Projective dimension in filtrated K-theory

0.3 MB·English

by  Rasmus Bentmann

PROJECTIVE DIMENSION IN FILTRATED K-THEORY
RASMUSBENTMANN
Abstract. Undermildassumptions,wecharacterisemoduleswithprojective
resolutions of length n ∈ N in the target category of filtrated K-theory over
2 a finite topological space in terms of two conditions involving certain Tor-
1 groups. WeshowthatthefiltratedK-theoryofanyseparableC∗-algebraover
0 any topological space with at most four points has projective dimension2 or
2 less. We observe that this impliesa universal coefficient theorem for rational
equivariant KK-theory over these spaces. As a contrasting example, we find
t a separable C∗-algebra in the bootstrap class over a certain five-point space,
c
O the filtrated K-theory of which has projective dimension 3. Finally, as an
applicationofourinvestigations,weexhibitCuntz-Kriegeralgebraswhichhave

7 projectivedimension2infiltratedK-theoryovertheirrespectiveprimitiveideal
1 space.

]
A
O
1. Introduction
.
h A far-reaching classification theorem in [7] motivates the computation of Eber-
at hard Kirchberg’s ideal-related Kasparov groups KK(X;A,B) for C∗-algebras A
m and B over a non-Hausdorff topological space X by means of K-theoretic invari-
[ ants. We are interested in the specific case of finite spaces here. In [9,10], Ralf

MeyerandRyszardNestlaidoutatheoreticframeworkthatallowsforageneralisa-
1
tion of Jonathan Rosenberg’s and Claude Schochet’s universal coefficient theorem
v
5 [15] to the equivariant setting. Starting from a set of generators of the equivariant
8 bootstrapclass,theydefineahomologytheorywithacertainuniversalityproperty,
7 which computes KK(X)-theory via a spectral sequence. In order for this universal
4 coefficient spectral sequence to degenerate to a short exact sequence, it remains
.
0 to be checked by hand that objects in the range of the homology theory admit
1 projective resolutions of length 1 in the Abelian target category.
2
Generalisingearlierresultsfrom[3,10,14]theverificationoftheabove-mentioned
1
condition for filtrated K-theory was achievedin [2] for the case that the underlying
:
v space is a disjoint union of so-called accordion spaces. A finite connected T -space
0
i
X X is an accordion space if and only if the directed graph corresponding to its
specialisation pre-order is a Dynkin quiver of type A. Moreover, it was shown in
r
a [2,10]that,ifX isafiniteT -spacewhichisnotadisjointunionofaccordionspaces,
0
thentheprojectivedimensionoffiltratedK-theoryoverX isnot boundedby1and
objects in the equivariant bootstrap class are not classified by filtrated K-theory.
TheassumptionoftheseparationaxiomT isnotalossofgeneralityinthiscontext
0
(see [11, §2.5]).
Therearetwonaturalapproachestotackletheproblemarisingfornon-accordion
spaces: one can either try to refine the invariant—this has been done with some
success in [10] and [1]; or one can hold onto the invariant and try to establish
projective resolutions of length 1 on suitable subcategories or localisations of the
categoryKK(X), in which X-equivariantKK-theory is organised. The latter is the
course we pursue in this note. We state our results in the next section.
The author was supported by the Danish National Research Foundation through the Centre
forSymmetryandDeformationandbytheMarieCurieResearchTrainingNetworkEU-NCG.
1
2 RASMUSBENTMANN
Parts of this paper are based on the author’s Diplom thesis [1] which was su-
pervised by Ralf Meyer at the University of Göttingen. I would like to thank my
PhD-supervisors, Søren Eilers and Ryszard Nest, for helpful advice and Takeshi
Katsura for pointing out a mistake in an earlier version of the paper.
2. Statement of Results
The definition of filtrated K-theory and related notation are recalled in §3.
Proposition1. Let X bea finitetopological space. Assumethat the ideal NT ⊂
nil
NT∗(X) is nilpotent and that the decomposition NT∗(X)=NT ⋊NT holds.
nil ss
Fix n∈N. For an NT∗(X)-module M, the following assertions are equivalent:
(i) M has a projective resolution of length n.
(ii) TorNT∗(X)(NT ,M) is a free Abelian group and TorNT∗(X)(NT ,M)
n ss n+1 ss
vanishes.
