Chapter 2 a Homology Theory for Etale Groupoids

Chapter 2

A homology theory for etale

group oids

2.1 Intro duction

In this chapter weintro duce a homology theory for etale group oids. Etale group oids

serve as mo del for structures like leaf spaces of foliations, orbifolds, and orbit spaces of

actions by discrete groups. In this sense, etale group oids should b e viewed as general-

ized spaces.

In the literature one nds, roughly sp eaking, two di erent approaches to the study

of etale group oids. One approach is based on the construction of the convolution alge-

bras asso ciated to an etale group oid, in the spirit of Connes' non-commutative geometry

[25, 29 ], and involves the study of cyclic and Ho chschild homology and cohomology

of these algebras [20 , 29 ]. The other approach uses metho ds of algebraic top ology

such as the construction of the classifying space of an etale group oid and its sheaf

cohomology groups [13 , 57 , 72 ].

Our motivation in this chapter is twofold. First, wewant to give a more complete

picture of the second approach, by constructing a suitable homology theory which com-

plements the existing cohomology theory. Secondly,we use this homology theory as

the main to ol to relate the two approaches see Theorems 4.6.3, 4.6.4, 4.6.5 of chapter

Let us b e more explicit: In the second approach, one de nes for any etale group oid

G natural cohomology groups with co ecients in an arbitrary G -equivariant sheaf.

These were intro duced in a direct wayby Hae iger [55]. As explained in [72], they can

b e viewed as a sp ecial instance of the Grothendieck theory of cohomology of sites [2],

and agree with the cohomology groups of the classifying space of G [74]. Moreover,

these cohomology groups are invariant under Morita equivalences of etale group oids.

This invariance is of crucial imp ortance, b ecause the construction of the etale group oid

mo deling the leaf space of a given foliation involves some choices which determine the

group oid only up to Morita equivalence, cf 1.3.3. We complete this picture by con-

structing a homology theory for etale group oids, again invariant under Morita equiva-

lence, which is dual in the sense of Verdier duality to the existing cohomology theory.

Thus, one result of our work is the extension of \the six op erations of Grothendieck"[2] from spaces to leaf spaces of foliations.

42 Chapter 2: A homology theory for etale group oids

Our homology theory of the leaf space of a foliation re ects some geometric prop-

erties of the foliation. For example, byintegration along the b ers leaves it is related

to the leafwise cohomology theory studied by Alvarez Lop ez, Hector and others see

[1] and the references cited there. It also shows that the Ruelle-Sullivan currentof

a measured foliation see [25] lives in Ha iger's closed cohomology. The results of

Chapter 4 imply that our homology is also the natural target for the lo calized Chern

character.

The homology theory also plays a central role in explaining the relation b etween

the sheaf theoretic and the convolution algebra approaches to etale group oids, already

referred to ab ove. Indeed, the various cyclic homologies of etale group oids can b e shown

to b e isomorphic to the homology of certain asso ciated etale group oids; it extends the

previous results of Burghelea, Connes, Feigin, Karoubi, Nistor, Tsygan. This connec-

tion explains several basic prop erties of the cyclic and p erio dic homology groups, and

leads to explicit calculations see Chapter 4. The previous work on the Baum-Connes

conjecture for discrete groups, or for prop er actions of discrete groups on manifolds,

suggest that this homology will play a role in the Baum-Connes conjecture for etale

group oids.

From an algebraic p oint of view, our homology theory is an extension of the homol-

ogy of groups, while from a top ological p oint of view it extends compactly supp orted

cohomology of spaces. In this context, we should emphasize that even in the simplest

examples, the etale group oids which mo del leaf spaces of foliations involve manifolds

which are neither separated nor paracompact. Thus, an imp ortant technical ingredient

of our work is a suitable extension of the notions related to compactly supp orted section

of sheaves to non-separated non-paracompact manifolds. For example, as a sp ecial

case of our results one obtains the Verdier and Poincar e duality for non-separated

manifolds. Our notion of compactly supp orted sections is also used in the construction

of the convolution algebra of a non-separated etale group oid. We b elieve that this

extension to non-separated spaces has a much wider use that the one in this thesis, and

wehave tried to give an accessible presentation of it in the app endix. The results in

the app endix also play a central role in the calculation concerning the cyclic homology

of etale group oids in Chapter 4, and make it p ossible to extend the results of [20] for

separated group oids to the non-separated case.

We conclude this intro duction with a brief outline.

In section 2.2, we present the de nition of our homology theory and mention some

of its immediate prop erties.

In section 2.3, a covariant op eration ' for any map ' between etale group oids

is intro duced, which can intuitively b e thought of as a kind of \integration along the

b er" at the level of derived categories. We then prove a Leray sp ectral sequence for

this op eration. This sp ectral sequence is extremely useful. For example, we will use

it to prove the Morita invariance of homology. It also plays a crucial role in many

calculations in Chapter 4.

