Tightening a proper Ore monoid action

In this section, we fix a groupoid G, an Ore monoid P and an action (X_p, σ_p,q) ofP onGby proper correspondences. As a preparation, we start with some results about proper correspondences.

Lemma 3.22. Let D be a groupoid and let X, Y be proper D-correspondences.

Then the map rX,Y:X ◦ Y/D → X/D given by rX,Y[(x, y)] := [x]

is proper.

Proof. The correspondenceX ◦ Y is proper by Proposition2.27. That is, the map induced by the range map rX ◦Y∗:X ◦ Y/D → D⁰ is proper. We have rX ◦Y∗ = rX ∗◦rX,Y. It follows thatrX,Y is a proper map sinceX ◦ Y/Dis Hausdorff andrX ∗

is proper and continuous.

Let Dbe a groupoid and let X andY be properD-correspondences. For x∈ X we let

Cx:={z∈ X ◦ Y |rX,Y([z]) = [x]}.

Let C_x⁰ := p⁻¹(Cx) ⊂ X ×_s,D0,rY, where p: X ×_s,D0,rY → X ◦ Y be the orbit space projection.

3.3 Actions of Ore monoids by proper correspondences

Lemma 3.23. LetD be a groupoid and letX andY be proper D-correspondences.

For all x∈ X, the restriction of the quotient map p:X ×_s,D0,rY → X ◦ Y toC_x⁰ is injective.

Proof. Let (x, ξ₁),(x, ξ₂) ∈ X ×_G0 Y be such that p(x, ξ₁) = p(x, ξ₂). It follows that there is an element g ∈ D_s(x) such that (x·g, g⁻¹ ·ξ1) = (x, ξ2). Hence x·g=xand g⁻¹·ξ₁ =ξ₂. By our assumption, the right action ofD onX is free.

This implies g=s(x) and hence ξ₁=ξ₂.

Finally, letSx,Y:Cx⊂ X ◦ Y → Y be the map given by

Sx,Y(ξ) := pr₂◦p⁻¹(ξ) (3.26) for allξ ∈Cx, where pr₂:X ×_s,D0,rY → Y is the projection map.

Corollary 3.24. Let D be a groupoid and let X: D → D and Y: D → D be proper correspondences. For all x∈ X, the map Sx,Y:Cx → Y is injective and continuous.

Proof. The mapSx,Y is continuous since it is the composition of two continuous maps. It is injective byLemma 3.23.

Now we turn back to our monoid action (X_p, σ_p,q). The isomorphismsσ_p,q induce homeomorphismsσ_p,q∗:X_pq/G → X_p◦_GX_q/G. We abbreviate

r_p,q :=rX_p,X_q ◦σ_p,q∗:X_pq/G → X_p/G (3.27) for allp, q∈P. Lemma 3.22implies thatr_p,q is proper for allp andq∈P. Notice thatr1,p is the map induced by the range maprXp on the quotient spaceX_p/G by our assumption thatX₁ is the identity correspondenceG:G → G. Moreover, since the mapsσ_p,q satisfy(3.1)we obtain

r_p,qr_pq,t=r_p,qt (3.28)

for all p, q, t ∈ P. Thus (X_p/G, r_p,q) is a projective system over the directed categoryC_P (see Definition 3.1). We denote its projective limit by

H⁰ := lim←−

CP

(X_p/G, r_p,q). (3.29)

A pointη ∈ H⁰ is given by (ηp)_p∈P with rp,q(ηpq) =ηp for all p, q∈P.

Lemma 3.25. The space H⁰ is locally compact and Hausdorff. The maps π_p:H⁰→ X_p/G, (η_p)_p∈P 7→η_p,

are proper for allp∈P.

Proof. By Proposition 2.17, the spaces X_p/G are locally compact Hausdorff for all p∈P. The mapsrp,q are proper for allp, q∈P. The proof then follows using similar arguments as in [1, Lemma 4.9].