ThebasicideaofthispaperistocomputetheTor-groupsabovebywritingdown
projective right-module resolutions for the fixed module NT .
ss
LetZ bethe(m+1)-pointspaceontheset{1,2,...,m+1}suchthatY ⊆Z
m m
isopenifandonlyifY ∋m+1orY =∅. AC∗-algebraoverZ isaC∗-algebraA
m
withaminimaldistinguishedidealsuchthatthecorrespondingquotientdecomposes
as a direct sum of m orthogonal ideals. Let S by the set {1,2,3,4} equipped with
the topology {∅,4,24,34,234,1234}. A C∗-algebraover S is a C∗-algebratogether
with two distinguished ideals which need not satisfy any further conditions, see
[11, Lemma 2.35].
Proposition2. LetX beatopologicalspacewithatmost 4points. LetM =FK(A)
for some C∗-algebra A over X. Then M has a projective resolution of length 2 and
TorNT∗(NT ,M)=0.
2 ss
Moreover, we can find explicit formulas for TorNT∗(NT ,M); for instance,
1 ss
TorNT∗(Z3)(NT ,M) is isomorphic to the homology of the complex
1 ss
i −i 0
3 −i 0 i 3
0 i −i (iii)
(1) M(j4)−−−−−−−−→ M(1234\k)−−−−→M(1234).
(cid:16) (cid:17)
j=1 k=1
M M
A similar formula holds for the space S, see (6).
The situation simplifies if we consider rational KK(X)-theory, whose morphism
groups are given by KK(X;A,B) ⊗ Q; see [6]. This is a Q-linear triangulated
category which can be constructed as a localisation of KK(X); the corresponding
localisation of filtrated K-theory is given by A 7→ FK(A)⊗Q and takes values in
the category of modules over the Q-linear category NT∗(X)⊗Q.
Proposition 3. Let X be a topological space with at most 4 points. Let A and B
be C∗-algebras over X. If A belongs to the equivariant bootstrap class B(X), then
there is a natural short exact universal coefficient sequence
Ext1 FK (A)⊗Q,FK (B)⊗Q ֌KK (X;A,B)⊗Q
NT∗(X)⊗Q ∗+1 ∗ ∗
(cid:0) ։HomNT∗(X)⊗Q FK∗((cid:1)A)⊗Q,FK∗(B)⊗Q .
In [6], a long exact sequence is construc(cid:0)ted which in our setting, b(cid:1)y the above
proposition, reduces the computation of KK (X;A,B), up to extension problems,
∗
to the computation of a certain torsion theory KK (X;A,B;Q/Z).
∗
Thenextpropositionsaysthattheupperboundof2fortheprojectivedimension
in Proposition 2 does not hold in general.
PROJECTIVE DIMENSION IN FILTRATED K-THEORY 3
Proposition 4. There is an NT∗(Z )-module M of projective dimension 2 with
4
free entries and TorNT∗(NT ,M) 6= 0. The module M ⊗ Z/k for k ∈ N has
2 ss Z ≥2
projective dimension 3. Both M and M ⊗ Z/k can be realised as the filtrated
Z
K-theory of an object in the equivariant bootstrap class B(X).
As an application of Proposition 2 we investigate in §10 the obstruction term
TorNT∗ NT ,FK(A) forcertainCuntz-Kriegeralgebraswithfour-pointprimitive
1 ss
ideal spaces. We find:
(cid:0) (cid:1)
Proposition 5. There is a Cuntz-Krieger algebra with primitive ideal space Z
3
which fulfills Cuntz’ condition (II) and has projective dimension 2 in filtrated K-
theory. The same holds for the space S.
Therelevanceofthisobservationliesinthe following: if Cuntz-Kriegeralgebras
had projective dimension at most 1 in filtrated K-theory over their primitive ideal
space, this would lead to a strengthened versionof Gunnar Restorff’s classification
result [13] with a proof avoiding reference to results from symbolic dynamics.
3. Preliminaries
Let X be a finite topological space. A subset Y ⊆X is called locally closed if it
is the difference of two open subsets of X. The set of locally closed subsets of X is
denotedbyLC(X). ByLC(X)∗,wedenotethe setofnon-empty, connected locally
closed subsets of X.
Recall from [11] that a C∗-algebra over X is pair (A,ψ) consisting of a C∗-al-
gebra A and a continuous map ψ: Prim(A) → X. A C∗-algebra (A,ψ) over X is
called tight if the map ψ is a homeomorphism. A C∗-algebra (A,ψ) over X comes
with distinguished subquotients A(Y) for every Y ∈LC(X).
