Algebraic K-Theory in Low Degrees
Markus J. Pflaum
January 29, 2011

## 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 × MM and a distinguished element 0 such that the following axioms hold true.

(AMon 1)

The operation is associative that means m1m2m3=m1m2m3 for all m1,m2,m3M,

(AMon 2)

The element 0 is neutral with respect to that means 0m=m0=m for all mM,

(AMon 3)

The operation is commutative that means m1m2=m2m1 for all m1,m2M.

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

(MorMon)

For all m1,m2M the relation fm1m2=fm1~fm2 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:AGrpAMon.

### Construction of the Grothendieck group

###### Definition 1.1.

Let M be an abelian monoid. An abelian group K together with a morphism κ:MK 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:MA there exists a unique homomorphism of groups fK:KA such that the following diagram commutes.

 (1.1) Algebraic K-Theory in Low Degrees Diagram 1 Section 1 $M$ $\kappa$ $f$ $A$ $K$ ${f}_{K}$

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 FM be the free abelian group generated by the elements of M, and denote for every mM by m¯ the image of m in FM under the canonical injection MFM. Let RMFM be the (necessarily free) subgroup generated by all elements of the form m1m2¯-m¯1-m¯2, where m1,m2M and where - denotes subtraction within the abelian group FM. Then the following holds true.

###### Proposition 1.2.

For every abelian monoid M, the abelian group KGroM:=FM/RM together with the canonical morphism of monoids κMGro:MKGroA, mm¯+RM is a Grothendieck group for M.

###### Proof.

Let A be an abelian group and MA a morphism of monoids. By the universal property of FM, there exists a unique group homomorphism fFM:FMA such that the diagram

 (1.2) Algebraic K-Theory in Low Degrees Diagram 2 Section 1 $M$ $f$ $A$ $\mathrm{F}\left(M\right)$ ${f}_{\mathrm{F}\left(M\right)}$

commutes, where MFM is the canonical embedding. Observe now that for all m1,m2M

 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 fFM factorizes through the map FMKGroM. In other words this means that there exists a homomorphism fKGroM:KGroMA such that the diagram

 (1.3) Algebraic K-Theory in Low Degrees Diagram 3 Section 1 $M$ $f$ $A$ $\mathrm{F}\left(M\right)$ ${f}_{\mathrm{F}\left(M\right)}$ ${\mathrm{K}}^{\text{Gro}}\left(M\right)$ ${f}_{{\mathrm{K}}^{\text{Gro}}\left(M\right)}$

commutes. By the universal property of the free abelian group FM, the homomorphism fFM is uniquely determined by f. Since FMKGroM is an epimorphism, fKGroM is uniquely determined by fFM, hence uniqueness of fKGroM 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 mM in the Grothendieck group KGroM. As we will see later, the map MKGro, mm 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 λm1,n1=λm2,n2 for m1,m2,n1,n2M. The following observation is crucial for this.

###### Lemma 1.3.

For all m1,m2M one has m1=m2 in KGro if and only if there is an nM such that m1n=m2n.

###### Definition 1.4.

Two elements m1,m2 of an abelian monoid M are called stably equivalent, if there is an nM such that m1n=m2n.

###### 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 nM such that m1n=m2n. By construction of KGroM, there exist elements a1,,ak,a1,ak,b1,,bl,b1,blM for some k,l such that in FM the following relation holds true:

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

This implies that in FM, the following equation holds:

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

Since FM 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=1kaiaij=1lbjbj, one obtains m1n=m2n. This finishes the proof. ∎

Let us come back to our original problem and assume that λm1,n1=λm2,n2. Then one concludes

 m1+n2=m2+n1,

hence by the lemma there exists nM such that

 (1.4) m1+n2+n=m2+n1+n.

If one defines now m1,n1m2,n2 for m1,m2,n1,n2M if there exists nM such that Eq. (1.4) holds true, then the lemma implies that λm1,n1=λm2,n2 exactly when m1,n1m2,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,bM such that m1,n1m2,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,n1m2,n2 and m2,n2m3,n3. Then there exist n,nM 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 m1n2n=m2n1n, 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 KGroM.

###### Proof.

Since is a congruence relation, M × M/ inherits from M × M the structure of an abelian monoid. Moreover, since m,nn,m0,0, every element of M × M/ has an inverse, thus M × M/ is an abelian group. Since λm1,n1=λm2,n2 if and only if m1,n1m2,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:AMonAGrp. Assume to be given two abelian monoids M,N and a morphism of monoids f:MN. By the universal property of the Grothendieck group KGroM there exists a uniquely determined group homomorphism, which we denote KGrof, such that the following diagram commutes.

 (1.5) Algebraic K-Theory in Low Degrees Diagram 4 Section 1 $M$ ${\mathrm{K}}^{\text{Gro}}\left(M\right)$ $N$ ${\mathrm{K}}^{\text{Gro}}\left(N\right)$ ${\mathrm{K}}^{\text{Gro}}\left(f\right)$ ${\kappa }_{N}^{\text{Gro}}$ $f$ ${\kappa }_{M}^{\text{Gro}}$

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:M1M2 and f2:M2M3. 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:AMonAGrp is left adjoint to the forgetful functor ι:AGrpAMon.

