Liftings of diagrams of semilattices by diagrams of dimension groups

0.36 MB·English

by  Jiri Tuma

LIFTINGS OF DIAGRAMS OF SEMILATTICES
5 BY DIAGRAMS OF DIMENSION GROUPS
0
0
2 JIRˇ´IT˚UMAANDFRIEDRICHWEHRUNG

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a
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2 Introduction
2
Therearevariouswaystoobtaindistributivesemilatticesfromothermathemat-

] ical objects. Two of them are the following; we refer to Section 1 for more precise
M
definitions. Adimension group isadirected,unperforatedpartiallyorderedabelian
G group with the interpolation property, see also K.R. Goodearl [6]. With a dimen-
sion group G we can associate its semilattice of compact ( = finitely generated)
.
h ideals Id G. Because of the interpolation property the positive cone G+ has the
c
t
a refinementproperty,thusthecompactidealsemilatticeIdcGisdistributive. Aring
m R is locally matricial over a field K, if it is isomorphic to a direct limit of finite
[ productsoffullmatricialringsMn(K). IfRislocallymatricial,or,moregenerally,

1 regular (in von Neumann’s sense), then its semilattice IdcR of compact ideals is
distributive, see, for example, K.R. Goodearl [5].
v
6 These two different contexts are related as follows. With a locally matricial
7 ringR,wecanassociateits(partiallyordered)Grothendieck group K0(R). Itturns
3 out that K (R) is a dimension group, and the following relation holds:
0
1
0 IdcR∼=Idc(K0(R)), (0.1)
5
see [5, Corollary 15.21]. Therefore, the compact ideal semilattice of a locally ma-
0
tricial ring is the compact ideal lattice of a dimension group. By the results of
/
h K.R. Goodearl and D.E. Handelman [7], the converse holds for dimension groups
t
a of cardinality at most ℵ1, but it fails for larger dimension groups, see F. Wehrung
m [17, 18]. Nevertheless this makes it worthwhile to study the functor that with a
: dimension group G associates its semilattice of compact ideals IdcG. For a sur-
v
vey paper on this and related issues we refer the reader to K.R. Goodearl and F.
i
X Wehrung [8].
r ThecentralopenproblemthatmotivatesthepresentpaperisProblem10.1of[8],
a
that asks whether any distributive h∨,0i-semilattice of cardinality at most ℵ is
1
isomorphic to the compact ideal semilattice of some dimension group. Let us call
this the compact ideal representation problem. By [7], this is equivalent to asking
whether any distributive h∨,0i-semilattice of cardinality at most ℵ is isomorphic
1
tothe compactidealsemilatticeofsomelocallymatricialring. Althoughwedonot
solve this problem here, we settle related issues with some interest of their own,
2000MathematicsSubjectClassification. 06A12,06C20,06F20,15A03,15A24,15A48,16E20,
16E50,19A49,19K14.
Key words and phrases. Distributive semilattice, dimension group, locally matricial algebra,
compactideal,directlimit,flat,generic.
The authors were supported by the Barrande program and by the institutional grant
CEZ:J13/98:113200007a. ThefirstauthorwasalsopartiallysupportedbyGACR201/99andby
GAUK162/1999.
1
2 J.T˚UMAANDF.WEHRUNG
and,hopefully,thatwillprovideasteppingstonetowardsasolutionforthegeneral
problem.
We firstrecallthe answersto somerelatedquestions. As acorollaryofthe main
resultofG.M.Bergman[2]andatheoremofK.R.Goodearl[5],thefollowingresult
is obtained, see also [8] for different proofs.
Theorem 0.1. Every countable distributive h∨,0i-semilattice S is isomorphic to
Id G for some countable dimension group G.
c
Another result of [8] is the following.
Theorem 0.2. Every distributive lattice L with zero is isomorphic to Id G for
c
some dimension group G.
