Simplicial and dimension groups with group action and their realization

Proposition A.1 (Faithfulness).

Let Γ be any group, let K be a Γ-graded field, let R and S be graded matricial K-algebras, and let ϕ,ψ:R→S be graded algebra homomorphisms (not necessarily unit-preserving). The following assertions are equivalent:

1. K0Γ⁢(ϕ)=K0Γ⁢(ψ).

2. There exists an invertible element u∈S1Γ such that ϕ⁢(r)=u⁢ψ⁢(r)⁢u-1 for all r∈R.

Proof.

If the arguments related to the presence of an involution are not considered, the proof of [15, Theorem 3.4] can be used for the current proof: the relations on the standard graded matrix units hold without the requirements that Γ is abelian and Lemma 2.1 can be used instead of [15, Lemma 3.2]. We also emphasize that the maps in [15, Theorem 3.4] are not assumed to be unit-preserving. ∎

Theorem A.2 (Classification).

If Γ is any group, K is a Γ-graded field, R and S are graded ultramatricial K-algebras, and f:(K0Γ⁢(R),DR)→(K0Γ⁢(S),DS) is an isomorphism of the category OGΓD, then there is a graded algebra isomorphism ϕ:R→S such that K0Γ⁢(ϕ)=f. If f is order-unit-preserving, then ϕ can be chosen to be unit-preserving.

Proof.

We recall [15, Theorem 4.5] stating that if Γ is abelian and if K has an involution which satisfies some additional conditions, then ϕ can be obtained to be a graded *-algebra isomorphism. If the arguments related to the presence of an involution are not considered and Propositions 4.9 and A.1 are used instead of [15, Theorem 2.7 and Theorem 3.4], the proof of [15, Theorem 4.5] modifies to the proof of the theorem. ∎

Proposition B.1.

Let Γ be a group with a subgroup Δ. If G is an ordered and directed Γ-group with a generating interval D, then (G,D) has an ordered Γ-group extension by (Z⁢[Γ/Δ],Δ).

Proof.

Let H=G⊕ℤ⁢[Γ/Δ], let i:G→H be the natural injection, and let p:H→ℤ⁢[Γ/Δ] be the natural projection so that the diagram

is a short exact sequence in the category of ℤ⁢[Γ]-modules. Let us denote the set

by H+. It is direct to check that H+ is closed under the action of Γ and that (0,0)∈H+. Let (x,a⁢Δ),(y,b⁢Δ)∈H+ and d1,d2∈D be such that x+a⁢d1,y+b⁢d2∈G+. Since D is upwards directed, there is d∈D such that d≥d1, d≥d2, so that x+a⁢d≥x+a⁢d1≥0 and y+b⁢d≥y+b⁢d2≥0. Thus x+y+(a+b)⁢d≥0, and so H+ is additively closed. Hence, H+ is a cone in H. The cone H+ is strict since the cone G+ is strict. So, H+ defines a partial order on H.

Next, we show that (0,Δ) is an order-unit of H. For any (x,a⁢Δ)∈H, there is y∈G+ with x≤y since G is directed. Then y=∑i=1nbi⁢di for some n, di∈D and bi∈ℤ+⁢[Γ] for i=1,…,n. Since D is upwards directed, one can find d∈D such that di≤d for all i=1,…,n. Thus,

Let c∈ℤ+⁢[Γ] be such that c≥a+b. This implies that c-a≥b so that (c-a)⁢d≥b⁢d. The relation x≤b⁢d implies that -x+b⁢d≥0, and so -x+(c-a)⁢d≥-x+b⁢d≥0. Thus, we have that (-x,(c-a)⁢Δ)∈H+, and hence (x,a⁢Δ)≤(0,c⁢Δ). This demonstrates that (0,Δ) is an order-unit and also implies that H is directed.

By the definition of H+, we have that i and p are order-preserving. Since (0,a⁢Δ) is in H+ for every a∈ℤ+⁢[Γ], we have that p⁢(H+)=ℤ+⁢[Γ/Δ]. The relation i-1⁢(H+)=G+ holds since i⁢(x)∈H+ for x∈G if and only if x is in G+. The relation p⁢(0,Δ)=Δ holds by the definition of p. So, it remains to show that i-1⁢([(0,0),(0,Δ)])=D. For d∈D, we have (-d,Δ)∈H+ by the definition of H+. Hence (0,0)≤(d,0)≤(0,Δ), and so i⁢(d)∈[(0,0),(0,Δ)]. Conversely, if i⁢(x)∈[(0,0),(0,Δ)] for some x∈G, then (0,0)≤(x,0)≤(0,Δ), which implies that (x,0),(-x,Δ)∈H+. Hence, x∈G+ and -x+d∈G+ for some d∈D by the definition of H+. Thus 0≤x≤d, which implies that x∈D by the convexity of D. ∎

Proposition B.2.

Let Δ be a normal subgroup of Γ. If G is a simplicial Γ-group with a simplicial Γ-basis stabilized by Δ and a generating interval D, then (G,D) has an ordered Γ-group extension (H,u) by (Z⁢[Γ/Δ],Δ) such that H satisfies (SDPΔ) and that Δ⊆Stab⁡(H).

Proof.

Let H=G⊕ℤ⁢[Γ/Δ] and u=(0,Δ). Then (H,u) is an ordered Γ-group extension of (G,D) by (ℤ⁢[Γ/Δ],Δ) by definition and by Proposition B.1. By Proposition 3.5, Stab⁡(H)=Δ. So, it is sufficient to show that H satisfies (SDPΔ). Let Xi=(xi,bi⁢Δ) be in H+ for i=1,…,n and assume that ∑i=1nai⁢Xi=(0,0) for some ai∈ℤ⁢[Γ]. This implies that ∑i=1nai⁢xi=0, that the elements bi can be chosen in ℤ+⁢[Γ], that ∑i=1nπ⁢(ai⁢bi)=0 where π is the natural map π:ℤ⁢[Γ]→ℤ⁢[Γ/Δ], and that there are di∈D such that xi+bi⁢di≥0. Since D is upwards directed, there is d∈D such that di≤d for all i=1,…,n, and hence xi+bi⁢d≥xi+bi⁢di≥0.

