Multirelative K-Theory

and Axioms for the K-Theory of Rings

Frans Keune

Received: July 17, 1996

Communicated by Ulf Rehmann

Abstract. K-groups are defined for a special type of m-tuples of ideals in a ring. It is shown that some of the properties of this multirelative K-theory characterize the K-theory of rings.

1991 Mathematics Subject Classification: Primary 19D99.

Introduction

Multirelative K-groups K_{n}(R, 𝔞_{1}, …, 𝔞_{m}) of an m-tuple (𝔞_{1}, …, 𝔞_{m}) of ideals of a ring R are recently used to derive properties of the absolute K-groups, e.g. by Levine [4] and by Bloch and Lichtenbaum [1]. Here it is shown how K-theory as defined in [3] can easily be extended to the multirelative case and that some of its properties can be taken as axioms for the K-theory of rings. Special types of m-tuples of ideals— the ‘normal’ m-tuples—play a crucial role. In fact we will only define multirelative K-groups for such m-tuples. The notion of normal m-tuple of ideals is introduced in Section 2. It already appeared in 1981 in a paper by Dayton and Weibel [2] on the K-theory of affine glued schemes under the name of ‘condition (CRT)’ (= Chinese Remainder Theorem).

In Section 4 we review briefly higher K-theory as defined in [3]. In Section 6 multirelative K-groups are defined, and in Section 7 it is shown that from some of their properties one can reconstruct the K-theory of rings.

1 Notations

In this paper ‘ring’ stands for a non-unital ring. Non-unital rings form a category which is denoted by R.

Since the functors GL, E and K_{1} are product preserving functors from unital rings to groups, they can be extended to functors defined on R in the usual way: if T is one of these functors, then put

T (R) := Ker(T (R^{+}) → T (ℤ)),

where R^{+} = R × ℤ with multiplication given by

(r, k)(s, l) = (rs + ks + lr, kl)

is a ring with (0, 1) as unity element.

Here ‘ideal’ will always stand for ‘twosided ideal’.

By A we will denote the category of Abelian groups, by G the category of all groups, and by S the category of sets. The category of simplicial objects in a category C is denoted by sC.

2 m-cubes and normal m-tuples

In this section the notion of normality of an m-tuple of ideals is considered. Only the group structure is involved in its definition, and since we can use later a similar notion for groups instead of rings we give a more general definition. By \underline{m} we will denote the set {1, …, m}.

Definition 1. An m-tuple (B_{1}, …, B_{m}) of normal subgroups of a group A—also denoted as (A, B_{1}, …, B_{m})—is called normal if for all subsets I and J of \underline{m}

⋂_{i∈I} B_{i} · ∏_{j∈J} B_{j} = ⋂_{i∈I}(B_{i} · ∏_{j∈J} B_{j}).

The condition is trivially fulfilled when I ∩ J ≠ ∅. In the case of Abelian groups it reads in the additive notation as

⋂_{i∈I} B_{i} + ∑_{j∈J} B_{j} = ⋂_{i∈I}(B_{i} + ∑_{j∈J} B_{j}).

Note that in the special case of an m-tuple of ideals in a commutative ring the condition is a local one since it involves only intersections and sums of ideals.

The subsets of \underline{m} are ordered by inclusion. This ordered set determines in the usual way a category C_{m}. For every pair (I, J) of subsets with I ⊆ J there is the unique morphism ρ_{J}^{I} from I to J in C_{m}.

Definition 2. Let D be a category. An m-cube in D is a functor

D: C_{m} → D, I ↦ D_{I}, ρ_{J}^{I} ↦ r_{J}^{I}.

The morphisms in C_{m} are generated by the ρ_{J}^{I}, where #J = #I + 1. An m-cube in a category D is a commutative diagram in D having the shape of an m-dimensional cube. The edges of the cube correspond to the images of these generating morphisms.

Definition 3. Let D: C_{m} → D be an m-cube in D. It is said to be a split m-cube if for every pair of subsets (I, J) of \underline{m} satisfying I ⊆ J there is a morphism s_{I}^{J}: D_{J} → D_{I} in D such that

(S1) s_{I}^{J}s_{J}^{K} = s_{I}^{K} for all I ⊆ J ⊆ K,

Multirelative K-Theory 279

(S2) r_{J}^{I}s_{I}^{J} = 1_{D_{J}} for all I ⊆ J, (S3) r_{J}^{I∩J}s_{I∩J}^{I} = s_{J}^{I∪J}r_{I∪J}^{I} for all I and J.

(Of course such a split m-cube can also be seen as a functor defined on a category which is obtained from C_{m} by adjoining extra morphisms σ_{I}^{J}: J → I.)

In condition (S3) one only needs the case where #(I \ J) = #(J \ I) = 1. It then reads

(S3’) r_{I∪{k}}^{I}s_{I}^{I∪{j}} = s_{I∪{k}}^{I∪{j,k}}r_{I∪{j,k}}^{I∪{j}} for all j, k ∉ I with j ≠ k.