Onthe other hand,P.R˚uˇziˇcka[14]constructsa distributive h∨,0i-semilattice of
cardinality ℵ that is not isomorphic to Id G for any dimension group G.
2 c
These results leave open the case of distributive h∨,0i-semilattices of cardinal-
ity ℵ . By P. Pudla´k [13], every distributive h∨,0i-semilattice is isomorphic to
1
the direct union of its finite distributive h∨,0i-subsemilattices. A variant of this
result, stating that every distributive h∨,0i-semilattice is a direct limit of finite
Boolean h∨,0i-semilattices, is proved in [8]. Thus it is natural to try an inductive
approachto the compact ideal representationproblem for distributive semilattices
of cardinality ℵ : find a simultaneous representation of a suitable limit system of
1
finite Boolean semilattices that converges to a given distributive semilattice S of
cardinalityℵ . ByA.P.Huhn[10],onecanassumethatthelimitsystemisindexed
1
by an infinite lattice in which every principal ideal is a finite dismantlable lattice.
After preliminary results, most notably about the existence of the so-called λ-
generic maps in Sections 4 and 5, we prove in Section 6 our main results:
Theorem 6.4. Every finite dismantlable diagram of finite Boolean h∨,0i-semi-
lattices has a simultaneous representation with respect to the functor Id .
c
Theorem 7.1. For any countable distributive h∨,0i-semilattice S and any count-
able dimension vector space H, every h∨,0i-homomorphism f: S → Id H can be
c
lifted by a positive homomorphism f: G→H for some dimension vector space G.
We also state consequences of Theorem 7.1, as well in ring theory as in lattice
theory, see Corollaries 7.4 and 7.5.
The remaining sections contain various counterexamples to other strategies for
a positive solution of the compact ideal representation problem for distributive
semilattices of cardinality ℵ .
1
1. Basic concepts
Forbasicconceptsaboutpartiallyorderedabeliangroups,wereferthereaderto
K.R. Goodearl [6]. We recall some of the definitions here. For a partially ordered
abelian group G, we denote by G+ = {x∈G|0≤x} the positive cone of G, and
we say that G is directed, if G = G++(−G+). We put G++ = G+\{0}, and we
note N=Z++. A subgroup H of G is convex, if 0≤x≤a implies that x∈H, for
all a ∈ H and x ∈ G. We say that H is an ideal of G, if it is a directed, convex
subgroup of G, and a compact ideal of G, if it is an ideal generated by a finite
subset of G.
LIFTING SEMILATTICE DIAGRAMS 3
AseveryidealofGisdirected,anyideal(resp.,compactideal)ofGisgenerated
(as an ideal) by a subset of G+ (resp., finite subset of G+). In the latter case,
one may replace a finite subset {a ,...,a } of G+ by the singleton {a}, where
1 n
a= n a , so the compact ideals of G are exactly the subsets of G of the form
i=1 i
PG(a)= x∈G|∃n∈Z+ such that −na≤x≤na , for a∈G+.
An order-unit(cid:8)of G is an element a∈G+ such that G(a)=(cid:9) G. We denote by IdG
thelatticeofidealsofG,andbyId Gtheh∨,0i-semilatticeofcompactidealsofG.
c
Observe that IdG is an algebraic lattice and that Id G is its h∨,0i-semilattice of
c
compact elements, see G. Gr¨atzer [9] for unexplained terminology. For elements
a, b ∈ G+, we write a ∝ b, if there exists n ∈ N such that a ≤ nb, and a ≍ b, if
a∝b∝a. Hence
Id G= G(a)|a∈G+ ,
c
with containment and equality among the G(a)-s determined by
(cid:8) (cid:9)
G(a)⊆G(b) iff a∝b and G(a)=G(b) iff a≍b,
for all a, b ∈ G+. Hence, Id G is isomorphic to ∇(G+), the maximal semilattice
c
quotient of G+, see [8].