There is an appropriate version KK(X) of bivariant K-theory for C∗-algebras
over X (see [7,11]). The corresponding category, denoted by KK(X), is equipped
with the structure of a triangulated category; moreover, there is an equivariant
analogue B(X)⊆KK(X) of the bootstrap class [11].
As defined in [10], for Y ∈ LC(X), we let FK (A) := K A(Y) denote the
Y ∗
Z/2-gradedK-group of the subquotient of A associated to Y.
(cid:0) (cid:1)
LetNT(X)bethe Z/2-graded,pre-additivecategorywhoseobjectsetisLC(X)
and whose space of morphisms from Y to Z is NT (X)(Y,Z) – the Z/2-graded
∗
Abelian group of all natural transformations FK ⇒ FK . Let NT∗(X) be the
Y Z
full subcategory with object set LC(X)∗. We often abbreviate NT∗(X) by NT∗.
Every open subset of a locally closed subset of X gives rise to an extension of
distinghuishedsubquotients. Thecorrespondingnaturalmapsintheassociatedsix-
term exactsequence yield morphisms in the categoryNT, whichwe briefly denote
by i, r and δ.
A(left-)moduleoverNT(X)isagrading-preserving,additivefunctorfromNT(X)
to the category AbZ/2 of Z/2-graded Abelian groups. A morphism of NT(X)-
modules is a natural transformations of functors. Left-modules over NT∗(X)
are defined similarly. By Mod NT∗(X) we denote the category of countable
c
NT∗(X)-modules.
(cid:0) (cid:1)
FiltratedK-theory isthefunctorKK(X)→Mod NT∗(X) takingaC∗-algebra
c
AoverX tothecollection K (A(Y)) withtheobviousNT∗(X)-module
∗ Y∈LC(X)∗ (cid:0) (cid:1)
structure.
(cid:0) (cid:1)
LetNT ⊂NT∗ betheidealgeneratedbyallnaturaltransformationsbetween
nil
different objects, and let NT ⊂ NT∗ be the subgroup spanned by all identity
ss
transformations idY for objects Y ∈ LC(X)∗. The subgroup NT is in fact a
Y ss
subring of NT∗ isomorpic to ZLC(X)∗. We say that NT∗ decomposes as semi-
direct product NT∗ = NT ⋊NT if NT∗ as an Abelian group is the inner
nil ss
4 RASMUSBENTMANN
direct sum of NT and NT , see [2,10]. We do not know if this fails for any
nil ss
finite space.
We define right-modules over NT∗(X) as contravariant, grading-preserving,ad-
ditive functors G: NT∗(X) → AbZ/2. If we do not specify between left and right,
then we always mean left-modules.
The subring NT ⊂ NT∗ is regarded as an NT∗-right-module by the obvi-
ss
ous action: The ideal NT ⊂ NT∗ acts trivially, while NT acts via right-
nil ss
multiplicationinNT ∼=ZLC(X)∗. ForanNT∗-moduleM,wesetM :=M/NT ·
ss ss nil
M.
ForY ∈LC(X)∗wedefinethefreeNT∗-left-moduleonY byP (Z):=NT(Y,Z)
Y
for all Z ∈LC(X)∗ andsimilarly for morphisms Z →Z′ in NT∗. Analogously,we
definethefreeNT∗-right-moduleonY byQ (Z):=NT(Z,Y)forallZ ∈LC(X)∗.
Y
AnNT∗-left/right-moduleiscalledfreeifitisisomorphictoadirectsumofdegree-
shifted free left/right-modules on objects Y ∈ LC(X)∗. It follows directly from
Yoneda’s Lemma that free NT∗-left/right-modules are projective.
An NT-module M is called exact if the Z/2-graded chain complexes
···→M(U)−iYU→M(Y)−r−YY−\→U M(Y \U)−δ−YU−\→U M(U)[1]→···
are exact for all U,Y ∈ LC(X) with U open in Y. An NT∗-module M is called
exact if the corresponding NT-module is exact (see [2]).
In[10],thefunctorsFK areshowntoberepresentable,thatis,thereareobjects
Y
R ∈∈ KK(X) and isomorphisms of functors FK ∼= KK(X;R ,␣). Recall that
Y Y Y
the opposite category KK(X)op inherits a canonical triangulation from the trian-
gulated structure on KK(X). We let FK denote the stable homological functor on
KK(X)op represented by the same set of objects {R |Y ∈LC(X)∗}. We remark
Y
that KK(X;A,R ) does not identifycwith the K-homology of A(Y).