Let {y1,…,ym}⊆G+ be a simplicial Γ-basis stabilized by Δ. Thus, for xi+bi⁢d, d∈G+, there are bi⁢j,cj∈ℤ+⁢[Γ] for i=1,…,n, j=1,…,m such that

The condition

implies that

Since Δ is normal in Γ, the map π is both a left and a right ℤ⁢[Γ]-module homomorphism. Therefore, ∑i=1nπ⁢(ai⁢bi)=0 implies that ∑i=1nπ⁢(ai⁢bi⁢cj)=0 for all j=1,…,m, and hence

Let Yj=(yj,0) for j=1,…,m and let Ym+1=(-d,Δ). Then Yj and Ym+1 are in H+ by Proposition B.1. For all i=1,…,n, j=1,…,m, let Bi⁢j=bi⁢j∈ℤ+⁢[Γ] and Bi⁢(m+1)=bi∈ℤ+⁢[Γ]. For any i=1,…,n,

For any j=1,…,m,

This finishes the proof. ∎

Proposition B.3.

Let Δ be a normal subgroup of Γ. If (G,D) is a direct limit of a directed system of simplicial Γ-groups with simplicial Γ-bases stabilized by Δ in the category OGΓD, then (G,D) has an ordered Γ-group extension (H,u) by (Z⁢[Γ/Δ],Δ) such that H satisfies (SDPΔ) and such that Δ⊆Stab⁡(H).

Proof.

Let I be a directed set, let (G,D) be a direct limit of simplicial Γ-groups ((Gi,Di),gi⁢j), i,j∈I, i≤j, with simplicial Γ-bases stabilized by Δ, and let gi, i∈I, be the translational maps. By Proposition B.2, we can find ordered Γ-group extensions (Hi,ui), i∈I, of (Gi,Di) by (ℤ⁢[Γ/Δ],Δ) which satisfy (SDPΔ) and such that Δ⊆Stab⁡(Hi). Let ιi denote the inclusion of (Gi,Di) into (Hi,ui) and let pi denote the projection (Hi,ui)→(ℤ⁢[Γ/Δ],Δ). If hi⁢j=gi⁢j⊕1ℤ⁢[Γ/Δ] where the second term is the identity map on ℤ⁢[Γ/Δ], then the system ((Hi,ui),hi⁢j), i,j∈I, i≤j, is a directed system in 𝐎𝐆Γu. Note that hi⁢j⁢ιi=ιj⁢gi⁢j and pj⁢hi⁢j=pi for all i≤j by the proof of Proposition B.2. Let (H,u) be a direct limit of this directed system which exists by Proposition 4.1 and let hi=gi⊕1ℤ⁢[Γ/Δ]. Since all Γ-groups Hi satisfy (SDPΔ) and are stabilized by Δ, H satisfies (SDPΔ) and it is stabilized by Δ.

Define the maps ι:G→H and p:H→ℤ⁢[Γ/Δ] by ι⁢(gi⁢(xi))=hi⁢ιi⁢(xi) and p⁢(hi⁢(yi))=pi⁢(yi), respectively. We claim that we obtain the required properties from the commutative diagram below:

Indeed, one checks that the maps ι and p are well-defined, that ι is injective, and that p is surjective using properties of a direct limit. Then one checks that ker⁡p is equal to the image of ι, that ι and p are order-preserving, that ι-1⁢(H+)=G+, that p⁢(u)=Δ, and that p⁢(H+)=ℤ+⁢[Γ/Δ] using the definitions of the maps. It remains to check that ι-1⁢([0,u])=D. If d∈D=⋃i∈Igi⁢(Di), then there are i∈I and di∈Di=ιi-1⁢([0,ui]) such that d=gi⁢(di). Hence, the relation 0≤ιi⁢(di)≤ui holds in Hi. Applying hi to this relation, we obtain that

Hence, d=gi⁢(di)∈ι-1⁢([0,u]). Conversely, if x∈ι-1⁢([0,u]), let ι⁢(x)=hi⁢(yi) for some yi∈Hi. Since

we obtain yi=ιi⁢(xi) for some xi∈Gi. Thus,

which implies x=gi⁢(xi) since ι is injective. The relation 0≤ι⁢(x)=hi⁢ιi⁢(xi)≤u=hi⁢(ui) implies that there is j≥i such that

and so gi⁢j⁢(xi)∈ιj-1⁢([0,uj])=Dj. Thus,

as desired. ∎

Theorem B.4.

If Γ is such that Z⁢[Γ] is Noetherian and (G,D) is a dimension Γ-group satisfying (SDPΔ) for some subgroup Δ of Γ such that Δ⊆Stab⁡(G), then (G,D) has an ordered Γ-group extension (H,u) by (Z⁢[Γ/Δ¯],Δ¯), where Δ¯ is the normal closure of Δ in Γ, such that H satisfies (SDPΔ¯) and such that Δ¯⊆Stab⁡(H).

Proof.

By Theorem 5.12, we can find a directed system of simplicial Γ-groups with simplicial Γ-bases stabilized by Δ¯ such that (G,D) is a direct limit of this directed system in 𝐎𝐆ΓD. By Proposition B.3, an ordered Γ-group extension (H,u) by (ℤ⁢[Γ/Δ¯],Δ¯) exists and it satisfies the required properties. ∎