The assignment G7→Id G can be naturally extended to a functor, by defining,
c
for a positive homomorphism (that is, an order-preserving group homomorphism)
f: G→H ofpartiallyorderedabeliangroupsandacompactidealI ofG,(Id f)(I)
c
as the compact ideal of H generated by the image f[I] of I under f. Hence,
(Id f)(G(a)) = H(f(a)), for all a ∈ G+, and it is an easy exercise to verify that
c
the functor Id thus defined preserves direct limits.
c
Mostof the partiallyorderedabeliangroupswe shalldealwith will be equipped
with an additional structure of vector space, always over the field Q of rational
numbers. Any partially ordered vector space E has the additional property that
mx ≥ 0 implies that x ≥ 0, for all m ∈ N and all x ∈ E, we say that E is
unperforated, see [6]. We observe that all group homomorphisms between vector
spaces preserve the vector space structure.
A refinement monoid (see A. Tarski [15], F. Wehrung [16]) is a commutative
monoidM suchthat for any positive integers m, n and elements a , ..., a , b ,
0 m−1 0
..., b of M such that a = b , there are elements c (for i < m
n−1 i<m i j<n j i,j
and j < n) such that a = c , for all i < m, and b = c , for all
i P j<n i,jP j i<m i,j
j <n.
P P
A partially ordered abelian group G is an interpolation group (see [6]) if for all
positive integers m, n and elements a , ..., a , b , ..., b of G such that
0 m−1 0 n−1
a ≤ b , for all i < m and j < n, there exists x ∈ G such that a ≤ x ≤ b , for
i j i j
all i < m and j < n. Equivalently, the positive cone G+ is a refinement monoid
(see [6, Proposition 2.1]). In addition, we say that G is a dimension group (see
[6]) if G is directed and unperforated. If G is an interpolation group, then Id G
c
is a refinement semilattice. Refinement semilattices are usually called distributive
semilattices, see [9].
We can prove right away the following very elementary lifting result:
Proposition 1.1. For every h∨,0i-semilattice S, there exists a partially ordered
vector space E such that Id E ∼=S.
c
Proof. We first observe that S can be h∨,0i-embedded into some powerset semi-
lattice P(X), for a set X. For example, take X = S and use the embedding
4 J.T˚UMAANDF.WEHRUNG
that with any s ∈ S associates j(s) = {x∈S |s(cid:2)x}. We then let F denote the
vector space of all maps f: X → Q with finite range. For any f ∈ F, we put
suppf ={x∈X |f(x)6=0}, and we put
M ={f ∈F |f(x)≥0, for all x∈X and suppf ∈S}.
Then M is an additive submonoid of F (in particular, it is cancellative), M ∩
(−M)={0}, and M is closed under multiplication by nonnegative scalars. Hence
M is the positive cone of a structure of directed partially ordered vector space on
E =M+(−M). Let π: M →S be the mapdefined by the ruleπ(f)=suppf, for
all f ∈M. Then π(f)≤π(g) (resp., π(f)=π(g)) iff f ∝g (resp., f ≍g) in E, for
all f, g ∈ M, whence π induces an isomorphism ϕ from Id E onto S by the rule
c
ϕ(E(f))=suppf, for all f ∈M. (cid:3)
Unfortunately, even if S is distributive, the partially ordered vector space E
constructed above may not have interpolation.
Throughout the paper we shall often formulate our results in basic categorical
language. Among the main categories we shall be working with are the following:
— The category E of pairs A = (A ,1 ), where A is a partially ordered
0 A 0
vector space (that we will subsequently identify with A) and 1 is an
A
order-unit of A . For objects A and B of E, a homomorphism, or positive
0
homomorphism, from A to B is a positive homomorphism f: A → B .
0 0
Observe that we do not require that f(1 ) = 1 . We say that a homo-
A B
morphism f: A → B is normalized, if f(1 ) = 1 . In particular, we will
A B
slightlyabuseterminologybyusingisomorphismsonly inthe ‘normalized’
sense,thatis,anisomorphismf: A→Bsatisfiestheequalityf(1 )=1 .
