Algebraic K-Theory in Low Degrees

Markus J. Pflaum

1 The Grothendieck group of an abelian monoid

Prerequisites

Abelian monoids

Recall that an abelian monoid is a set M together with a binary operation ⊕:M × M→M and a distinguished element 0 such that the following axioms hold true.

(AMon 1): The operation ⊕ is associative that means m1⊕m2⊕m3=m1⊕m2⊕m3 for all m1,m2,m3∈M,
(AMon 2): The element 0 is neutral with respect to ⊕ that means 0⊕m=m⊕0=m for all m∈M,
(AMon 3): The operation ⊕ is commutative that means m1⊕m2=m2⊕m1 for all m1,m2∈M.

The category AMon of abelian monoids is a full subcategory of the category of monoids. Morphisms of AMon are given by maps f:M→M~ between abelian monoids M and M~ with binary operations ⊕ and ⊕~, respectively, such that the following axiom holds true.

(MorMon): For all m1,m2∈M the relation f⁢m1⊕m2=f⁢m1⁢⊕~⁢f⁢m2 holds true.

Objective

The category AGrp of abelian groups is a full subcategory of AMon. The main goal of the following considerations is to construct a left adjoint to the embedding functor ι:AGrp↪AMon.

Construction of the Grothendieck group

Definition 1.1.

Let M be an abelian monoid. An abelian group K together with a morphism κ:M→K of monoids is called a Grothendieck group of M, if the following universal property is satisfied:

(Gro)

For every abelian group A and every morphism of monoids f:M→A there exists a unique homomorphism of groups fK:K→A such that the following diagram commutes.

(1.1)

Clearly, if a Grothendieck group exists for M, then it is unique up to isomorphism by the universal property. Let us show that for every ablian monoid M there exists a Grothendieck group. To this end let F⁢M be the free abelian group generated by the elements of M, and denote for every m∈M by m¯ the image of m in F⁢M under the canonical injection M↪F⁢M. Let R⁢M⊂F⁢M be the (necessarily free) subgroup generated by all elements of the form m1⊕m2¯-m¯1-m¯2, where m1,m2∈M and where - denotes subtraction within the abelian group F⁢M. Then the following holds true.

Proposition 1.2.

For every abelian monoid M, the abelian group KGro⁢M:=F⁢M/R⁢M together with the canonical morphism of monoids κMGro:M→KGro⁢A, m↦m¯+R⁢M is a Grothendieck group for M.

Proof.

Let A be an abelian group and M→A a morphism of monoids. By the universal property of F⁢M, there exists a unique group homomorphism fF⁢M:F⁢M→A such that the diagram

(1.2)

commutes, where M→F⁢M is the canonical embedding. Observe now that for all m1,m2∈M

	fF⁢M⁢m1⊕m2¯-m1¯-m2¯	=fF⁢M⁢m1⊕m2¯-fF⁢M⁢m1¯-fF⁢M⁢m2¯
		=f⁢m1⊕m2-f⁢m1-f⁢m2
		=fm1+fm2-fm1-fm2=0,

hence fF⁢M factorizes through the map F⁢M→KGro⁢M. In other words this means that there exists a homomorphism fKGro⁢M:KGro⁢M→A such that the diagram

(1.3)

commutes. By the universal property of the free abelian group F⁢M, the homomorphism fF⁢M is uniquely determined by f. Since F⁢M→KGro⁢M is an epimorphism, fKGro⁢M is uniquely determined by fF⁢M, hence uniqueness of fKGro⁢M follows. This proves the claim, since the composition of the two vertical arrows in Diagram (1.3) coincides with κMGro. ∎

From now on, we will denote by m the equivalence class of an element m∈M in the Grothendieck group KGro⁢M. As we will see later, the map M→KGro, m↦m need not be injective, in general.

Another representation of the Grothendieck group

Next, let us provide a second representation for KGro. To this end consider the map

λ:M × M→KGro,m1,m2↦m1-m2.

By construction of KGro, this map must be surjective. Note that M × M inherits the structure of a commutative monoid from M. Let us determine, when λ⁢m1,n1=λ⁢m2,n2 for m1,m2,n1,n2∈M. The following observation is crucial for this.