This can easily be seen as follows. Put K = I ∩ J, I \ K = {i_{1}, …, i_{p}} and J \ K = {j_{1}, …, j_{q}}. Then the result follows from the diagram

D_{K} ⟶ D_{K∪{i_{1}}} ⟶ ⋯ ⟶ D_{I}
↑  ↑  ↑
D_{K∪{j_{1}}} ⟶ D_{K∪{i_{1},j_{1}}} ⟶ ⋯ ⟶ D_{I∪{j_{1}}}
↑  ↑  ↑
⋮  ⋮  ⋮
↑  ↑  ↑
D_{J} ⟶ D_{J∪{i_{1}}} ⟶ ⋯ ⟶ D_{I∪J}

where the horizontal maps are r-maps and the vertical maps are s-maps.

Definition 4. An m-tuple T = (A, B_{1}, …, B_{m}) of normal subgroups determines an m-cube in G:

I ↦ T_{I} = A / ∏_{i∈I} B_{i}.

When I ⊆ J, then ∏_{i∈I} B_{i} ⊆ J and 1_{A} induces a grouphomomorphism r_{J}^{I}: T_{I} → T_{J}. This m-cube is said to be induced by the m-tuple T. Similarly for an m-tuple of ideals in a ring.

Proposition 2.1. Let D: C_{m} → D be an m-cube in G, which is split as an m-cube in S. Then D is induced by a normal m-tuple of normal subgroups of D_{∅}.

Proof. For i ∈ \underline{m} put

B_{i} = Ker(r_{{i}}^{∅}: D_{∅} → D_{{i}}).

We will first show that the cube is induced by the m-tuple (D_{∅}, B_{1}, …, B_{m}). Since the cube splits in S, the homomorphisms D_{∅} → D_{I} are surjective. To show that for each I ⊆ \underline{m}

Ker(D_{∅} → D_{I}) = ∏_{i∈I} B_{i}.

This can be done by induction on #(I). For #(I) = 0 it is trivial. Let #(I) > 0. Choose k ∈ I. By induction hypothesis

Ker(D_{∅} → D_{I\{k}}) = ∏_{i∈I\{k}} B_{i}.

Since the cube splits in S we have a commutative diagram with exact rows and columns:

1  1  1
↓  ↓  ↓
1 ⟶ B_{k} ∩ ∏_{i∈I\{k}} B_{i} ⟶ B_{k} ⟶ Ker(r_{I}^{I\{k}}) ⟶ 1
↓  ↓  ↓
1 ⟶ ∏_{i∈I\{k}} B_{i} ⟶ D_{∅} ⟶ D_{I\{k}} ⟶ 1
↓  ↓  ↓
1 ⟶ Ker(r_{I}^{{k}}) ⟶ D_{{k}} ⟶  D_{I}  ⟶ 1
↓  ↓  ↓
1  1  1

Hence

Ker(r_{I}^{∅})/B_{k} ≅ ∏_{i∈I\{k}} B_{i} / (B_{k} ∩ ∏_{i∈I\{k}} B_{i}) ≅ ∏_{i∈I}(B_{i}/B_{k}),

and therefore,

Ker(r_{I}^{∅}) = ∏_{i∈I} B_{i}.

For the normality of the m-tuple let I, J ⊆ \underline{m} and consider the commutative square

D_{∅}  ⟶^{(r_{{i}}^{∅})} ×_{i∈I} D_{∅}/B_{i}
↓^{r_{J}^{∅}}  ↓^{(r_{J∪{i}}^{{i}})}
D_{∅} / ∏_{j∈J} B_{j} ⟶^{(r_{J∪{i}}^{J})} ×_{i∈I} D_{∅} / ∏_{j∈J∪{i}} B_{j}.

Since the m-cube is split in S the vertical homomorphisms have compatible sections in S. So r_{J}^{∅} induces a surjective homomorphism on the kernels of the horizontal homomorphisms. This holds for all I, J ⊆ \underline{m}. Therefore, the m-tuple (D_{∅}, B_{1}, …, B_{m}) is normal.

For the Abelian case we also prove the converse.

Proposition 2.2. Let T = (A, B_{1}, …, B_{m}) be a normal m-tuple of subgroups of an Abelian group A. Then the induced m-cube is split in the category S.

Proof. By taking kernels of the surjective homomorphisms in the induced m-cube it can be extended to a diagram of 3^{m} Abelian groups. We will give a detailed description of this diagram and show how a splitting of the cube can be obtained from it.

For each pair (I, J) of disjoint subsets of \underline{m} define

C_{J}^{I} = ⋂_{i∈I} B_{i} + ∑_{j∈J} B_{j} / ∑_{j∈J} B_{j}.

Then for each such pair (I, J) and each k ∉ I ∪J we have a surjective homomorphism C_{J}^{I} → C_{J∪{k}}^{I}, induced by r_{J∪{k}}^{J}: A_{J} → A_{J∪{k}}, where we use the notation

A_{J} = A / ∑_{j∈J} B_{j}.