Y
We assume that the reader is familiar with standard notions from homological
algebra, such as Ext- and Tor-functors and their computation via projective resol-
utions.
We occasionally use terminology from [9,10] concerning homological algebra in
KK(X) relative to the ideal I := ker(FK) of morphisms in KK(X) inducing trivial
module maps on FK. An object A ∈∈ KK(X) is called I-projective if I(A,B) = 0
for every B ∈∈ KK(X). We recall from [9] that FK restricts to an equivalence
of categories between the subcategories of I-projective objects in KK(X) and of
projective objects in Mod NT∗(X) . Similarly, the functor FK induces an equi-
c
valence of projective objects in KK(X)op relative to the ideal I := ker(FK) and
(cid:0) (cid:1)
projective NT∗-right-modules. c
b c
4. Proof of Proposition 1
Recall the following result from [10].
Lemma 1 ([10, Theorem 3.12]). Let X be a finite topological space. Assume that
the ideal NT ⊂ NT∗(X) is nilpotent and that the decomposition NT∗(X) =
nil
NT ⋊NT holds. Let M be an NT∗(X)-module. The following assertions are
nil ss
equivalent:
(1) M is a free NT∗(X)-module.
(2) M is a projective NT∗(X)-module.
(3) M is a free Abelian group and TorNT∗(X)(NT ,M)=0.
ss 1 ss
Now we prove Proposition 1. We consider the case n = 1 first. Choose an
epimorphismf: P ։M for someprojectivemodule P, andletK be its kernel. M
has a projective resolution of length 1 if and only if K is projective. By Lemma
PROJECTIVE DIMENSION IN FILTRATED K-THEORY 5
1, this is equivalent to K being a free Abelian group and TorNT∗(NT ,K)= 0.
ss 1 ss
WehaveTorNT∗(NT ,K)=0ifandonlyifTorNT∗(NT ,M)=0becausethese
1 ss 2 ss
groupsareisomorphic. WewillshowthatK isfreeifandonlyifTorNT∗(NT ,M)
ss 1 ss
is free. The extension K ֌P ։M induces the following long exact sequence:
0→TorNT∗(NT ,M)→K →P −∼=→M →0.
1 ss ss ss ss
Assume that K is free. Then its subgroup TorNT∗(NT ,M) is free as well.
ss 1 ss
Conversely, if TorNT∗(NT ,M) is free, then K is an extension of free Abelian
1 ss ss
groups and thus free. Notice that P is free because P is projective. The general
ss
case n ∈ N follows by induction using an argument based on syzygies as above.
This completes the proof of Proposition 1.
5. Free Resolutions for NT
ss
TheNT∗-right-moduleNT decomposesasadirectsum S ofthe
ss Y∈LC(X)∗ Y
simple submodules S which are given by S (Y) ∼= Z and S (Z) = 0 for Z 6= Y.
Y Y LY
We obtain
TorNT∗(NT ,M)= TorNT(S ,M).
n ss n Y
Y∈LMC(X)∗
Our task is then to write down projective resolutions for the NT∗-right-modules
S . Thefirststepiseasy: wemapQ ontoS bymappingtheclassoftheidentity
Y Y Y
inQ (Y)tothegeneratorofS (Y). Extendedbyzero,thisyieldsanepimorphism
Y Y
Q ։S .
Y Y
In order to surject onto the kernel of this epimorphism, we use the indecompos-
able transformations in NT∗ whose range is Y. Denoting these by η : W → Y,
i i
1≤i≤n, we obtain the two step resolution
n
Q −(−η−1−η−2−··−·−η−n→) Q ։S .
Wi Y Y
i=1
M
In the notation of [10], the map n Q → Q corresponds to a morphism
i=1 Wi Y
φ: R → n R of I-projectives in KK(X). If the mapping cone C of φ is
Y i=1 Wi L φ
again I-projective, the exact triangle ΣC → R −→φ n R → C yields the
L φ Y i=1 Wi φ
projective resolution
L
n n
···→Q →Q [1]→ Q [1]→Q [1]→Q → Q →Q ։S ,
Y φ Wi Y φ Wi Y Y
i=1 i=1
M M
where Q =FK(C ). We denote periodic resolutions like this by
φ φ
◦
Q tt // n Q // Q →S .