Lemma 1.3.

For all m1,m2∈M one has m1=m2 in KGro if and only if there is an n∈M such that m1⊕n=m2⊕n.

Definition 1.4.

Two elements m1,m2 of an abelian monoid M are called stably equivalent, if there is an n∈M such that m1⊕n=m2⊕n.

Proof of the Lemma.

If m1 and m2 are stably equivalent, the relation m1=m2 follows immediately:

m1=m1⊕n-n=m2⊕n-n=m2.

It remains to show that m1=m2 implies the existence of an n∈M such that m1⊕n=m2⊕n. By construction of KGro⁢M, there exist elements a1,…,ak,a1′,…⁢ak′,b1,…,bl,b1′,…⁢bl′∈M for some k,l∈ℕ such that in F⁢M the following relation holds true:

m1-m2=∑i=1kai⊕ai′-ai-ai′-∑j=1lbi⊕bi′-bi-bi′.

This implies that in F⁢M, the following equation holds:

m1+∑i=1kai+ai′+∑j=1lbj⊕bj′=m2+∑i=1kai⊕ai′+∑j=1lbj+bj.

Since F⁢M is free on elements of M, one concludes that the summands appearing on the left side of the equation are a permutation of the summands appearing on the right side. Hence

m1⊕⨁i=1kai⊕ai′⊕⨁j=1lbj⊕bj′=m2⊕⨁i=1kai⊕ai′⊕⨁j=1lbj⊕bj.

Putting n:=⨁i=1kai⊕ai′⊕⨁j=1lbj⊕bj′, one obtains m1⊕n=m2⊕n. This finishes the proof. ∎

Let us come back to our original problem and assume that λ⁢m1,n1=λ⁢m2,n2. Then one concludes

m1+n2=m2+n1,

hence by the lemma there exists n∈M such that

(1.4)

m1+n2+n=m2+n1+n.

If one defines now m1,n1∼m2,n2 for m1,m2,n1,n2∈M if there exists n∈M such that Eq. (1.4) holds true, then the lemma implies that λ⁢m1,n1=λ⁢m2,n2 exactly when m1,n1∼m2,n2.

Lemma 1.5.

The relation ∼ on M×M is a congruence relation. This means in particular that for all m1,m2,n1,n2,a,b∈M such that m1,n1∼m2,n2 the relation

m1⊕a,n1⊕b∼m2⊕a,n2⊕b

holds true.

Proof.

Clearly, the relation ∼ is symmetric and reflexive. let us show that it is transitive. To this end, assume m1,n1∼m2,n2 and m2,n2∼m3,n3. Then there exist n,n′∈M such that

m1⊕n2⊕n=m2⊕n1⊕n⁢ and ⁢m2⊕n3⊕n′=m3⊕n2⊕n′.

Adding the two equalities, one obtains

m1⊕n3⊕m2⊕n2⊕n⊕n′=m3⊕n1⊕m2⊕n2⊕n⊕n′,

which proves that ∼ is transitive. If m1⊕n2⊕n=m2⊕n1⊕n, then

m1⊕a⊕n2⊕b⊕n=m2⊕a⊕n1⊕b⊕n,

which entails that ∼ is even a congruence relation. ∎

Proposition 1.6.

For every commutative monoid M, the quotient space M×M/∼ of equivalence classes of the congruence relation ∼ is an abelian group which is canonically isomorphic to KGro⁢M.

Proof.

Since ∼ is a congruence relation, M × M/∼ inherits from M × M the structure of an abelian monoid. Moreover, since m,n⊕n,m∼0,0, every element of M × M/∼ has an inverse, thus M × M/∼ is an abelian group. Since λ⁢m1,n1=λ⁢m2,n2 if and only if m1,n1∼m2,n2 and since λ is surjective, it follows immediately that the quotient map λ¯:M × M/∼→KGro(M) is well-defined and an isomorphism. ∎

Functorial properties

Sofar, we have defined KGro only on objects of the category of abelian monoids. Let us now extend KGro to a functor KGro:AMon→AGrp. Assume to be given two abelian monoids M,N and a morphism of monoids f:M→N. By the universal property of the Grothendieck group KGro⁢M there exists a uniquely determined group homomorphism, which we denote KGro⁢f, such that the following diagram commutes.