Thus A_{J} = C_{J}^{∅}. The kernel of the surjective homomorphism C_{J}^{I} → C_{J∪{k}}^{I} is

(⋂_{i∈I} B_{i} + ∑_{j∈J} B_{j}) ∩ (B_{k} + ∑_{j∈J} B_{j}) / B_{k} + ∑_{j∈J} B_{j}.

We have the inclusions

⋂_{i∈I∪{k}} B_{i} + ∑_{j∈J} B_{j} ⊆ (⋂_{i∈I} B_{i} + ∑_{j∈J} B_{j}) ∩ (B_{k} + ∑_{j∈J} B_{j}) ⊆ ⋂_{i∈I∪{k}}(B_{i} + ∑_{j∈J} B_{j}).

By normality these groups are equal, so we have a short exact sequence

0 → C_{J}^{I∪{k}} → C_{J}^{I} → C_{J∪{k}}^{I} → 0.

For each pair (I, J) of disjoint subsets of \underline{m} satisfying I ∪ J = \underline{m} choose a section

t_{J}^{I}: C_{J}^{I} → C_{∅}^{I}(⊆ C_{∅}^{∅} = A)

of the map C_{∅}^{I} → C_{J}^{I} induced by r_{J}^{∅}: A → A_{J} and satisfying t_{J}^{I}(0) = 0. Next define maps t_{J}^{I}: C_{J}^{I} → C_{∅}^{I} for every disjoint pair (I, J) using induction to the number of elements of the complement of I ∪ J. So, let (I, J) be a disjoint pair of subsets of \underline{m} with #(I ∪ J) = n < m and assume that sections t_{L}^{K}: C_{L}^{K} → C_{∅}^{K} have already been defined for pairs (K, L) with K ∪ L having more than n elements.

Choose k ∈ \underline{m} \ (I ∪ J). Let x ∈ C_{J}^{I}, then for y = r_{J}^{∅}t_{J∪\{k\}}^{I}r_{J∪\{k\}}^{J}(x) we have

r_{J∪{k}}^{J}(y) = r_{J∪{k}}^{∅}t_{J∪{k}}^{I}r_{J∪{k}}^{J}(x) = r_{J∪{k}}^{J}(x),

so, x − y ∈ C_{J}^{I∪{k}}. Now define t_{J}^{I} by

t_{J}^{I}(x) = t_{J}^{I∪{k}}(x − y) + t_{J∪{k}}^{I}r_{J∪{k}}^{J}(x).

It easily verified that this map is a section of r: C_{∅}^{I} → C_{J}^{I}. Furthermore it is independent of the choice of k: if also l ∉ I ∪ J, then in both cases the image of an x ∈ C_{J}^{I} under t_{J}^{I} is determined in the same way by the images of the same elements in the following four groups

C_{J}^{I∪{l,k}}, C_{J∪{k}}^{I∪{l}}, C_{J∪{l}}^{I∪{k}}, and C_{J∪{k,l}}^{I} :

0  0  0
↓  ↓  ↓
0 ⟶ C_{J}^{I∪{l,k}} ⟶ C_{J}^{I∪{l}} ⟶ C_{J∪{k}}^{I∪{l}} ⟶ 0
↓  ↓  ↓
0 ⟶ C_{J}^{I∪{k}} ⟶ C_{J}^{I} ⟶ C_{J∪{k}}^{I} ⟶ 0
↓  ↓  ↓
0 ⟶ C_{J∪{l}}^{I∪{k}} ⟶ C_{J∪{l}}^{I} ⟶ C_{J∪{k,l}}^{I} ⟶ 0
↓  ↓  ↓
0  0  0

Thus we obtain a splitting of the cube, where the sections s_{I}^{J} of the homomorphisms r_{J}^{I}, where I ⊆ J, are the maps r_{I}^{∅}t_{J}^{∅}. In particular, condition (S3’) follows from the above diagram for I = ∅.

3 Operations on normal m-tuples of ideals

By R_{m} we will denote the category of all normal m-tuples of ideals. Such an mtuple is denoted as (R, 𝔞_{1}, …, 𝔞_{m}), where R is a ring and 𝔞_{1}, …, 𝔞_{m} are ideals of R. A morphism ϕ: (R, 𝔞_{1}, …, 𝔞_{m}) → (S, 𝔟_{1}, …, 𝔟_{m}) is just a ringhomomorphism ϕ: R → S satisfying ϕ(𝔞_{i}) ⊆ 𝔟_{i} for all i ∈ \underline{m}.

The following notations will simplify notations for long exact sequences of multirelative K-theory. Another advantage will be that they are useful to indicate funtoriality properties.

For each m ≥ 1 the functor D: R_{m} → R_{m−1} is the functor that deletes the last ideal:

D(R, 𝔞_{1}, …, 𝔞_{m}) = (R, 𝔞_{1}, …, 𝔞_{m−1})

and which has no effect on morphisms.