φ i=1 Wi Y Y
If the mapping cone Cφ is notLI-projective, the situation has to be investigated
individually. Wewillseeexamplesofthisin§7and§9. Theresolutionsweconstruct
in these cases exhibit a certain six-term periodicity as well. However, they begin
with a finite number of “non-periodic steps” (one in §7 and two in §9), which can
beconsideredassymptomsofthedeficiencyoftheinvariantfiltratedK-theoryover
non-accordionspaces from the homological viewpoint.
6. Tensor Products with Free Right-Modules
Lemma2. FixY,Z ∈LC(X)∗ andη ∈NT (Y,Z). LetM beanNT∗-left-module.
∗
Then there is an isomorphism
(QY −η→∗ QZ)⊗NT∗ M ∼= M(Y)−M−−(η→) M(Z)
(cid:0) (cid:1)
6 RASMUSBENTMANN
of diagrams in the category of Z/2-graded Abelian groups.
Proof. The direct sum over all Z ∈LC(X)∗ of the maps
Q (Z)×M =NT(Z,Y)×M →M(Y), (µ,m)7→µ·m
Y
yields an NT∗-bilinear map Q ×M →M(Y). This induces a grouphomomorph-
Y
ism QY ⊗NT∗ M → M(Y). An inverse map M(Y) → QY ⊗NT∗ M is given by
m7→idY⊗mform∈M(Y). WehavefoundanisomorphismQY⊗NT∗M ∼=M(Y),
which is easily seen to be natural in Y. (cid:3)
Lemma 3. Let Q = mi Q ⊕ ni Q [1] for 1 ≤ i ≤ 3, m ,n ∈ N and
i j=1 Yi k=1 Zi i i
j k
Yi,Zi ∈ LC(X)∗ be free NT∗-right-modules. Suppose there is an exact sequence
j k L L
of module maps of the form
α β
Q //Q //Q
1 2 3
OO
(2) γ[1] γ
(cid:15)(cid:15)
Q [1]oo Q [1]oo Q [1].
3 2 1
β[1] α[1]
Then there is an exact triangle
(3) ΣR −γ→∗ R −α−→∗ R −β→∗ R ,
(3) (1) (2) (3)
in KK(X)op, where R = mi R ⊕ ni ΣR , such that FK applied to (3)
(i) j=1 Yi k=1 Zi
j k
yields a long exact sequence isomorphic to (2). In particular, if M = FK(A) for
L L
some A∈∈KK(X), then the induced sequence c
Q1⊗NT∗ M α∗ //Q2⊗NT∗ M β∗ // Q3⊗NT∗ M
OO
(4) γ∗[1] γ∗
(cid:15)(cid:15)
Q3⊗NT∗ M[1]oo Q2⊗NT∗ M[1]oo Q1⊗NT∗ M[1]
β∗[1] α∗[1]
is exact as well.
Proof. ByYoneda’slemma,thenaturaltransformationsinducingthemodulehomo-
morphisms (2) induce morphisms in KK(X)op between the respective representing
objects. We obtain the candidate triangle (3). The split-exactness of the sequence
(2) shows that it is also isomorphic to the six-term sequence obtained by applying
FK to the mapping cone triangle
(5) ΣR −→C −→R −β→∗ R .
c (3) β∗ (2) (3)
Inparticular,FK(C )isprojective. Since,moreover,C belongsto the localising
β∗ β∗
subcategory generated by the I-projective objects, it is itself I-projective by [9,
Theorem3.41].cNow the equivalence of categoriesbetween the I-projective objects
in KK(X)op and the projectivebNT∗-right-modules shows thatbthe triangles (3)
and (5) are isomorphic. Hence (3) is an exact triangle. Applyibng the homological
functor KK(X;␣,A) to the correspondingexact triangle in KK(X) yields the exact
six-term sequence (4). (cid:3)
7. Proof of Proposition 2
We may restrict to connected T -spaces. In [10], a list of isomorphism classes
0
of connected T -spaces with three or four points is given. If X is a disjoint union
0
of accordion spaces, then the assertion follows from [2]. The remaining spaces fall
into two classes:
PROJECTIVE DIMENSION IN FILTRATED K-THEORY 7
(1) all connected non-accordion four-point T -spaces except for the pseudo-
0
circle;
(2) the pseudocircle (see §7.2).
