E_{rs}(a), so 1_{g} is a bounded isomorphism when restricted to a band around j_{s} = 0, hence this last element is equivalent to [A[t, t^{−1}], [α, p_{t}^{s}]] which represents λ_{t}^{s}([A, α]). Since λ_{t}^{s} is a monomorphism, we are done. We now only need to see how g, which acts on ℤ^{i+1} by multiplying the sth coordinate by −1, acts. A typical element [α] ∈ K′_{−i}(R) may be written (by Theorem 1.9)

[figure: j_{s}; =; …; −2; −1; 0; 1; 2; …; =; α; …; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; p; B; 1−p; …; B; B; B; B; B]

which is mirrored about j_{s} = 0 by g to give α^{−1}, so g acts by −1. This completes the proof. □

2.18. Corollary. λ_{t}^{s+1} = −λ_{t}^{s} : K′_{−i}(R) → K′_{−i+1}(R[t, t^{−1}]) and μ_{t}^{s+1} = −μ_{t}^{s} : K′_{−i+1}(R[t, t^{−1}]) → K′_{−i}(R).

Proof. Let τ be the transposition that interchanges the sth and s + 1th coordinate. Then λ_{t}^{s} · τ = λ_{t}^{s+1} and μ_{t}^{s+1} = τ · μ_{t}^{s}. □

In view of 2.18 we shall denote λ_{t}^{1}, μ_{t}^{1} by λ_{t}, μ_{t}, respectively, and note that

μ_{t}^{s} = (−1)^{s−1}μ_{t},  λ_{t}^{s} = (−1)^{s}λ_{t}.  (2.19)

Proof of Main Theorem. One possible definition of K_{−i−1}(R) is by induction as the intersection of the images of the Bass-Heller-Swan homomorphisms

K_{−i}(R[t_{1}, t_{1}^{−1}]) → K_{−i+1}(R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}])

and

K_{−i}(R[t_{2}, t_{2}^{−1}]) → K_{−i+1}(R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}]).

Our proof will be by induction on the statement K′_{−i}(R) ≅ K_{−i}(R) by an isomorphism under which the image of λ_{t} : K′_{−i}(R) → K_{−i+1}(R[t, t^{−1}]) is sent to the image of the usual Bass-Heller-Swan homomorphism. The start of the induction is proposition 2.6. Denoting λ_{t_{1}}, λ_{t_{2}}, μ_{t_{1}}, μ_{t_{2}} by λ_{1}, λ_{2}, μ_{1}, μ_{2} respectively, we have a homomorphism λ_{2} · λ_{1} : K′_{−i−1}(R) → K_{−i+1}(R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}]). Theorem 2.5 implies λ_{2}λ_{1} = −λ_{1}λ_{2} so im(λ_{2}λ_{1}) is contained in (im λ_{1} ∩ (im λ_{1}) = K_{−i−1}(R) (by induction hypothesis). λ_{2}λ_{1} is a monomorphism by Proposition 2.13. To show it is epic let a ∈ K_{−i+1}(R). Consider a as an element of K_{−i+1}(R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}]). By induction hypothesis a = λ_{1}(b_{1}) = λ_{2}(b_{2}) and, by 2.13, b_{1} = μ_{1}(a), b_{2} = μ_{2}(a). The diagram

\xymatrix{ & K′_{−i−1}(R) \ar[dr]^{λ_{t_{1}}^{s}} \\ K′_{−i}R[t_{2}, t_{2}^{−1}] \ar@{-->}[ur]^{μ_{t_{2}}^{s+1}} \ar@{-->}[dr]_{λ}_{t_{1}}^{s} &  & K′_{−i}(R[t_{1}, t_{1}^{−1}]) \\  & K′_{−i+1}R[t_{1}, t_{1}^{−1}, t_{2}, t_{2}^{−1}] \ar@{-->}[ur]_{μ_{t_{2}}^{s}}}  (2.20)

is commutative since one way sends [α] to [α, p_{t_{1}}^{s}] and then turns t_{2}-powers into a grading, whereas the other way around turns t_{2}-powers into a grading and then takes the commutator with p_{t_{1}}^{s}. Hence μ_{2}λ_{1} = −λ_{1}μ_{2} and we get λ_{2}λ_{1}(−μ_{2}μ_{1}(a)) = λ_{2}μ_{2}λ_{1}μ_{1}(a) = a, so λ_{2}λ_{1} is onto. This completes the proof of the Main Theorem. □

3. Final Remarks

In geometric applications one is usually not considering an automorphism α : A → A, but rather an isomorphism α : A → B. This, however, only makes a difference in the category C_{0}(R) since if p_{−}^{s} : A → A is the projection 1.7 we may consider αp_{−}^{s}α^{−1}. The restriction to a band around j_{s} = 0 gives a projection, which is 1 if j_{s} ≪ 0 and 0 if j_{s} ≫ 0. Hence the width of the band only matters in case i = 0; in all other cases we have a well-defined invariant in K_{−i}(R). For i = 0 we have to divide out by identity projections to get a well-defined invariant.

This amounts to getting an invariant in K̃_{0}(R). With the obvious notion of a contractible chain-complex in the category C_{i+1}(R) we thus get an associated K̃_{−i}(R)-invariant (K̃_{−i}(R) = K_{−i}(R) for i > 0). Also associated to a homotopy projection of a C_{i}(R) chain complex, we get a K̃_{−i}(R)-invariant (using the methods of [4], see [3] for a proof). These ideas are further developed in [3]. Altogether a number of results due to Quinn can be given “standard” proofs using this description of the K_{−i}-functors.

References

1. H. Bass, Algebraic K-theory, Benjamin, 1968.

2. H. Bass, A. Heller, and R. G. Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 61–80.

3. E. K. Pedersen, K_{−i}-invariants of chain complexes, Topology (Leningrad, 1982), Lecture Notes in Mathematics, vol. 1060, Springer, Berlin, 1984, pp. 174–186.

4. A. A. Ranicki, The algebraic theory of finiteness obstructions, Math. Scand. 57 (1985), 105–126.

5. R. Swan, Algebraic k-theory, Lecture Notes in Mathematics, vol. 76, Springer, 1968.

Matematisk Institut, Odense Universitet, Odense Denmark