For each m ≥ 1 the functor M: R_{m} → R_{m−1} is the functor that deletes the last ideal and that takes the ring and the other ideals modulo this ideal:

M(R, 𝔞_{1}, …, 𝔞_{m}) = (R/𝔞_{m}, \overline{𝔞}_{1}, …, \overline{𝔞}_{m−1}),

where \overline{𝔞}_{j} = 𝔞_{j} + 𝔞_{i}/𝔞_{i}, and which maps a morphism to the induced morphism.

A functor morphism ϕ: D → M of the functors D, M: R_{m} → R_{m−1} is defined as follows: let A = (R, 𝔞_{1}, …, 𝔞_{m}), then ϕ_{A}: D(A) → M(A) is the canonical ringhomomorphism R → R/𝔞_{m}.

Every A ∈ R_{m} has an underlying ideal I(A), which is defined as the intersection of the m ideals in A: when A = (R, 𝔞_{1}, …, 𝔞_{m}), then

I(A) = 𝔞_{1} ∩ ⋯ ∩ 𝔞_{m}.

Thus defined, I(A) is functorial in A.

4 Higher K-theory of rings

In [3] the definition of higher K-groups is as follows. Let R ∈ R. Choose a simplicial ring R with an augmentation ε: R → R such that

• R is aspherical, i.e. π_{n}(R) = 0 for all n ≥ 1,

• R_{m} is free for all m ≥ 0, say R_{m} is free on a set X_{m} of generators,

• the sets X_{m} of free generators are stable under degeneracies: s_{j}(X_{m}) ⊆ X_{m+1} for all m ≥ 0,

• the augmentation ε induces an isomorphism π_{0}(R) ⟶^{∼} R.

Then for n ≥ 3 the group K_{n}(R) is defined as the (n − 2)nd homotopy group of the simplicial group GL(R), and the groups K_{1}(R) and K_{2}(R) are given by the exactness of

0 → K_{2}(R) → π_{0}(GLR) → GL(R) → K_{1}(R) → 0.

The groups K_{n}(R) for n ≥ 3 are Abelian because GL(R) is a simplicial group. The group K_{1}(R) is Abelian since it is the cokernel of GL(R_{0}) → GL(R), and K_{2}(R) is Abelian because it is the cokernel of GL(R_{1}) → GL(Z_{0}), where Z_{0} = { (x_{0}, x_{1}) | ϵ(x_{0}) = ϵ(x_{1}) }. In [3] it is shown using a comparison theorem that the higher Kgroups are thus well-defined and that they are actually functors. For the purpose of this paper we will confine to a functorial resolution Fr(R) of a ring R, which we now describe. Let F: S → R the free ring functor and let U: R → S be the underlying set functor, then the functor F U: R → R together with the obvious functor morphisms ν: F U → (F U)^{2} and η: F U → I is a cotriple. Put

Fr_{n} = (F U)^{n+1}.

Face and degeneracy morphisms are given by

d_{i} = (F U)^{i}η(F U)^{n−1−i} and s_{j} = (F U)^{i}ν(F U)^{n−1−i}.

The augmentation is then given by η.

A property of this functorial resolution is that, when applied to a surjective ringhomomorphism R → S, it gives a dimensionwise surjective homomorphism FrR → FrS of simplicial rings, and since the ringhomomorphisms are dimensionwise split it also gives a surjective simplicial grouphomomorphism GL(FrR) → GL(FrS). This is often convenient when considering homotopy fibres, because surjective simplicial grouphomomorphisms are fibrations themselves. So instead of taking a homotopy fibre one just takes a fibre, i.e. the kernel of the simplicial group homomorphism.

5 Cubes in a simplicial group

Let A be a simplicial group with augmentation d_{0} : A_{0} → A. It is a contravariant functor A: Ω_{+}^{op} → G from the category Ω_{+} of finite ordered sets

[n] = {0, …, n} (n ≥ −1)

(where [−1] = ∅) and monotone (= order preserving) maps to the category of groups. (Here we use the notation A_{−1} = A.) We will show that A determines an m-cube of groups for every nonnegative integer m. In stead of the ordered set of subsets of \underline{m} for the description of an m-cube the ordered set of subsets of [m − 1] will be used for this purpose.

Let Ω(m) be the category of injective monotone maps

α: [k] → [m − 1].

A morphism from α: [k] → [m − 1] to β: [l] → [m − 1] is a monotone map γ: [k] → [l] such that βγ = α. It exists if and only if Im(α) ⊆ Im(β), and it is unique if it exists.

For each I ⊆ [m − 1] there is a unique injective monotone map

α_{I}: [k] → [m − 1],

where k = m − 1 − #(I) and Im(α_{I}) = [m − 1] \ I. If I ⊆ J ⊆ [m − 1], then Im(α_{I}) ⊇ Im(α_{J}) , so then there is a unique

γ_{I}^{J}: α_{J} → α_{I},

i.e. a monotone γ_{I}^{J}: [m − 1 − #(J)] → [m − 1 − #(I)] such that α_{I}γ_{I}^{J} = α_{J}.