A B
— ThefullsubcategoryE ofEconsistingofalldimension vector spaces inE.
d
— The category S of h∨,0i-semilattices and h∨,0i-homomorphisms.
— The fullsubcategoryS ofS whoseobjectsarethe finite Booleansemilat-
fb
tices.
Inparticular,Id defines,byrestriction,afunctor fromEto Sthatpreservesdirect
c
limits.
Every partially ordered set (P,≤) can be viewed as a category with objects
x ∈ P and a unique morphism ε : x → y whenever x ≤ y in P. Let E′ be a
x,y
subcategory of E. We say that a diagram Φ: (P,≤) → S of semilattices (i.e., a
functorfromP,viewedasacategory,toS)hasalifting, withrespecttothefunctor
Id , to the category E′, if there is a diagram Ψ: (P,≤) → E′ such that the two
c
functors Φ and Id ◦Ψ are naturally equivalent, that is, if there are isomorphisms
c
ι : Φ(x)→Id (Ψ(x)), for x∈P, such that the following diagram commutes,
x c
Id (Ψ(ε ))
c x,y
Id (Ψ(x)) // Id (Ψ(y))
c c
OO OO
ιx ιy
Φ(x) //Φ(y)
Φ(ε )
x,y
for all x ≤ y in P. We will usually omit the phrase with respect to Id , since lifts
c
with respect to other functors will not be investigated.
For a set I, we denote by P(I) the powerset of I, and, for any i∈I, we put
P (I)={X ∈P(I)|i∈X}, P∗(I)={X ∈P(I)|i∈/ X}.
i i
LIFTING SEMILATTICE DIAGRAMS 5
2. Pseudo-simplicial spaces
For a nonempty, finite set X, let Q denote the partially ordered vector space
X
with underlying vector space QX and partial ordering defined by
f ≤g ⇔ either f =g or f(x)<g(x), for all x∈X,
endowed with the canonical order-unit 1X, defined as the constant function on X
with value 1. Then Q is a dimension vector space. It is simple, that is, Id Q ∼=
X c X
2. We denote by (1X) the canonical basis of QX, so that 1X = 1X.
x x∈X x∈X x
We observe that while 1X ∈ (Q )++, the vectors 1X, for x ∈ X, do not belong
X x P
to (Q )++ unless X is a singleton. Nevertheless they keep a certain positivity
X
character,captured by the following definition:
Definition 2.1. For a partially ordered abelian group G, we define a partial pre-
ordering ≤arch on G by the rule
x≤arch y ⇐⇒∃u∈G such that nx≤ny+u, for all n∈Z+, (2.1)
for all x, y ∈ G. Then we define the partially preordered abelian group G =
arch
(G,≤arch), the archimedean quotient of G.
In particular, for any finite set X, the archimedean quotient of Q is QX, and
X
0<arch 1X, for all x∈X.
x
Definition 2.2. An object A of E is
(i) simple pseudo-simplicial, if A∼=Q for some nonempty finite set X,
X
(ii) pseudo-simplicial, if A = A for some natural number n and some
i<n i
simple pseudo-simplicial spaces A , for i<n.
i
L
The formula in (ii) above means that A = A as vector spaces, 1 =
i<n i A
1 , and the canonical bijection from A to A is an isomorphism in E
i<n Ai i<Ln i
(thatis,forthevectorspacestructureaswellasthe—componentwise—partialorder
P Q
structure).
In particular, simplicial vector spaces, that is, spaces of the form Qn with the
positive cone (Qn)+ consisting of all vectors with non-negative coordinates, are
particularcasesofpseudo-simplicialspaces. Observethatforanypseudo-simplicial
space E, the space E is simplicial.
arch
Forasimplepseudo-simplicialspaceA,the canonical basis ofAisdefinedasthe
setT ofminimalelementstofA++ suchthatt∧(1 −t)=0,where∧denotesthe
arch A
meetoperationinthelattice-orderedgroupA . HenceT isa(finite)vectorspace
arch
basis of A and the map from Q to A that with any sequence (x ) associates
T t t∈T
x t is an isomorphism from Q onto A. It sends 1T to t, for all t ∈ T, and
t∈T t T t
1T to 1 .