###### 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.
1. Consider the abelian monoid of natural numbers (,+) with addition as binary operation. Then KGro(,+)=(,+). On the other hand, one has KGro(,)={0}, but KGro(*,)=(>0,).

2. If A is an abelian group, then by the universal property of the Grothendieck group one immediately obtains KGroA=A.

3. Consider the set of non-zero integres * with multiplication as binary operation. Then KGro(*,)=(,).

4. Let X be a compact topological space, and VecX 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 × n, the category of isomorphism classes of complex vector budnles over X is small. Denote by IsoVecX its set of objects. Then the direct sum of vector bundles over X induces the structure of an abelian monoid on IsoVecX. The isomorphism class of the trivial vector bundle X × 0 of fiber dimension 0 serves as the zero element in IsoVecX. The K-theory of the space X (in degree 0) is now defined as the Grothendieck-group of IsoVecX 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 IsoR-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 IsoR-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, and the finitely generated projective left R-modules are characterized by this property. From this, it follows immediately that IsoR-Modfp is small. To check the second claim, let f:M1M2 and g:N1N2 be two isomorphisms in R-Modfp. Then f,g:M1N1M2N2 is an isomorphism as well, hence descends to a binary operation on IsoR-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 IsoR-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 K0R, 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 K0R if and only if MRnNRn for some n.

###### Proof.

Clearly, if MRnNRn and M, N denote the equivalence classes of M respectively N in K0R, 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 KGroIsoR-Modfp there exist finitely generated projective left R-modules Ai,Ai,Bi,Bi, i=1,,k such that in FIsoR-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 FIsoR-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 IsoR-Modfp, the following equality holds true.

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

Hence we obtain MPNP 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 PQRn for some n. This entails

 M⊕Rn≅M⊕P⊕Q≅N⊕P⊕Q≅N⊕Rn,

and the claim follows. ∎

###### Remark 2.4.
1. Note that since a finitely projective R-module is a direct sum of some Rn, the relation MRnNRn 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.

2. Sometimes, one writes K0algR instead of K0R 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 K0A will not lead to any confusion.

### Basic examples

###### Example 2.5.
1. 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.
2. Let X be a compact topological space. Recall that by the Serre–Swan Theorem the category VecX of complex vector bundles over X is equivalent to the category of finitely generated projective modules over the algebra 𝒞X 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 𝒞X 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 𝒞X 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*. Recall that by GLnR𝔐n × nR one denotes the group of invertible n × n-matrices with entries in R. For natural nm>0 one has a natural embedding ιnm:GLmRGLn which is defined by the requirement that r=rij1i,jmGLmR is mapped to the matrix ιnmrGLnR with entries

 ιn⁢m⁢ri⁢j:=ri⁢j,if 1≤i,j≤m,1,if i=j    and   mm or j>m.

By definition, GLnRn*,ιnmmn then forms a direct system of groups. It has a direct limit which is denoted by GLR and which can be represented as the set of all matrices r=riji,j with entries rijR for which there is an 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~GLR is given by

 r⋅r~:=s, where ⁢si⁢j:=∑k∈ℕri⁢l⋅r~k⁢j.

It is immediate to check that rr~ is an element of GLR. The unit element in GLR is given by the matrix e with components

 ei⁢j:=1,if i=j,0,if i≠j.

The set GLR of matrices riji,j satisfying (3.1) together with the product froms a group indeed. Moreover, for every n* there is a natural embedding ιn:GLnRGLR which is defined by the requirement that r=rij1i,jnGLnR is mapped to the matrix ιnrGLR with entries

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

It is straightforward to prove that GLR,ιnn* is the direct limit of GLnRn*,ιnmmn indeed. Sometimes, one calls GLR 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 λR, n* and all integers ij with 1i,j<n+1 by eijnλ the matrix in GLnR 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

A matrix of the form eijnλ is called an elementary matrix over R of order n. The subgroup of GLnR 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 EnR. By slight abuse of language one sometimes calls ER the group of elementary matrices over R.

It is immediate to check that under the group homomorphism ιnm from above with 0<mn<, the group EmR is mapped into EnR, and that EnRn*,ιnmmn is a direct system of groups. The direct limit of this direct systems is ER as one easily checks.

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

###### Proposition 3.1.

The elementary matrics eijλ, eijμ satisfy for all λ,μR the following relations.

1. eijλeijμ=eijλ+μ, if ij,

2. eijλ,eklμ=1, if ij, kl, jk, and il,

3. eijλ,ejlμ=eilλμ, if ij, il, and jl,

4. eijλ,ekiμ=ekj-μλ, if ij, ik, and jk.

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

###### Lemma 3.2.

For 1i,j,k,ln with ij, kl and il or jk 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 δrs denotes the Kronecker symbol, i.e. δrs=1 for r=s and δrs=0 for rs.

###### Proof of the Lemma.

Let us compute the components of the matrix eijnλeklnμ.

 (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. ∎

### Definition and fundamental properties of K1alg

###### Definition 3.3.

Let R be a unital ring. Then K1algR, 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 K1algR and results from elementary matrix theory as stated in the prerequisites. ∎

## References

• M. F. Atiyah (1989)
K-Theory,