(1.5)

This in particular entails that

KGro⁢i⁢dM=i⁢dKGro⁢M⁢ and ⁢KGro⁢f2∘f1=KGro⁢f2∘KGro⁢f1

for abelian monoids M,M1,M2,M3 and morphisms f1:M1→M2 and f2:M2→M3. Hence KGro is a functor from the category of abelian monoids to the category of abelian groups, indeed. One sometimes calls this functor the Grothendieck K-functor.

Theorem 1.7.

The Grothendieck-functor KGro:AMon→AGrp is left adjoint to the forgetful functor ι:AGrp↪AMon.

Proof.

Let M be an abelian monoid, A an abelian group, and consider the map

κMGro*:AGrp⁢KGro⁢M,A→AMon⁢M,ι⁢A,f↦f∘κMGro.

By Diagram (1.5), this map is natural in M. Naturality in A is obvious by definition. Moreover, since KGro satisfies the universal property (Gro) in Definition 1.1, κMGro* is even bijective. The claim follows. ∎

Basic examples

Remark 1.8.

Sometimes it happens that a set M carries two binary operations ⊕ and ⊗ which both induce on M the structure of an abelian monoid. To distinguish the corresponding two, possibly different, Grothendieck groups we denote them in such a situation by KGro(M,⊕) and KGro(M,⊗), respectively.

Example 1.9.

Consider the abelian monoid of natural numbers (ℕ,+) with addition as binary operation. Then KGro(ℕ,+)=(ℤ,+). On the other hand, one has KGro(ℕ,⋅)={0}, but KGro(ℕ*,⋅)=(ℚ>0,⋅).
If A is an abelian group, then by the universal property of the Grothendieck group one immediately obtains KGro⁢A=A.
Consider the set of non-zero integres ℤ* with multiplication ⋅ as binary operation. Then KGro(ℤ*,⋅)=(ℚ,⋅).
Let X be a compact topological space, and Vecℂ⁢X the category of complex vector bundles over X. Since every complex vector bundle over X is isomorphic to a subbundle of some trivial bundle X × ℂn, the category of isomorphism classes of complex vector budnles over X is small. Denote by Iso⁢Vecℂ⁢X its set of objects. Then the direct sum of vector bundles over X induces the structure of an abelian monoid on Iso⁢Vecℂ⁢X. The isomorphism class of the trivial vector bundle X × 0 of fiber dimension 0 serves as the zero element in Iso⁢Vecℂ⁢X. The K-theory of the space X (in degree 0) is now defined as the Grothendieck-group of Iso⁢Vecℂ⁢X that means as the abelian group

K0⁢X:=KGro⁢Iso⁢Vecℂ⁢X.

For further reading on the K-theory of compact topological spaces see [1, 2].

2 The functor K0 for a unital ring

Definition and fundamental properties

Let R be a unital (but possibly noncommutative) ring, and R⁢-⁢Modfp the category of finitely generated projective left modules over R.

Proposition 2.1.

The category of isomorphism classes of finitely generated projective left R-modules is small. Denote by Iso⁢R⁢-⁢Modfp the set of isomorphism classes of finitely generated projective left R-modules. Then the direct sum in the abelian category R⁢-⁢Mod induces on Iso⁢R⁢-⁢Modfp the structure of an abelian monoid.

Proof.

Every finitely generated projective left R-module is isomorphic to a direct summand of some Rn, n∈ℕ, and the finitely generated projective left R-modules are characterized by this property. From this, it follows immediately that Iso⁢R⁢-⁢Modfp is small. To check the second claim, let f:M1→M2 and g:N1→N2 be two isomorphisms in R⁢-⁢Modfp. Then f,g:M1⊕N1→M2⊕N2 is an isomorphism as well, hence ⊕ descends to a binary operation on Iso⁢R⁢-⁢Modfp which we will denote by the same symbol:

⊕:Iso(R-Modfp) × Iso(R-Modfp)→Iso(R-Modfp).