Definition 5. Let A be an augmented simplicial group and let m be a nonnegative integer. Then the m-cube of A is the m-cube A(m): C_{m} → G with

\begin{cases}A(m)_{I} = A_{[m−1−#(I)]} & for all I ⊆ [m − 1], \\ r_{J}^{I} = A(γ_{I}^{J}): A(m)_{I} → A(m)_{J} & for all I ⊆ J ⊆ [m − 1]. \end{cases}

Lemma 5.1. Let the augmentation d_{0} : A_{0} → A_{−1} induce a surjective homomorphism π_{0}(A) → A_{−1}. Then for all integers i, j, m such that 0 ≤ j < i ≤ m

d_{i}^{(m)}(Ker(d_{j}^{(m)})) = Ker(d_{j}^{(m−1)}).

Proof. Let x ∈ Ker(d_{j}^{(m)}). Then, since i > j, d_{j}d_{i}(x) = d_{i−1}d_{j}(x) = 1. So

d_{i}(Ker(d_{j})) ⊆ Ker(d_{j}). Now, let y ∈ Ker(d_{j}^{(m−1)}). There is an x ∈ A_{m} such that d_{j}(x) = 1 and d_{i}(x) = y. For m > 1 this is the case because a simplicial group is a Kan-complex, while for m = 1 it follows from the condition on the augmentation.

Proposition 5.1. Let A be a simplicial group with an augmentation d_{0} : A → A that induces an isomorphism π_{0}(A) → A. Then for all m ≥ 1 the m-cube A(m) is induced by the m-tuple

(A_{m−1}, Ker(d_{0}), …, Ker(d_{m−1})).

Proof. All face maps are surjective, so it remains to show that for all J ⊆ [m − 1]

Ker(r_{J}^{∅}) = ∏_{j∈J} Ker(d_{j}^{(m−1)}).

For J = ∅ this is trivially true. Let J be nonempty and proceed by induction. Let x ∈ Ker(r_{J}^{∅}). Let k ∈ J be maximal. Then r_{{k}}^{∅}(x) = d_{k}(x) ∈ Ker(r_{J}^{{k}}). By

induction this group is equal to ∏_{j∈J′} Ker(d_{j}^{(m−2)}), where J′ = J \ {k}. (Here we used the maximality of k in J and the same result for the (m − 1)-cube A(m − 1).) By the lemma we have

d_{k}( ∏_{j∈J′} Ker(d_{j}^{(m−1)})) = ∏_{j∈J′} Ker(d_{j}^{(m−2)}).

Choose y ∈ ∏_{j∈J′} Ker(d_{j}^{(m−1)} such that d_{k}(y) = d_{k}(x). Then xy^{−1} ∈ Ker(d_{k}). It follows that

Ker(r_{J}^{∅}) ⊆ ∏_{j∈J} Ker(d_{j}^{(m−1)}).

For the other inclusion note that d_{j} = r_{{j}}^{∅} and

r_{J}^{{j}}r_{{j}}^{∅} = r_{J}^{∅}.

Proposition 5.2. Let A be as in Proposition 5.1 and assume moreover that A is aspherical. Then the m-tuple

(A_{m−1}, Ker(d_{0}), …, Ker(d_{m−1}))

is normal.

Proof. The edges of the m-cube are face maps of the simplicial group (A). Normality means that these maps preserve intersections of (the images of) the normal subgroups Ker(d_{0}), …, Ker(d_{m−1}). By induction it suffices to show this for the face maps d_{i}^{(m−1)}. Let J ⊆ [m − 1]. Then to show that

d_{i}( ⋂_{j∈J} Ker(d_{j})) = ⋂_{j∈J} d_{i}(Ker(d_{j})).

for i ∉ J. The inclusion of the left hand side in the right hand side is trivial. So let x ∈ ⋂_{j∈J} d_{i}(Ker(d_{j})). Then for j ∈ J there is an y_{j} ∈ Ker(d_{j}) such that x = d_{i}(y_{j}). For j < i it follows that d_{j}(x) = d_{j}d_{i}(y_{j}) = d_{i−1}d_{j}(x_{j}) = 1. Similarly for j > i we have d_{j−1}(x) = 1. So, since a simplicial group is a Kan-complex and for J = [m − 1] since A is aspherical, there is a y ∈ A_{m−1} such that d_{j}(y) = 1 for all j ∈ J and d_{i}(y) = x. This shows that x ∈ d_{i}(⋂_{j∈J} Ker(d_{j})).

6 Multirelative K-theory

A normal m-tuple of ideals A = (R, 𝔞_{1}, …, 𝔞_{m}) induces an m-cube in R

A: I ↦ R / ∑_{i∈I} 𝔞_{i},

which by Proposition 2.2 is split in S. Application of Fr to this m-cube gives an m-cube of simplicial rings which is dimensionwise split in R. Put

Fr(R, 𝔞_{i}) := Ker(Fr(R) → Fr(R/𝔞_{i})).