The spaces in the first class have the following in common: If we fix two of them,
sayX,Y,thenthereisanungradedisomorphismΦ: NT∗(X)→NT∗(Y)between
thecategoriesofnaturaltransformationsontherespectivefiltratedK-theoriessuch
that the induced equivalence of ungraded module categories
Φ∗: Modungr NT∗(Y) →Modungr NT∗(X)
c c
restricts to a bijective corresp(cid:0)ondence b(cid:1)etween exact(cid:0)ungraded(cid:1)NT∗(Y)-modules
and exact ungraded NT∗(X)-modules. Moreover, the isomorphism Φ restricts to
isomorphisms from NT (X) onto NT (Y) and from NT (X) onto NT (Y).
ss ss nil nil
In particular, the assertion holds for X if and only if it holds for Y.
The above is a consequence of the investigations in [1,2,10]; the same kind of
relationwas foundin [2] for the categoriesofnaturaltransformationsassociatedto
accordion spaces with the same number of points. As a consequence, it suffices to
verify the assertion for one representant of the first class—we choose Z —and for
3
the pseudocircle.
7.1. ResolutionsforZ . Wereferto[10]foradescriptionofthecategoryNT∗(Z ),
3 3
which in particular implies, that the space Z satisfies the conditions of Proposi-
3
tion1. ForthelocallyclosedsubsetsY 6=1234,theproceduredescribedin§5yields
projective resolutions as follows:
◦
tt
Q [1] //Q // Q →S , and similarly for S , S ;
1 4 14 14 24 34
◦
rr
Q [1] //Q [1]⊕Q [1]⊕Q [1] // Q →S ;
1234 1 2 3 4 4
◦
tt
Q //Q // Q →S , and similarly for S , S ;
234 1234 1 1 2 3
◦
Q tt //Q ⊕Q // Q →S , and similarly for S , S .
4 14 24 124 124 134 234
Therefore, by Lemma 3, TorNT∗(S ,M)=0 for Y 6=1234 and n≥1.
n Y
As we know from [10], the subset 1234 of Z plays an exceptional role. In the
3
notationofloc.cit.(withthedirectionofthearrowsreversedbecausewearedealing
(iii)
with a right-module), the kernel of the homomorphism Q ⊕Q ⊕Q −−−−→
124 134 234
Q is of the form
1234
Zoo 0 Z[1]
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ]]❀(cid:1)(cid:1)✄❀✄❀✄❀✄❀✄❀✄❀✄❀✄ ]]❀❀❀❀❀❀❀❀ (cid:127)(cid:127)⑦⑦⑦⑦⑦⑦⑦⑦ aa❉❉❉❉◦❉❉❉❉
Z2 oo Z 0 oo 0oo Z[1] oo ◦ Z2 .
^^❃❃❃❃❃❃❃❃ ]]❀(cid:1)(cid:1)✄❀✄❀✄❀✄❀✄❀✄❀✄❀✄ (cid:1)(cid:1)✄✄✄✄✄✄✄✄ __❅❅❅❅❅❅❅❅ }}③③③③◦③③③③
Zoo 0 Z[1]
It is the image of the module homomorphism
i −i 0
−i 0 i
0 i −i
Q ⊕Q ⊕Q −−−−−−−−→Q ⊕Q ⊕Q ,
14 24 34 (cid:16) (cid:17) 124 134 234
8 RASMUSBENTMANN
the kernel of which, in turn, is of the form
0 oo Z[1] Z[1]
(cid:1)(cid:1)✄✄✄✄✄✄✄✄ __❅(cid:127)(cid:127)⑦❅⑦❅⑦❅⑦❅⑦❅⑦❅⑦❅⑦ bb❊❊❊❊❊❊❊❊ ||②②②②②②②② aa❇❇❇❇◦❇❇❇❇
Zoo 0 Z[1]oo Z[1]3 oo Z[1]oo ◦ Z.
]]❀❀❀❀❀❀❀❀ (cid:127)(cid:127)⑦__❅⑦❅⑦❅⑦❅⑦❅⑦❅⑦❅⑦❅ ||②②②②②②②② bb❊❊❊❊❊❊❊❊ }}⑤⑤⑤⑤◦⑤⑤⑤⑤
0 oo Z[1] Z[1]
A surjection from Q4 ⊕Q1234[1] onto this module is given by δ1124i34 0i 0i , where
δ14 :=δ14◦r3 . The kernel of this homomorphism has the fo(cid:16)rm (cid:17)
1234 3 1234
Z[1]oo Z[1] 0
}}③③③③③③③③ }}③aa❉③❉③❉③❉③❉③❉③❉③❉ __❅❅❅❅❅❅❅❅ (cid:1)(cid:1)✄✄✄✄✄✄✄✄ ^^❂❂❂❂◦❂❂❂❂
Z[1] oo Z[1] Z[1]oo 0oo 0 oo ◦ 0.