It is immediate to prove that ⊕ is associative and commutative on Iso⁢R⁢-⁢Modfp, and that the equivalence class of the zero module serves as neutral element. This proves the proposition. ∎

Definition 2.2.

For every unital ring R one defines K0⁢R, the K-theory of order 0 of R, by

K0⁢R:=KGro⁢Iso⁢R⁢-⁢Modfp.

Proposition 2.3.

Two finitely generated projective left R-modules M and N represent the same element in K0⁢R if and only if M⊕Rn≅N⊕Rn for some n∈ℕ.

Proof.

Clearly, if M⊕Rn≅N⊕Rn and M, N denote the equivalence classes of M respectively N in K0⁢R, then the equation

M=M⊕Rn-Rn=N⊕Rn-Rn=N

follows immediately. It remains to show the converse. Assume that M=N. Then, by definition of KGro⁢Iso⁢R⁢-⁢Modfp there exist finitely generated projective left R-modules Ai,Ai′,Bi,Bi′, i=1,…,k such that in F⁢Iso⁢R⁢-⁢Modfp, the free abelian group over the set of isomorphism classes of finitely generated projective left R-modules, the equality

M¯-N¯=∑i=1kAi⊕Ai′¯-Ai¯-Ai′¯-∑i=1kBi⊕Bi′¯-Bi¯-Bi′¯

holds true, where we have denoted by M¯ the image of M in F⁢Iso⁢R⁢-⁢Modfp and likewise for the other left R-modules. This implies that

M¯+∑i=1kBi⊕Bi′¯+∑i=1kAi¯+Ai′¯=N¯+∑i=1kAi⊕Ai′¯+∑i=1kBi¯+Bi′¯

which means that the R-modules appearing as summands on the left hand side are permutations of the summands appearing on the right hand side. Thus, in Iso⁢R⁢-⁢Modfp, the following equality holds true.

M⊕⨁i=1kBi⊕Bi′⊕⨁i=1kAi⊕Ai′=N⊕⨁i=1kAi⊕Ai′⊕⨁i=1kBi⊕Bi′

Hence we obtain M⊕P≅N⊕P for

P:=⨁i=1kAi⊕Ai′⊕⨁i=1kBi⊕Bi′.

Since P is a finitely generated projective left R-module, there exists a left R-module Q such that P⊕Q≅Rn for some n∈ℕ. This entails

M⊕Rn≅M⊕P⊕Q≅N⊕P⊕Q≅N⊕Rn,

and the claim follows. ∎

Remark 2.4.

Note that since a finitely projective R-module is a direct sum of some Rn, the relation M⊕Rn≅N⊕Rn holds true, if and only if M and N are stably equivalent in the sense of Definition 1.4. This observation also shows that Proposition 2.3 is a direct consequence of Lemma 1.3.
Sometimes, one writes K0alg⁢R instead of K0⁢R to emphasize that one considers the algebraic K-theory of the ring R and not a topological version of K-theory. Note, however, that for a Banach-algebra A the topological K-theory of A in degree 0 coincides with its algebraic K-theory as defined above. This means in particular, that in this case the not so precise notation K0⁢A will not lead to any confusion.

Basic examples

Example 2.5.

Let 𝕜 be a field. A finitely generated projective module over 𝕜 is a 𝕜-vector space of finite dimension. The isomorphism classes of finitely generated projective 𝕜-modules are therefore uniquely determined by dimension. Moreover, under this characterization, the isomorphism class of the direct sum of two finitely generated projective 𝕜-modules corresponds to the sum of the dimensions of the two modules. Hence, by Example 1.9.1.9 it follows that

K0⁢𝕜≅KGro⁢ℕ=ℤ.
Let X be a compact topological space. Recall that by the Serre–Swan Theorem the category Vecℂ⁢X of complex vector bundles over X is equivalent to the category of finitely generated projective modules over the algebra 𝒞⁢X of continuous functions on X, hence one has a natural isomorphism of monoids

Iso⁢Vecℂ⁢X≅Iso⁢𝒞⁢X⁢-⁢Modfp.