There is a map r:H⁰→ G⁰ given by

r(η) :=r_1,1(η₁)

for all η= (ηp)_p∈P ∈ H⁰. The associativity condition(3.28)implies that r(η) = (rXp)_∗(η_p) for all p ∈ P. Moreover, Lemma 3.25 implies that r:H⁰ → G⁰ is a proper map; it is the anchor map of a natural action ofG on H⁰. It is given by

g·(η_p)_p∈P = (gη_p)_p∈P (3.30) for all g∈ G and all η= (ηp)_p∈P ∈ H⁰ with r(η) =s(g). Let

G˜:=GnH⁰

be the transformation groupoid associated with this action. Let

X˜_p :=X_p×_sp,G0,rH⁰ (3.31) for all p ∈ P. We claim that ˜X_p is naturally a tight correspondence over the groupoid ˜G. First, we define the anchor maps ˜r_p,s˜_p: ˜G → H⁰ = ˜G⁰. Given a point η ∈ H⁰ and an element x_p ∈ X_p such that r(η) = s_p(x_p), we may concatenatexp toη and get a new pointxp·η ∈ H⁰, defined formally by

(x_p·η)_pt :=p_pt◦σ_p,t⁻¹(x_p, η_t⁰) (3.32) for all t ∈ P, where p_pt: X_pt → X_pt/G is the quotient map and η_t⁰ is any point inX_t such thatp_t(η_t⁰) =η_t. Ifb∈P is not of the formptfor somet∈P then by the Ore condition, we may find k1 and k2 ∈P with bk1 =pk2. We define

(x_p·η)_b :=r_b,k1((x_p·η)_pk2). (3.33) The point (xp·η)pt does not depend on the choice ofη_t⁰ sinceσ_p,t⁻¹ isG-equivariant.

Here we use implicitly that because the Ore monoid pP is cofinal in P there is a homeomorphism

H⁰ = lim

←−C_P

(X_p/G, r_p,q)→lim

←−C_pP

(X_pt/G, r_pk,pl). (3.34) It follows that, for allxp∈ X_p, we have a map

(H⁰)^sp(xp)→ H⁰, η 7→x_p·η.

3.3 Actions of Ore monoids by proper correspondences

We define the range and source maps ˜r_p,s˜_p: ˜X_p→ H⁰ as

r˜p(xp, η) :=xp·η, s˜p(xp, η) :=η (3.35) for all (x_p, η)∈ X_p×_sp,G0,rH⁰.

The right action of ˜G on ˜X is given by

(xp, η)·(g, ζ) := (xpg, g⁻¹η) (3.36) for all ((xp, η),(g, ζ))∈X˜_p×_s,H˜ 0,rG˜

G. And the left action is given by˜

(g, ζ)·(x_p, η) = (g·x_p, η) (3.37) for all ((g, ζ),(x_p, η))∈G ט _s

G˜,H⁰,˜rpX˜_p.

Lemma 3.26. The map s˜p: ˜X_p → H⁰ is a local homeomorphism for allp∈P. It is surjective ifs_p is surjective.

Proof. We may use the following pullback diagram X˜_p =X_p×_sp,G0,rH⁰ X_p

H⁰ G⁰

pr1

pr2= ˜sp r

sp

The source maps_p:X_p→ G⁰ is a local homeomorphism, hence so is the parallel map ˜s. Similarly, ˜sp is surjective if sp is.

Lemma 3.27. The map r˜p induces a homeomorphism r˜p∗: ˜X_p/G → H⁰ for allp∈P.