This is a simplicial ideal. The m-cube is then induced by the m-tuple

(Fr(R), Fr(R, 𝔞_{1}), …, Fr(R, 𝔞_{m})),

of simplicial ideals, an object of the category sR_{m} of normal m-tuples of simplicial ideals. We also define the simplicial ideal

Fr(R, 𝔞_{1}, …, 𝔞_{m}) := ⋂_{i=1}^{m} Fr(R, 𝔞_{i}).

Application of GL gives an m-cube of simplicial groups, which is dimensionwise split in G. This m-cube is induced by the m-tuple

(GLFr(R), GLFr(R, 𝔞_{1}), …, GLFr(R, 𝔞_{m}))

of simplicial normal subgroups. For n ≥ 3 we define multirelative K_{n} by

K_{n}(R, 𝔞_{1}, … 𝔞_{m}) := π_{n−2}(GLFr(R, 𝔞_{1}, …, 𝔞_{m})).

Multirelative K_{2} and K_{1} are then given by the exactness of

0 → K_{2}(R, 𝔞_{1}, … 𝔞_{m}) → π_{0}(GLFr(R, 𝔞_{1}, …, 𝔞_{m})) → GL(𝔞_{1} ∩ ⋯ ∩ 𝔞_{m}) → K_{1}(R, 𝔞_{1}, … 𝔞_{m}) → 0.

These multirelative K_{1} and K_{2} are Abelian groups for the same reason as in the absolute case.

Now let A ∈ R_{m} with m ≥ 1. Then ϕ_{∗}: GLFr(DA) → GLFr(MA) is a fibration with fibre GLFr(A). The long exact sequence of homotopy groups is a long exact sequence of multirelative K-groups which can easily be extended to include multirelative K_{2} and K_{1}.

Proposition 6.1. Let A ∈ R_{m} with m ≥ 1. Then we have a functorial exact sequence

⋯ → K_{n}(A) → K_{n}(DA) → K_{n}(MA) → K_{n−1}(A) → ⋯ → K_{1}(MA).

The connecting map K_{n}(MA) → K_{n−1}(A) will be denoted by δ and the map K_{n}(A) → K_{n}(DA) by ι. To put it in an even more functorial way, we have an exact sequence of functors and functor morphisms

⋯ → K_{n} ⟶^{ι} K_{n}D ⟶^{K_{n}(ϕ)} K_{n}M ⟶^{δ} K_{n−1} → ⋯ → K_{1}M.

In the remaining part of this section multirelative K_{0} is defined and the long exact sequence for multirelative K-theory is extended with multirelative K_{0}-groups.

Definition 6. For a normal m-tuple A of ideals we define

K_{0}(A) = K_{0}(IA).

Thus defined, K_{0} is a functor from R_{m} to A.

For m = 1 we take the long exact sequence to be the long exact sequence of an ideal in a ring. Now assume that m ≥ 1 and that we have an extended long exact sequence

⋯ → K_{1}D → K_{1}M → K_{0} → K_{0}D → K_{0}M

of functors R_{m} → A. We will show that there is also such a sequence of functors R_{m+1} → A.

Let A = (R, 𝔞_{1}, …, 𝔞_{m+1}) ∈ R_{m+1}. Put 𝔟 = IA = ⋂_{i=1}^{m+1} 𝔞_{i} . We have exact sequences for the following m-tuples of ideals

B = DA = (R, 𝔞_{1}, …, 𝔞_{m}),
\overline{B} = (R/𝔟, 𝔞_{1}/𝔟, …, 𝔞_{m}/𝔟)

and

(R, 𝔞_{1}, …, 𝔞_{m−1}, 𝔟).

These m-tuples are normal and their K-groups fit into a commutative diagram

[figure: K_{1}(DB); K_{1}(MB); K_{1}(B); K_{1}(D\overline{B}); K_{0}(B); K_{2}(MB); K_{1}(\overline{B}); K_{0}(𝔟); K_{0}(DA).]

Let the dashed arrow be the composition K_{1}(\overline{B}) → K_{1}(D\overline{B}) → K_{0}(𝔟). By an easy diagram chase we see that the sequence with the dashed arrow is exact as well. The identity on R is a morphism

(R, 𝔞_{1}, …, 𝔞_{m}, 𝔟) → A

in R_{m+1}. So we have a commutative diagram with exact rows:

K_{1}(R, 𝔞_{1}, …, 𝔞_{m}, 𝔟) ⟶ K_{1}(B) ⟶ K_{1}(\overline{B}) ⟶ K_{0}(𝔟)
↓  ↓^{1}  ↓^{α}
K_{1}(A)  ⟶ K_{1}(DA) ⟶ K_{1}(MA).

It now suffices to show that the morphism α in this diagram is an isomorphism. The (m + 1)-tuple (R/𝔟, 𝔞_{1}/𝔟, …, 𝔞_{m+1}/𝔟) induces an exact sequence

K_{1}(R/𝔟, 𝔞_{1}/𝔟, …, 𝔞_{m+1}/𝔟) → K_{1}(\overline{B}) → K_{1}(MA).