P A
We denote by E the full subcategory of E whose objects are the pseudo-sim-
ps
plicial spaces.
We state without proof the following lemma, that summarizes some elementary
properties of pseudo-simplicial spaces.
Lemma 2.3. Let m be a natural number, let A , ..., A be simple pseudo-sim-
0 m−1
plicial spaces. Put A = A , and identify A with its canonical image in A,
i<m i i
for all i<m. Then the following assertions hold.
L
(i) The ideals of A are exactly the subsets of the form A for a subset I
i∈I i
of {0,1,...,m−1}.
L
6 J.T˚UMAANDF.WEHRUNG
(ii) The simple ideals of A are exactly the A , for i<m.
i
(iii) Id A∼=2m, a finite Boolean semilattice.
c
For a partiallyorderedvectorspaceE, the two binaryrelations ∝ and≍ onE+
can be refined as follows. For a positive, rational number λ and a, b ∈ E+, we
introduce the following notations:
a∝ b, if a≤λb;
λ
a≍ b, if a∝ b and b∝ a;
λ λ λ
a∝arch b, if a∝ b, for all rational ε>0;
λ λ+ε
a≍arch b, if a≍ b, for all rational ε>0.
λ λ+ε
Hence a∝arch b (resp., a≍arch b) in E implies that a∝ b (resp., a≍ b) in E ;
λ λ λ λ arch
the converse holds if E is simple.
3. Basic facts about refinement
We start with an easy and useful lemma.
Lemma3.1. LetM bearefinementmonoid, letI andT , i∈I,befinitenonempty
i
sets, let a, a (for i∈I and j ∈T ) be elements of M such that
i,j i
a= a , for all i∈I.
i,j
jX∈Ti
Then there are elements x (for ϕ∈T = T ) of M such that
ϕ i∈I i
a= x and a = x , Qfor all i∈I and j ∈T . (3.1)
ϕ i,j ϕ i
ϕ∈T ϕ∈T
X X
ϕ(i)=j
Proof. An easy induction on the cardinality of I. There is nothing to prove if
|I| = 1. Now assume that the claim holds for any set I of cardinality n ≥ 1 and
I′ = I ∪{k} 6= I. By the induction hypothesis on I we get elements x ∈ M for
ϕ
ϕ∈T such that
a= x and a = x , for all i∈I and j ∈T .
ϕ i,j ϕ i
ϕ∈T ϕ∈T
X X
ϕ(i)=j
Since also a = a and M is a refinement monoid, we get that there are
l∈Tk k,l
elements x ∈M, for ϕ∈T and l ∈T , such that
ϕ,l k
P
x = x and a = x
ϕ ϕ,l k,l ϕ,l
lX∈Tk ϕX∈T
for any ϕ∈T and l ∈T . Thus
k
a= a = x .
k,l ϕ,l
lX∈Tk lX∈TkϕX∈T
SinceT×T = T ,theindices(ϕ,l)areinone-to-onecorrespondencewith
k i∈I∪{k} i
functions ψ ∈T ×T . Moreover,for every i∈I and j ∈T we get
k i
Q
a = x = x = x .
i,j ϕ ϕ,l ψ
ϕX∈T ϕX∈T lX∈Tk ψ∈XT×Tk
ϕ(i)=j ϕ(i)=j ψ(i)=j
LIFTING SEMILATTICE DIAGRAMS 7
Finally, for every l∈T we get
k
a = x = x . (cid:3)
k,l ϕ,l ψ
ϕX∈T ψ∈XT×Tk
ψ(k)=l
Lemma 3.2. Let λ ≥ 1 be a rational number, let E be a dimension vector space,
let I be a finite set, let a (for i ∈ I) be elements of E+. Then the following are
i
equivalent:
(i) a ≍ a , for all i, j ∈I.
i λ j
(ii) There are elements b (for X ∈P(I)) of E+ such that
X
a = b +λ· b , for all i∈I.
i X X
X∈XP∗i(I) X∈XPi(I)
Proof. (ii)⇒(i) is trivial.