By Example 1.9.1.9 the K-theory of 𝒞⁢X then has to coincide with the K-theory of the space X (in degree 0):

K0⁢𝒞⁢X≅K0⁢X.

Note that, if X is a smooth manifold, one even has

K0⁢𝒞∞⁢X≅K0⁢X,

where 𝒞∞⁢X denotes the algebra of smooth functions on X.

3 The functor K1alg for a unital ring

Prerequisites

Groups of invertible infinite matrices.

Let R be a unital ring, and n∈ℕ*. Recall that by GLn⁢R⊂𝔐n × n⁢R one denotes the group of invertible n × n-matrices with entries in R. For natural n≥m>0 one has a natural embedding ιn⁢m:GLm⁢R↪GLn which is defined by the requirement that r=ri⁢j1≤i,j≤m∈GLm⁢R is mapped to the matrix ιn⁢m⁢r∈GLn⁢R with entries

ιn⁢m⁢ri⁢j:=ri⁢j,if 1≤i,j≤m,1,if i=j and m<i≤n,0,if i≠j and i>m or j>m.

By definition, GLn⁢Rn∈ℕ*,ιn⁢mm≤n then forms a direct system of groups. It has a direct limit which is denoted by GL∞⁢R and which can be represented as the set of all matrices r=ri⁢ji,j∈ℕ with entries ri⁢j∈R for which there is an n∈ℕ such that

(3.1)

ri⁢j1≤i,j≤n∈GLn⁢R⁢ and ⁢ri⁢j=1,if i,j>n and i=j,0,if i>n or j>n and i≠j.

The product of two elements r,r~∈GL∞⁢R is given by

r⋅r~:=s, where ⁢si⁢j:=∑k∈ℕri⁢l⋅r~k⁢j.

It is immediate to check that r⋅r~ is an element of GL∞⁢R. The unit element in GL∞⁢R is given by the matrix e with components

ei⁢j:=1,if i=j,0,if i≠j.

The set GL∞⁢R of matrices ri⁢ji,j∈ℕ satisfying (3.1) together with the product ⋅ froms a group indeed. Moreover, for every n∈ℕ* there is a natural embedding ιn:GLn⁢R↪GL∞⁢R which is defined by the requirement that r=ri⁢j1≤i,j≤n∈GLn⁢R is mapped to the matrix ιn⁢r∈GL∞⁢R with entries

ιn⁢ri⁢j:=ri⁢j,if 1≤i,j≤n,1,if i=j and n<i,0,if i≠j and i>n or j>n.

It is straightforward to prove that GL∞⁢R,ιnn∈ℕ* is the direct limit of GLn⁢Rn∈ℕ*,ιn⁢mm≤n indeed. Sometimes, one calls GL∞⁢R the group of invertible infinite matrices over R.

Groups of elementary matrics.

Let us now recall the definition and basic properties of the group of elementary matrices. To this denote for λ∈R, n∈ℕ*∪∞ and all integers i≠j with 1≤i,j<n+1 by ei⁢jn⁢λ the matrix in GLn⁢R having entry λ at the i-th row and j-th column, entry 1 at all diagonal elements, and 0 at all other places. In other words, this means

ei⁢jn⁢λk⁢l:=1,if k=l and 1≤k<n+1,λ,if k=i and l=j,0,if k≠l, k,l≠i,j, and 1≤k,l<n+1.

A matrix of the form ei⁢jn⁢λ is called an elementary matrix over R of order n. The subgroup of GLn⁢R generated by all elementary matrices over R of order n is called the group of elementary matrix over R of order n and is denoted by En⁡R. By slight abuse of language one sometimes calls E∞⁡R the group of elementary matrices over R.

It is immediate to check that under the group homomorphism ιn⁢m from above with 0<m≤n<∞, the group Em⁡R is mapped into En⁡R, and that En⁡Rn∈ℕ*,ιn⁢mm≤n is a direct system of groups. The direct limit of this direct systems is E∞⁡R as one easily checks.