Proof. To see that the map ˜r_p∗ is surjective, letη = (η_p)p∈P ∈ H⁰. Let η_p⁰ ∈ X_p be such that pp(η⁰_p) = ηp. For all t∈ P we write ζt :=S_η⁰_p,Xt(ηpt). Recall from Equation (3.26)thatS_η⁰_p,Xt(η_pt) = pr₂◦p⁻¹(η_pt), wherep:C_ηp⁰ ⊂ X_p×_s

p,G⁰,rtX_t→ X_p◦_GX_t is the quotient map. Letq ∈P. Thenrq,t(ζqt) =rq,t◦Sη⁰_p,Xqt(ηpqt). But r_q,t◦S_η⁰_p,Xqt =S_ηp⁰,Xq ◦r_pq,t. This implies that

rq,t(ζqt) =Sη⁰_p,Xq◦rpq,t(ηpqt) =Sη⁰_p,Xq(ηpq) =ζq

and therefore ζ ∈ H⁰. By construction, (η⁰_p·ζ)_pt =η_pt for all t∈P. Let b∈P.

SinceP is an Ore monoid we may find k1 and k2 ∈P withbk1=pk2. We define (η⁰_p·ζ)_b=r_b,k1(η⁰_p·ζ)_pk2=r_b,k1(η_pk2) =r_b,k1(η_bk1) =η_b.

Thusη_p⁰·ζ =η and ˜r_p∗is surjective. To see that ˜r_p∗ is injective, let (x_p, ξ),(y_p, η)∈ X˜_p with xp ·ξ = yp ·η. So ppt◦σ⁻¹_p,t(yp, η_t⁰) = ppt ◦σ_p,t⁻¹(xp, ξ_t⁰) for all t ∈ P. Henceσ⁻¹_p,t(y_p, η_t⁰) =σ_p,t⁻¹(x_p, ξ_t⁰)h for an elementh∈ G. This means that there is an elementg∈ G such thatr(g) =sp(xp) and

(y_pg, g⁻¹η⁰_t) = (x_p, ξ⁰_t·h).

Consequently, the points (x_p, ξ) and (y_p, η) represent the same point in ˜X_p/G.

Finally, the map ˜rp∗ is open and continuous since the space H⁰ is endowed with the projective limit topology and ˜r_p∗ is given by the maps σ_t⁻¹ and p_t, see Equation (3.32), which are both open and continuous for all t∈P.

Lemma 3.28. X˜_p is a tight correspondence over the groupoid G˜ for all p∈P. The left action ofG˜ onX˜_p is free if the left action of Gon X_p is free.

Proof. The defining formulas of the actions (3.37), (3.36)show that the right and left actions of the groupoid ˜G on ˜X_p commute; the right action of ˜G on ˜X_p is free and proper since the right action of G onX_p is free and proper and the left action of ˜G on ˜X_p is free if the left action of G on X_p is free. The source map on ˜X is a local homeomorphism ByLemma 3.26. The map ˜rp∗: ˜X_p/G → H⁰ is a homeomorphism by Lemma 3.27.

Next we show that the isomorphisms σ_p,q: X_pq → X_p ◦_G X_q lift to isomor-phisms ˜σp,q: ˜X_pq →X˜_p◦_G˜X˜_q. Letτp,q: ˜X_p◦_G˜X˜_q→X˜_pq be the map given by

τ_p,q([x_p, ξ, y_q, η]) := (σ⁻¹[x_p, y_q], η) (3.38) for all [x_p, ξ, y_q, η]∈X˜_p◦_G˜X˜_q.

Lemma 3.29. τ_p,q: ˜X_p◦_G˜X˜_q→X˜_pq is an isomorphism for all p, q∈P.

Proof. First we have to check that Equation (3.38) gives a well-defined map.

If [x_p, ξ, y_q, η] = [x¹_p, ξ¹, y¹_q, η¹]∈X˜_p◦_G˜X˜_q, then there is an element (g, ζ)∈G˜such that gζ = ξ and (x_pg, g⁻¹ξ, g⁻¹y_p, η) = (x¹_p, ξ¹, y_q¹, η¹) as elements in the space X_p×_sp,G0,rH⁰×_˜sp,H0,˜rq X_q×_sq,G0,rH⁰. Thusη¹ =η,xpg=x¹_p, g⁻¹yq=y_q¹. Then

τp,q([x¹_p, ξ¹, y¹_q, η¹]) = (σ_p,q⁻¹[xpg, g⁻¹yq], η)