aa❉❉❉❉❉❉❉❉ aa❉}}③❉③❉③❉③❉③❉③❉③❉③ (cid:127)(cid:127)⑦⑦⑦⑦⑦⑦⑦⑦ ]]❀❀❀❀❀❀❀❀ (cid:0)(cid:0)✁✁✁✁◦✁✁✁✁
Z[1]oo Z[1] 0
This module is isomorphic to the shifted version of ker(Q ։S ). Therefore,
1234 1234
we end up with the projective resolution
◦
rr
Q ⊕Q [1] //Q ⊕Q ⊕Q //Q ⊕Q ⊕Q //Q →S .
4 1234 14 24 34 124 134 234 1234 1234
ThehomomorphismfromQ124⊕Q134⊕Q234toQ4⊕Q1234[1]isgivenby 00−δ2434 ,
i i i
where δ4 :=δ4◦r2 . (cid:16) (cid:17)
234 2 234
Computing the tensor product of this complex with M as described in §6 and
taking homology, we obtain TorNT∗(NT ,M)=0 for n≥2, and that
n ss
TorNT∗(NT ,M)=TorNT∗(S ,M)
1 ss 1 1234
is the homology of the complex (1).
Example 1. For the filtrated K-module with projective dimension 2 constructed in
[10, §5] we get TorNT∗(NT ,M)∼=Z/k.
1 ss
Remark 1. As explicated in the beginning of this section, the category NT∗(S)
corresponding to the four-point space S defined in the introduction is isomorphic in
an appropriate sense to the category NT∗(Z ). As has been established in [1], the
3
indecomposable morphisms in NT∗(S) are organised in the diagram
δ
③③③r③δ③③③③==12i⑤❇⑤❇⑤❇⑤r❇◦❇⑤❇⑤❇⑤❇//>> 34❉❉❉δ❉❉i❉❉❉!! ✇✇✇r✇i✇✇✇✇✇;;2 ❍❍❍❍❍r❍i❍❍❍##
123 ◦ // 4 1 ◦ // 234 //1234 //123.
❉❉❉❉❉r❉❉❉!! ❇⑤r❇❇⑤i❇⑤⑤δ⑤❇⑤❇⑤❇⑤ >> ③③③i③③③③③== ●●●●●r●●●●## ✈✈✈✈✈i✈✈✈✈✈;;
13 ◦ //24 3
PROJECTIVE DIMENSION IN FILTRATED K-THEORY 9
In analogy to (1), we have that TorNT∗(S)(NT ,M)is isomorphic tothe homology
1 ss
of the complex
δ −r 0
−i 0 i
0 r −δ
(6) M(12)[1]⊕M(4)⊕M(13)[1]−−−−−−−−−→M(34)⊕M(1)[1]⊕M(24)
(cid:16) (cid:17)
(iδ i)
−−−−→M(234),
where M =FK(A) for some separable C∗-algebra A over X.
7.2. Resolutions for the pseudocircle. Let C = {1,2,3,4} with the partial
2
order defined by 1 < 3, 1 < 4, 2 < 3, 2 < 4. The topology on C is thus given by
2
{∅,3,4,34,134,234,1234}. Hence the non-empty, connected, locally closed subsets
are
LC(C )∗ ={3,4,134,234,1234,13,14,23,24,124,123,1,2}.
2
The partial order on C corresponds to the directed graph
2
4 2
• //•
❄❄❄❄ ⑧⑧⑧??
⑧
⑧⑧⑧⑧❄❄❄❄
⑧ (cid:31)(cid:31)
• //•
3 1.
The space C is the only T -space with at most four points with the property
2 0
that its order complex (see [10, Definition 2.6]) is not contractible; in fact, it is
homeomorphictothecircleS1. Therefore,bytherepresentabilitytheorem[10,§2.1]
we find
NT (C ,C )∼=KK (X;R ,R )∼=K R (C ) ∼=K∗ S1 ∼=Z⊕Z[1],
∗ 2 2 ∗ C2 C2 ∗ C2 2
that is, there are non-trivial odd natural tra(cid:0)nsformati(cid:1)ons FK(cid:0)C2(cid:1)⇒ FKC2. These
r δ i
are generated, for instance, by the composition C −→ 1 −→ 3 −→ C . This follows
2 2
from the description of the categoryNT∗(C ) below. Note that δC2◦δC2 vanishes
2 C2 C2
because it factors through r1 ◦i13 =0.