Now assume that (i) holds. By applying interpolation to the system of inequal-
ities λ−1a ≤a , for all i, j ∈I, we obtain a∈E such that λ−1a ≤a≤a , for all
i j i j
i, j ∈I. Hence, a≤a ≤λa, for all i∈I, so that there are a′, a′′ ∈E+ such that
i i i
a =a+a′ and(λ−1)a=a′+a′′,foralli∈I. ByLemma 3.1appliedtoM =E+
i i i i
and J ={0,1}, there are elements b of E+, for X ∈P(I), such that
X
a′ =(λ−1) b , a′′ =(λ−1) b , for all i∈I;
i X i X
X∈XPi(I) X∈XP∗i(I)
a= b .
X
XX∈P(I)
Therefore, for all i∈I,
a = b +(λ−1) b
i X X
XX∈P(I) X∈XPi(I)
= b +λ· b . (cid:3)
X X
X∈XP∗i(I) X∈XPi(I)
4. Flat and generic homomorphisms with simple target
Definition 4.1. Let m be a natural number, let A , ..., A , B be simple
0 m−1
objects of E , let f: A → B be a positive homomorphism, let λ ≥ 1 be a
d i<m i
rational number. We say that f is
L
(i) λ-flat,iff(1 )≍arch f(1 ),foralli,j <msuchthatf(1 ),f(1 )6=0.
Ai λ Aj Ai Aj
(ii) λ-generic, if for any simple pseudo-simplicial space C and for any λ-flat
homomorphism g: A →C such that
i<m i
f[Ai]6={L0}⇔g[Ai]6={0}, for all i<m,
there exists a homomorphism h: B →C such that g =h◦f.
The following simple result summarizes some of the basic properties of flat and
generic homomorphisms:
Lemma 4.2. Let m be a natural number, let A be a pseudo-simplicial space, let
A , ..., A , B, G be simple objects of E . Let α, β, λ be rational numbers such
0 m−1 d
that 1≤α≤β and 1≤λ.
(i) Every homomorphism f: A →B is λ-flat for some λ≥1.
i<m i
L
8 J.T˚UMAANDF.WEHRUNG
(ii) Every α-flat homomorphism from a pseudo-simplicial space to a simple
pseudo-simplicial space is β-flat.
(iii) Everyβ-generichomomorphism from apseudo-simplicial spacetoasimple
pseudo-simplicial space is α-generic.
(iv) Suppose that G is pseudo-simplicial, let f: A → G and h: A → B be
homomorphisms, andletg: Id G→Id B beah∨,0i-homomorphismsuch
c c
that Id h=g◦Id f. If f is λ-generic and h is λ-flat, then there exists a
c c
homomorphism g: G→B such that g =Id g and h=g◦f.
c
The situation of (iv) above may be summarized by the following commutative
diagrams:
vvvgvvvvvvI;;dcOO B (cid:127)g(cid:127)(cid:127)(cid:127)??BOO
IdcGIdccHHHHfHHHHH Idch G__??f?????? h
c
IdcA A
Proof. To prove (i) observe that if f(1 ) 6= 0, then it is an order-unit in B.
Ai
Thus for any i, j < m such that both f(1 ) and f(1 ) are nonzero, there exists
Ai Aj
λ ∈Q++ suchthatf(1 )≤λ f(1 ). Nowset λany positive rationalnumber
i,j Ai i,j Aj
larger than 1 and all λ .
i,j
Claims (ii) and (iii) are straightforward.