= (σ_p,q⁻¹[xp, yq], η)

=τ_p,q([x_p, ξ, y_q, η]),

so the mapτp,q is well defined. Now let [xp, ξ, yq, η],[x¹_p, ξ¹, y_q¹, η¹]∈X˜_p◦_G˜X˜_q be such thatτ_p,q([x_p, ξ, y_q, η]) =τ_p,q([x¹_p, ξ¹, y¹_q, η¹]). Then

(σ⁻¹_p,q[x_p, y_q], η) = (σ_p,q⁻¹[x¹_p, y_q¹], η¹).

3.3 Actions of Ore monoids by proper correspondences

Henceη =η¹ andσ⁻¹_p,q[x_p, y_q] =σ_p,q⁻¹[x¹_p, y¹_q]. Sinceσ_p,q is an isomorphism, there is an elementg∈ G such that (xpg, g⁻¹yq) = (x¹_p, y_q¹) inX_p×_sp,G0,rqX_q. Thusxpg= x¹_p,g⁻¹yq =y_q¹. In addition,ξ¹ =y_q¹·η =g⁻¹yq·η=g⁻¹ξ. So

[x¹_p, ξ¹, y¹_q, η] = [xpg, g⁻¹ξ, g⁻¹yq, η] = [xp, ξ, yq, η]. (3.39) Thenτ_p,q is injective. Next let (x_pq, η)∈X˜_pq. Sinceσ_p,q is an isomorphism, there is (x¹_p, x²_q)∈ X_p×_sp,G0,rqX_q withσ(x_pq) = [x¹_p, x²_q]. Then (x¹_p, x²_q·η)∈X˜_p, (x²_q, η)∈ X˜_q, (x¹_p, x²_q·η, x²_q, η)∈ X_p×_sp,G0,rH⁰×_˜sp,H0,˜rqX_q×_sq,G0,rH⁰andτ(x¹_p, x²_q·η, x²_q, η) = (σ_p,q⁻¹[x¹_p, x²_q], η) = (x_pq, η). Then τ_p,q is surjective. It is also open and continuous sinceσ_p,q⁻¹ is a homeomorphism. To see thatτ_p,q is ˜G-equivariant, let [x_p, ξ, y_q, η]∈ X ◦˜ _G˜X˜_q and let (g, ζ)∈G˜satisfygζ =η. We have

τp,q([xp, ξ, yq, η])·(g, ζ) = (σ_p,q⁻¹[xp, yq], η)·g

= (σ_p,q⁻¹[xp, yq]g, g⁻¹η) = (σ_p,q⁻¹[xp, yqg], g⁻¹η)

=τp,q([xp, ξ, yqg, g⁻¹η]) =τp,q([xp, ξ, yq, η]·(g, ζ)).

Similarly, (g, ζ)·τp,q([xp, ξ, yq, η]) = τp,q((g, ζ)[xp, ξ, yq, η]) for all [xp, ξ, yq, η] ∈ X ◦˜ _G˜X_qand all (g, ζ)∈G˜withζ =σ_p,q⁻¹[x_p, y_q]·η. Sinceξ=y_q·η, we have ˜r_p(x_p, ξ) = xp·ξ=σ⁻¹_p,q[xp, yq]·η=ζ.

For allp, q∈P, we set ˜σ_p,q:=τ_p,q⁻¹. The maps ˜σ_p,q satisfyEquation (3.1), since the maps σp,q satisfy the same condition, being part of an action ofP onG by proper correspondences. Moreover, the correspondence ˜X_p is tight for all p∈P byLemma 3.28. It follows that ( ˜X_p,σ˜_p,q) defines an action of P on ˜G by tight correspondences. Hence it has a colimit byTheorem 3.15. It remains to show that the colimit of this action is a colimit of the original diagram.

Theorem 3.30. The diagrams (X_p, σ_p,q) and ( ˜X_p,σ˜_p,q) have the same colimit in Gr.