The group K_{1}(R/𝔟, 𝔞_{1}/𝔟, …, 𝔞_{m+1}/𝔟) is a quotient of GL((𝔞_{1}/𝔟) ∩ ⋯ ∩ (𝔞_{m+1}/𝔟)) = {1}, so α is injective. On the other hand, since the (m + 1)-tuple A of ideals is normal, the identity on R induces an isomorphism I(\overline{B}) → I(MA) and hence also an isomorphism

GL(I(\overline{B})) ⟶^{∼} GL(I(MA)).

Since the multirelative K_{1} is a quotient of the general linear group of the underlying ideal, the map α is surjective. This proves:

Theorem 1. Let A ∈ R_{m} for m ≥ 1. Then we have a functorial exact sequence

⋯ → K_{n}(A) → K_{n}(DA) → K_{n}(MA) → K_{n−1}(A) → ⋯ → K_{0}(MA).

7 Axioms for multirelative K-theory

It will be shown in this section that an axiomatic approach to multirelative K-theory is possible. We take some of the properties of multirelative K-groups as axioms and show that they determine all of multirelative K-theory.

Axioms

Multirelative K-theory consists of functors

K_{n}: R_{m} → A for m and n integers ≥ 0,

morphisms

δ: K_{n+1}M → K_{n} (for m and n integers ≥ 0)

of functors R_{m+1} → A and morphisms

ι: K_{n} → K_{n}D (for m and n integers ≥ 0)

of functors R_{m+1} → A, such that

(MK1) the following sequence is an exact sequence of functors R_{m+1} → A for all non-negative integers m and n

K_{n+1}D ⟶^{K_{n+1}ϕ} K_{n+1}M ⟶^{δ} K_{n} ⟶^{ι} K_{n}D ⟶^{K_{n}ϕ} K_{n}M.

(MK2) K_{n}(R) = 0 for all n ≥ 0 and all free associative non-unital rings R,

(MK3) K_{0}(A) = K_{0}(IA) for all A ∈ R_{m} for all m.

Loosely speaking, the multirelative K-groups are only defined for normal mtuples of ideals and they fit into exact sequences the way one can expect, the (absolute) K-groups of free non-unital rings are trivial and the multirelative K_{0} is just the Grothendieck group of the intersection of the ideals.

Let (R, 𝔞_{1}, …, 𝔞_{m}) be a normal m-tuple of ideals. It induces an m-cube

I ↦ R_{I} = R / ∑_{i∈I} 𝔞_{i},

which is split in S. Application of Fr gives an m-cube

I ↦ Fr(R_{I})

of aspherical simplicial rings, which is dimensionwise split in R.

Proposition 7.1. Let m and n be positive integers. Then the (m + n)-tuple

(Fr(R)_{n−1}, Fr(R, 𝔞_{1})_{n−1}, …, Fr(R, 𝔞_{m})_{n−1}, Ker(d_{0}^{(n−1)}), …, Ker(d_{n−1}^{(n−1)}))

is normal.

Proof. First we show that the induced (m + n)-cube is

(I_{1}, I_{2}) ↦ Fr(R_{I_{1}})_{n−1−#(I_{2})},

where the cube is indexed by pairs of subsets of \underline{m} and [n − 1]. This set of pairs is ordered by componentwise inclusion:

(I_{1}, I_{2}) ≤ (J_{1}, J_{2}) ⟺ I_{1} ⊆ J_{1} and I_{2} ⊆ J_{2}.

The homomorphism

Fr(R)_{n−1} → Fr(R_{I_{1}})_{n−1−#(I_{2})}

is the composition

Fr(R)_{n−1} → Fr(R_{I_{1}})_{n−1} → Fr(R_{I_{1}})_{n−1−#(I_{2})},

the first map being induced by ∅ ⊆ I_{1} and the second by [n − 1] \ I_{2} ⊆ [n − 1]. Both homomorphisms are surjective. The first one has kernel ⋂_{i∈I_{1}} Fr(R, 𝔞_{i})_{(n−1)} and the second one ⋂_{i∉I_{2}} Ker(d_{i}), where the d_{i} are face maps of Fr(R_{I_{1}}). Since Fr(R) and Fr(R_{I_{1}}) are both aspherical, elements of the second kernel can be lifted to elements of ⋂_{i∉I_{2}} Ker(d_{i}), where the d_{i} are face maps of Fr(R).

For the (m + n)-tuple to be normal it suffices that the intersections of the images of the m+n ideals are preserved under the maps on the edges of the induced (m+n)- cube. These are the homomorphisms

Fr(R_{J})_{l} → Fr(R_{J∪{k}})_{l},

where J ⊆ \underline{m}, k ∈ \underline{m} \ J and l ∈ [n − 1], and also the face maps

d_{i}: Fr(R_{J})_{p} → Fr(R_{J})_{p−1},

where p ∈ [n − 1] and 0 ≤ i ≤ p. Without loss of generality we may assume that J = \underline{m}, l = n − 1 and p = n − 1.