To prove(iv)observefirstthatg has alifting, thatis,a homomorphismg: G→
B such that Id g = g. Indeed, if g = 0, take g = 0, while if g is the unique
c
isomorphism from Id G to Id B, any nonzero positive homomorphism from G
c c
to B is a lifting of g, for example, if G= Q and if b ∈B++, the map g: G→B
m
defined by the rule g(x ,...,x )= x b, for all x , ..., x ∈Q.
0 m−1 i<m i 0 m−1
Now, for (iv) above, if h = 0, then any lifting g of g satisfies the required
(cid:0)P (cid:1)
conditions. So suppose that h 6= 0. Then, by the λ-genericity of f and the λ-
flatness of h, there exists a homomorphism g: G → B such that h = g ◦f. In
particular,g 6=0. Similarly,fromId h=g◦Id f andId h6=0followsthatg 6=0.
c c c
Therefore, since both G and B are simple, g lifts g. (cid:3)
We shallnow define the canonicalgenericmaps. We aregivena pseudo-simplic-
ialspaceAandah∨,0i-homomorphismf: Id A→2. We shallconstructasimple
c
pseudo-simplicial space G = Gen(A,f) and a lifting f: A → G of f with respect
to the Id functor.
c
Put A= A , for a natural number m and simple pseudo-simplicialspaces
i<m i
A , ..., A . We shall define a pseudo-simplicial space G = Gen(A,f) and, for
0 m−1
L
every rational number λ ≥ 1, a lifting f : A → G of f with respect to the Id
λ c
functor.
For all i < m, let T denote the canonical basis of A (see Section 2). Hence
i i
the T , for i < m, are mutually disjoint finite sets and T is a vector space
i i<m i
basis of A. Let I = {i<m|f(A )=1} (observe that A is a compact ideal of A,
i i
S
thus it belongs to the domain of f, see Lemma 2.3). Furthermore, we put
T = T , F =P(I)×T, and G=Q .
i F
i∈I
Y
LIFTING SEMILATTICE DIAGRAMS 9
ObservethatbothT andF arefinite,nonemptysets. Forarationalnumberλ≥1,
we define a linear map f : A→Q by its action on the elements of T . For
λ F i<m i
i<m and t∈T , we define f (t)=0 if i∈/ I, while, if i∈I, we put
i λ
S
f (t)= 1F +λ· 1F . (4.1)
λ (X,ϕ) (X,ϕ)
(XX,ϕ)∈F (XX,ϕ)∈F
i∈/X, ϕ(i)=t i∈X, ϕ(i)=t
For all i<m, let f denote the restriction of f to A .
i,λ λ i
Lemma 4.3. The map f is a positive homomorphism from A to Q , for all
i,λ i F
i<m, and it is nonzero iff i∈I.
Proof. For i ∈/ I, f = 0. Suppose now that i ∈ I. It is trivial that f defines
i,λ i,λ
a positive homomorphism from (Ai)arch ∼= QTi to QF. To prove that it is also
a nonzero positive homomorphism from A to Q , it is sufficient to prove that
i F
f (1 ) is an order-unit of QF. By observing that 1 = t and by using
i,λ Ai Ai t∈Ti
(4.1), we can compute:
P
f (1 )= 1F +λ· 1F .
i,λ Ai (X,ϕ) (X,ϕ)
(XX,ϕ)∈F (XX,ϕ)∈F
i∈/X i∈X
Since all the components of this vector relatively to all 1F , for (X,ϕ)∈ F, are
(X,ϕ)
positive, fi,λ(1Ti) is indeed an order-unit of QF. (cid:3)
ItfollowsfromLemma4.3thatf isapositivehomomorphismfromAtoQ . It
λ F
followsimmediatelyfromLemma4.3thatf : A→Q isaliftingofthesemilattice
λ F
mapf: Id A→2. Indeed,ifwetakethecanonicalisomorphismfromId Q to2,
c c F
the following diagram
can.