Proof. The colimit of the diagram ( ˜X_p,σ˜p,q) exists by Theorem 3.15. We set H:= colim( ˜X_p,σ˜_p,q). (3.40) We will show that the groupoidH given by Equation (3.40)is also a colimit of the diagram (X_p, σp,q). This is sufficient since the colimit of a diagram is unique (up to isomorphism) if it exists.

Let D be a groupoid and let (Y, ϕ_p) be a transformation from (X_p, σp,q) to const_D. For allp ∈P, the isomorphism ϕ⁻¹_p :Y → X_p◦_GY, composed with the projectionrXp,Y:X_p◦_GY → X_p/G, gives a mapψ_p:Y → X_p/G. On the one hand, we have

rp,qrXpq,Y(σ_p,q⁻¹×_Gid_Y)(id_Xp×_Gϕ⁻¹_q ) =rXp,Y

for all p, q∈P. So

r_p,qrXpq,Y(σ⁻¹_p,q×_Gid_Y)(id_Xp×_Gϕ⁻¹_q )ϕ⁻¹_p =rXp,Yϕ⁻¹_p

for all p, q ∈ P. On the other hand, the maps ϕ_p satisfy the coherence condi-tion (2.27). Therefore,

(σ⁻¹_p,q×_Gid_Y)(id_Xp×_Gϕ⁻¹_q )ϕ⁻¹_p =ϕ⁻¹_pq. Thus

rp,qψpq =ψp. (3.41)

Equation (3.41) says that (ψ_p)_p∈P is an inverse limit map fromY to (X_p/G, r_p,q).

Hence it induces a continuous map ψ∞= lim

←−ψ_p:Y → H⁰= lim

←−X_p/G. We define a left action of ˜G on Y, using ψ∞ as a left anchor map, by setting

(g, ξ)·y:=g·y, (3.42)

for all (g, ξ)∈G,˜ y∈ Y withψ∞(y) =ξ. This action is well defined and commutes with the right action of D. This implies thatY is also a correspondence from ˜G toD.

Let p ∈ P. We want to extend the isomorphism ϕp: X_p ◦_G Y → Y to an isomorphism ˜ϕ_p: ˜X_p◦_G˜Y → Y, so that ( ˜X_p,ϕ˜_p) becomes a natural transformation from ( ˜X_p,σ˜_p,q) to const_D. We claim that the map

ϕ˜_p: ˜X_p◦_G˜Y → Y, ((x_p, ξ), y)7→ϕ_p(x_p, y), (3.43) achieves this.

Lemma 3.31. For all p ∈ P, ϕ˜p: ˜X_p ◦_G˜ Y → Y is an isomorphism of corre-spondences. (Y,ϕ˜_p) is a transformation from ( ˜X_p,σ˜_p,q) to the constant diagram const_D.

Proof. First, let ((xp, ξ), y) ∈ X˜_p ×_s

G˜,G˜⁰,ψ∞ Y and let (g, η) ∈ G˜ with ψ∞(y) = gη=rG˜(g, η). Then

ϕ˜p((xp, ξ)(g, η),(g, η)⁻¹y) = ˜ϕp((xpg, g⁻¹ξ), g⁻¹y)

=ϕ_p(x_pg, g⁻¹y) =ϕ_p(x_p, y)

= ˜ϕp((xp, ξ), y).

It follows that ˜ϕp is a well defined map. Now let ((xp, ξ), y) ∈ X˜_p ×_s

G˜,G˜⁰,ψ∞

Y and let ((x⁰_p, ξ⁰), y⁰) ∈ X˜_p ×_s

G˜,G˜⁰,ψ∞ Y with ˜ϕp((xp, ξ), y) = ˜ϕp((x⁰_p, ξ⁰), y⁰).