Recall that for two elements g,h of a group G one denotes by g,h the commutator g⁢h⁢g-1⁢h-1. With this notation, the following holds true.

Proposition 3.1.

The elementary matrics ei⁢j∞⁢λ, ei⁢j∞⁢μ satisfy for all λ,μ∈R the following relations.

ei⁢j∞⁢λ⋅ei⁢j∞⁢μ=ei⁢j∞⁢λ+μ, if i≠j,
ei⁢j∞⁢λ,ek⁢l∞⁢μ=1, if i≠j, k≠l, j≠k, and i≠l,
ei⁢j∞⁢λ,ej⁢l∞⁢μ=ei⁢l∞⁢λ⋅μ, if i≠j, i≠l, and j≠l,
ei⁢j∞⁢λ,ek⁢i∞⁢μ=ek⁢j∞⁢-μ⋅λ, if i≠j, i≠k, and j≠k.

The essential tool for the proof of the proposition is the following result.

Lemma 3.2.

For 1≤i,j,k,l≤n with i≠j, k≠l and i≠l or j≠k one has

	ei⁢jn⁢λ⋅ek⁢ln⁢μ=	1-δi⁢l⁢ei⁢ln⁢λ⁢μ⁢δj⁢k+λ⁢δj⁢l+μ⁢δi⁢k+
		+1-δj⁢k⁢δi⁢l⁢1+
		+1-δi⁢k⁢ek⁢ln⁢μ-1+1-δj⁢l⁢ei⁢jn⁢λ-1,

where δr⁢s denotes the Kronecker symbol, i.e. δr⁢s=1 for r=s and δr⁢s=0 for r≠s.

Proof of the Lemma.

Let us compute the components of the matrix ei⁢jn⁢λ⁢ek⁢ln⁢μ.

	(ei⁢jn(λ)	ek⁢ln(μ))r⁢s=∑t(ei⁢jn(λ))r⁢t(ek⁢ln(μ))t⁢s=
		=∑tδr⁢t⁢δt⁢s=δr⁢s,for r≠i, s≠l,∑tδr⁢t⁢δt⁢s+λ⁢δj⁢s=δi⁢s,for r=i, s≠l, j=l,∑tδr⁢t⁢δt⁢s+λ⁢δj⁢s=δi⁢s+λ⁢δj⁢s,for r=i, s≠l, j≠l,∑tδr⁢t⁢δt⁢s+μ⁢δr⁢k=δr⁢l,for r≠i, s=l, j=k,∑tδr⁢t⁢δt⁢s+μ⁢δr⁢k=δr⁢l+μ⁢δr⁢k,for r≠i, s=l, j≠k,λ⁢μ⁢δj⁢k+λ⁢δj⁢l+μ⁢δi⁢k,for r=i, s=l, i≠l,1,for r=i, s=l, i=l, j≠k.

The claim follows. ∎

Proof of the Proposition.

∎

Definition and fundamental properties of K1alg

Definition 3.3.

Let R be a unital ring. Then K1alg⁢R, the algebraic K-theory of degree 1 of R, is defined as the abelian group

K1alg⁢R:=GL∞⁢R/GL∞⁢R,GL∞⁢R.

Proposition 3.4.

For every unital ring R the following equality holds true.

K1alg⁢R=GL∞ab⁢R=GL∞⁢R/E∞⁢R,E∞⁢R=GL∞⁢R/E∞⁢R.

Proof.

The claim is immediate by definition of K1alg⁢R and results from elementary matrix theory as stated in the prerequisites. ∎

References

M. F. Atiyah (1989)
K-Theory,

Second edition, Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA.

Note: notes by D. W. Anderson

External Links: MathReview Entry.

Cited by: Example 1.9.
M. Karoubi (2008)
K-Theory,

Classics in Mathematics, Springer-Verlag, Berlin.

Note: An introduction, Reprint of the 1978 edition, With a new postface by the author and a list of errata

External Links: ISBN 978-3-540-79889-7, Document, MathReview Entry.

Cited by: Example 1.9.