Because the m-cube J ↦ Fr(R_{J}) is dimensionwise split we have short exact sequences

0 → ⋂_{i∈I∪\{k\}} Fr(R, 𝔞_{i}) → ⋂_{i∈I} Fr(R, 𝔞_{i}) → ⋂_{i∈I} Fr(R/𝔞_{k}, \overline{𝔞}_{i}) → 0

of aspherical simplicial rings. It follows that for all J ⊆ [n − 1] we have

⋂_{i∈I} Fr(R, 𝔞_{i})_{n−1} ∩ ⋂_{j∈J} Ker(d_{j}) = ⋂_{j∈J} Ker(d′_{j}),

where the d′_{j} are the face maps of ⋂_{i∈I} Fr(R, 𝔞_{i}). Under Fr(R) → Fr(R/𝔞_{k}) this maps onto

⋂_{j∈J} Ker(d′′_{j}) = ⋂_{i∈I} Fr(R/𝔞_{k}, \overline{𝔞}_{i})_{n−1} ∩ ⋂_{j∈J} Ker(d′′′_{j}),

where the d′′_{j} are the face maps of ⋂_{i∈I} Fr(R/𝔞_{k}, \overline{𝔞}_{i}) and d′′′_{j} those of ⋂_{i∈I} Fr(R/𝔞_{k}).

Because the simplicial rings ⋂_{i∈I} Fr(R, 𝔞_{i}) are aspherical also the face maps d_{i}: Fr(R)_{n−1} → Fr(R)_{n−2} preserve intersections

⋂_{i∈I} Fr(R, 𝔞_{i})_{n−1} ∩ ⋂_{j∈J} Ker(d_{j}).

Theorem 2. Let A = (R, 𝔞_{1}, …, 𝔞_{m}) ∈ R. Then for all n ≥ 0 it follows from the axioms (MK1) and (MK2) that K_{n}(A) is naturally isomorphic to K_{0} of the following object of R_{m+n}:

(Fr(R)_{n−1}, Fr(R, 𝔞_{1})_{n−1}, …, Fr(R, 𝔞_{m})_{n−1}, Ker(d_{0}), …, Ker(d_{n−1})).

From axiom (MK3) it then follows that K_{n}(A) is determined. So (MK1), (MK2) and (MK3) can be taken as axioms for the (multirelative) K-theory of rings.

Proof. The proof follows from the following three lemmas.

Lemma 7.1. Let m ≥ −1 and q, n ≥ 0. Then

K_{q}(Fr(R)_{n}, Fr(R, 𝔞_{1})_{n}, …, Fr(R, 𝔞_{m})_{n}) = 0.

Proof. Since for m ≥ 0 the (m − 1)-tuples D(A) and M(A) are of the same type, the proof reduces by (MK1) to the case m = −1. For m = −1 the lemma follows from (MK2).

Put

A[n, p] = (Fr(R)_{n}, Fr(R, 𝔞_{1})_{n}, …, Fr(R, 𝔞_{m})_{n}, Ker(d_{0}), …, Ker(d_{p})),

where −1 ≤ p ≤ n. It is an object of R_{m+p+1}.

Lemma 7.2. For all p < n and all q > 0 we have

K_{q}(A[n, p]) = 0.

Proof. For p ≥ 0 we have

D(A[n, p]) = A[n, p − 1] and M(A[n, p]) = A[n − 1, p − 1].

By (MK1) the problem reduces to the case p = −1, which is covered by the previous lemma.

Lemma 7.3. For all q, n ≥ 0 we have

K_{q}(A[n, n]) ≅ K_{q+1}(A[n − 1, n − 1]).

Proof. This follows from (MK1) and the previous lemma.

From this lemma the theorem follows:

K_{n}(A) = K_{n}(A[−1, −1]) ≅ K_{n−1}(A[0, 0]) ≅ ⋯ ≅ K_{0}(A[n − 1, n − 1]).

References

[1] Bloch, S. and S. Lichtenbaum, A Spectral Sequence for Motivic Cohomology, preprint (K-theory Preprint Archives 62, 1995 Mar 3).

[2] Dayton, B.H. and C.A. Weibel, A spectral sequence for the K-theory of affine glued schemes, in “Algebraic K-theory”, Proceedings Evanston 1980, Lecture Notes in Mathematics 854, Springer-Verlag, Berlin, 1981.

[3] Keune, Frans, Nonabelian derived functors and algebraic K-theory, in “Algebraic K-theory,” Vol. I, Proceedings Seattle 1972, Lecture Notes in Mathematics 341, Springer-Verlag, Berlin, 1973.

[4] Levine, Marc, The Weight Two K-Theory of Fields, K-Theory 9, 443–501 (1995).

Frans Keune

Department of Mathematics University of Nijmegen Toernooiveld 1 6525 ED Nijmegen The Netherlands keune@sci.kun.nl 292

Documenta Mathematica 1 (1996)