3.3 Actions of Ore monoids by proper correspondences

Thenϕ_p(x_p, y) =ϕ_p(x⁰_p, y⁰). Sinceϕ_p is an isomorphism, there is an elementg∈ G such that rG(g) =rY(y) and (x⁰_p, y⁰) = (xpg, g⁻¹y). This impliesx⁰_p =xpg,y⁰ = g⁻¹y and hence ξ⁰ =ψ∞(g⁻¹y) =g⁻¹ψ∞(y) =g⁻¹ξ. The element (g, g⁻¹ξ)∈G˜ satisfies

((x_p, ξ)(g, g⁻¹ξ),(g, g⁻¹ξ)⁻¹y) = ((x_pg, g⁻¹ξ), g⁻¹y) = ((x⁰_p, ξ⁰), y⁰).

Then [(x⁰_p, ξ⁰), y⁰] = [(xp, ξ), y]∈X˜_p◦_G˜Y. It follows that the map ˜ϕp is injective.

Next lety∈ Y and let (xp, z)∈ X_p×_sp,G0,rY Y be such thatϕp(xp, z) =y. Then the pointx= ((x_p, ϕ∞(z)), z) satisfies ˜ϕ_p([x]) =y. Thus the map ˜ϕ_p is surjective.

The definition of the left action of ˜G on ˜X_p and the fact that ϕp isG,D-equivariant imply that ˜ϕp is a ˜G,D-equivariant.

Finally, the map (ϕp)_p∈P satisfies condition (2.29). Therefore, the definition of the isomorphisms ˜σp,q implies that ( ˜ϕp)_p∈P satisfies the same condition. We have the pullback diagram

X˜_p◦_s˜

p,G˜⁰,rY X_p×_sp,G0,rY

Y Y

pr1×id_Y

˜ ϕp◦p

id

ϕp◦p

So ˜ϕ_p◦p is open and continuous sinceϕ_p◦p is. Hence ˜ϕ_p is open and continuous sincep is open and continuous. This concludes the proof of the lemma.

Conversely, let (Y,ϕ˜_p) be a transformation from ( ˜X_p,σ˜_p,q) to const_D. There is a natural action ofG onY. The right anchor map is given byr_Y^G(y) =rG(rY(y)).

The action is defined by

g·y= (g, rY(y))·y (3.44)

for ally∈ Y andg∈ G withr^G_Y(y) =s(g).

This action commutes with the right action ofDon Y. Next we define ϕ_p: ˜X_p×_s

p,G⁰,r_Y^G Y → Y, (x_p, y)7→ϕ˜_p((x_p, rY(y)), y). (3.45) Using similar arguments as in Lemma 3.31, we see that (ϕp,Y) is a trans-formation from (X_p, σp,q) to the constant diagram const_D. The two construc-tions (Y, ϕ_p)7→(Y,ϕ˜_p) and (Y,ϕ˜_p)7→(Y, ϕ_p) are inverse to each other. Moreover, a modification between two transformations (ϕp,Y) and (ϕ⁰_p,Y⁰) is an isomor-phismw:Y → Y⁰ ofG,D-correspondences that intertwinesϕp and ϕ⁰_p, see Equa-tion (2.30). Equation (3.42)says thatwis an isomorphism of ˜G,D-correspondences

andEquation (3.43)says that it intertwines ˜ϕ_p and ˜ϕ⁰_p. Consequently,wis a modi-fication from (Y,ϕ˜p) to (Y⁰,ϕ˜⁰_p). Similarly, byEquation (3.44)andEquation (3.45), a modification between two transformations (Y,ϕ˜p) to (Y⁰,ϕ˜⁰_p) is the same as a modification between (Y, ϕ_p) and (Y⁰, ϕ⁰_p). Thus we have an equivalence between the groupoidsCorr^P((X_p, σp,q),const_D) andCorr^P(( ˜X_p,σ˜p,q),const_D). Therefore, the diagrams ( ˜X_p,σ˜_p,q) and (X_p, σ_p,q) have the same colimit inGr. This colimit isH by Theorem 3.15.

Im Dokument A colimit construction for groupoids (Seite 66-74)