2 // Id Q
f OO uuuuuIuducufu::λc F
Id A
c
is commutative, since f [A ]6={0} iff f(A )=1, for all i<m.
λ i i
Lemma 4.4. The homomorphism f is λ-flat.
λ
Proof. For all i ∈ I, f (1 ) = f (1 ) is a linear combination of the vectors
λ Ai i,λ Ai
1F , for (X,ϕ) ∈ F, with coefficients either 1 or λ (see (4.1)). The conclusion
(X,ϕ)
follows. (cid:3)
We observeherethe relevanceofthe definitionofflatnessby using≍arch instead
of ≍: indeed, the vectors 1F , for (X,ϕ)∈F, do not belong to the positive cone
(X,ϕ)
of Q except if F is a singleton.
F
Now we come to the main result of this section:
Lemma 4.5. The homomorphism f is λ-generic.
λ
Proof. Let C be a simple pseudo-simplicialspace, let g: A →C be a λ-flat
i<m i
homomorphismsuchthatg[A ]6={0}ifff [A ]6={0}(thatis,i∈I),foralli<m.
i λ i
L
Set again A = Q for every i < m. We define elements a (for i < m) and a
i Ti i (t)
(for i<m and t∈T ) of C by
i
a =g(1 ), a =g(t).
i Ai (t)
10 J.T˚UMAANDF.WEHRUNG
Observe that a ∈ C+ while we can only say that a ∈ C+ . Moreover, a =
i (t) arch i
a . From the fact that g is λ-flat follows that a ≍arch a in C, for all i,
t∈Ti (t) i λ j
j ∈I, hence, by Lemma 3.2 applied to C (that has interpolation, because C is
arch
pPseudo-simplicial), there are elements b (for X ∈P(I)) of C+ such that
X arch
a = b +λ· b , for all i∈I. (4.2)
i X X
X∈XP∗i(I) X∈XPi(I)
Now, for fixed i∈I, we apply refinement to the equality
a = b +λ· b ,
(t) X X
tX∈Ti X∈XP∗i(I) X∈XPi(I)
which holds in C+ . We obtain elements c (for (X,i) ∈ P(I)×I and t ∈ T )
arch X,i,t i
of C+ such that
arch
a = c +λ· c (for all i∈I and t∈T ) (4.3)
(t) X,i,t X,i,t i
X∈XP∗i(I) X∈XPi(I)
b = c (for all X ∈P(I)). (4.4)
X X,i,t
tX∈Ti
Now we apply Lemma 3.1 to the system of equations (4.4) in C+ . We find
arch
elements d (for (X,ϕ)∈F) of C+ such that
(X,ϕ) arch
b = d (for all X ∈P(I)) (4.5)
X (X,ϕ)
ϕ∈T
X
c = d (for all X ∈P(I), i∈I, t∈T ). (4.6)
X,i,t (X,ϕ) i
ϕ∈T
X
ϕ(i)=t
We define a linear map h: QF →C by the rule
h(1F )=d , for all (X,ϕ)∈F. (4.7)
(X,ϕ) (X,ϕ)
HencehisapositivehomomorphismfromQF toC . Toverifythathisapositive
arch
homomorphismfrom Q to C, it suffices to verify that h(1F)∈C+. This is trivial
F
for I =∅ (then Q =Q). Suppose that I 6=∅. We compute:
F
h(1F)= h(1F )= d
(X,ϕ) (X,ϕ)
(XX,ϕ)∈F (XX,ϕ)∈F
= b (by (4.5)).
X
XX∈P(I)
Hence, it follows from (4.2) that h(1F) ≍ a in C for any i ∈ I. But a ∈ C+,
i arch i
thus h(1F)∈C+.
It remains to prove that h◦f (t) = g(t), for all i < m and all t ∈ T . This is
λ i
obvious if i∈/ I, in which case both sides of the equality are zero. So suppose that