13 3
Figure 1 displays a set of indecomposable transformations generating the cat-
egory NT∗(C ) determined in [1, §6.3.2], where also a list of relations generating
2
the relations in the category NT∗(C ) can be found. From this, it is straight-
2
forward to verify that the space C satisfies the conditions of Proposition1.
2
5513
r i
(cid:27)(cid:27)
i r r δ
3 // 13KK 4 //14 r 6612II 3 // 1KK ◦ // 3LL
i i r δ
i ◦
((
1234
66
i δ◦
i r
i
(cid:19)(cid:19) (cid:21)(cid:21) (cid:19)(cid:19) (cid:18)(cid:18)
4 i // 234 r //23 r ((124 r // 2 ◦δ // 4
DD
r i
))24
Figure 1. Indecomposable natural transformations in NT∗(C )
2
10 RASMUSBENTMANN
Proceeding as described in §5, we find projective resolutions of the following
form (we omit explicit descriptions of the boundary maps):
◦
ss
Q [1] //Q [1]⊕Q [1] // Q →S , and similarly for S ;
123 1 2 3 3 4
◦
ss
Q [1] // Q ⊕Q // Q →S , and similarly for S ;
1 3 4 134 134 234
◦
uu
Q //Q //Q →S , and similarly for S , S , S ;
4 134 13 13 14 23 24
◦
rr
Q ⊕Q // Q ⊕Q //Q →S ;
3 4 134 234 1234 1234
◦
rr
Q ⊕Q [1] // Q ⊕Q //Q ⊕Q ⊕Q →Q →S ,
4 123 134 234 1234 13 23 123 123
and similarly for S ;
124
◦
qq
Q ⊕Q [1] //Q ⊕Q ⊕Q // Q ⊕Q →Q →S ,
234 1 1234 23 24 123 124 1 1
and similarly for S .
2
We get TorNT∗(S ,M) = 0 for every Y ∈ LC(C )∗ \ {123,124,1,2}, and
1 Y 2
TorNT∗(S ,M)=0 for all Y ∈LC(C )∗ and n≥2. Therefore,
n Y 2
TorNT∗(NT ,M)∼= TorNT∗(S ,M).
1 ss 1 Y
Y∈{12M3,124,1,2}
The four groups TorNT∗(S ,M) with Y ∈ {123,124,1,2} can be described expli-
1 Y
citlyasin§7.1usingthe aboveresolutions. Thisfinishesthe proofofProposition2.
8. Proof of Proposition 3
We apply the Meyer-Nest machinery to the homological functor FK ⊗ Q on
the triangulated category KK(X) ⊗ Q. We need to show that every NT∗ ⊗ Q
module of the form M = FK(A)⊗Q has a projective resolution of length 1. It is
easy to see that analogues of Propositions 1 and 2 hold. In particular, the term
TorNT∗⊗Q(NT ⊗Q,M) always vanishes. Here we use that Q is a flat Z-module,
2 ss
so that tensoring with Q turns projective NT∗-module resolutions into projective
NT∗⊗Q-module resolutions. Moreover, the freeness condition for the Q-module
TorNT∗⊗Q(NT ⊗Q,M) is empty since Q is a field.
1 ss
9. Proof of Proposition 4
The computations to determine the categoryNT∗(Z ) arevery similar to those
4
for the category NT∗(Z ) which were carried out in [10]. We summarise its struc-
3
ture in Figure 2. The relations in NT∗(Z ) are generated by the following:
4
• the hypercube with vertices 5,15,25,...,12345 is a commuting diagram;
• the following compositions vanish:
i r i r
1235−→12345−→4, 1245−→12345−→3,
i r i r
1345−→12345−→2, 2345−→12345−→1,
δ i δ i δ i δ i
1−→5−→15, 2−→5−→25, 3−→5−→35, 4−→5−→45;
• the sum of the four maps 12345→5 via 1, 2, 3, and 4 vanishes.
This implies that the space Z satisfies the conditions of Proposition 1.
4
In the following, we will define an exact NT∗-left-module M and compute
TorNT∗(S ,M). Using the same techniques as in the previous examples, one
2 12345