In section 2.4, we prove that the op eration L' has a right adjoint ' at the level

of derived categories, thus establishing Verdier duality. The Poincar e dualitybetween

Hae iger cohomology and our homology of etale group oids is an immediate conse-

quence.

In an app endix, we showhow to adapt the de nition of the functor X ; A

assigning to a space X and a sheaf A the group of compactly supp orted sections in

Section 2.2: Homology/compactly supp orted cohomology 43

suchaway that all the prop erties as expressed in [16], say can b e proved without

using Hausdor ness and paracompactness of the space X .

2.2 Homology/compactly supp orted cohomology

In this section we will intro duce the homology groups H G ; A for any etale

group oid G and any G -sheaf A. Among the main prop erties to b e proved will b e the

invariance of homology under Morita equivalence.

For any Hausdor space X , the standard prop erties of the functor which assigns

to a sheaf S its group of compactly supp orted sections X ; S are well known and

can b e found in any b o ok on sheaf theory. In the app endix, we showhow to extend

this functor to the case where X is not necessarily Hausdor , while retaining all the

standard prop erties. We emphasize that throughout this thesis, will denote this

extended functor.

n 1 0

and hence the spaces G and G Let us x an etale group oid G . The spaces G

for n 0 are assumed to satisfy the general conditions of 1.1.14, but we will not assume

0 0

that G is Hausdor . We write d = cdimG for the cohomological dimension of G .

Thus, for any n 0 and any Hausdor op en set U G , the usual cohomological

dimension of U is at most d.

2.2.1 Bar complex. Let A be a G -sheaf, and assume that A is c-soft cf. 2.5.1

we will brie y say that A is a \c-soft G -sheaf ". For each n 0, as a sheaf on G

n n

consider the sheaf A = Aon G constructed by pull-back along : G !

n n

G ; x ::: x =x . It is again a c-soft sheaf b ecause is etale. The groups

n 0 n 0 n

G ; A of compactly supp orted sections, intro duced in the App endix, together

c n

form a simplicial ab elian group:

/ /

2 1 0

/ / /

B G ; A: ::: 2.1

G ; A G ; A G ; A

c 2 c 1 c 0 / / /

with face maps:

n n1

d : G ; A ! G ; A 2.2

i c n c n1

n n1

de ned as follows. First, for the face map d : G !G cf. 1.1.13 there is

an evident map isomorphism in fact A ! d A , whose stalk at =x

n n1 0

::: x is the identity map for i 6= 0 and the action by g :A = A !A =

n 0 n x x

0 1

d A if i = 0. The map d in 2.2 is now obtained from this by summation along

n1 i

the bres see 2.5.12:

n1 n

G ; A G ; A

c n1 c n

G ;

n n1

G ; d G ; d d A A

c c i ! n1 n1

i i

Using our notation 2.5.13, B G ; A is the chain complex asso ciated to the simplicial

; A with the structure maps: vector space n 7! G

ag j g ; :::; g if i =0

1 2 n

d a j g ; :::; g = a j g ; :::; g g ; :::; g if 1 i n 1 ;

i 1 n 1 i i+1 n

a j g ; :::; g if i = n

1 n1

44 Chapter 2: A homology theory for etale group oids

s a j g ; :::; g = a j :::; g ; 1;g ; ::: ;

i 1 n i i+1

The homology groups H G ; A are de ned as the homology groups of the simplicial

ab elian groups 2.1, or, equivalently, as those of the asso ciated chain complex given

by the alternating sum = 1 d .

Similarly,any b ounded b elowchain complex S of c-soft sheaves gives rise to a

double complex:

B G ; S 2.3

and we de ne the hyperhomology H G ; S to b e the homology of the asso ciated total

complex.

2.2.2 Lemma. Any quasi-isomorphism S !I between boundedbelow chain com-

plexes of c-soft G -sheaves induces an isomorphism

H G ; S ! H G ; I :

n n

P ro of: The sp ectral sequence of the double complex 2.3 takes the form

= H H G ; S = H G ; S ; E

p q p+q

p;q

1 p p

where the E -term is the homology H G ; S of the complex G ; S . The

p c

p;q

lemma thus follows from 2.5.7.

2.2.3 c-soft resolutions. Let A b e an arbitrary G -sheaf. There always exists a

resolution:

0 d

0 !A !S ! :: : !S ! 0 2.4

by c-soft G -sheaves. For example, since the category of G -sheaves has enough injectives,

one can takeany injective resolution 0 !A !I and take S to b e the truncation

I softness of S then follows as in [16 ], p.55. Or, one can use for I the abby

Go dement resolution of A on the space G with its natural G -action, and truncate it.

In the case of a smo oth etale group oid and working over R, one also has the standard

resolution:

0 1

0 !A!A !A ! :: :

obtained from the G -sheaves of di erential forms on G . Note that the last two

resolutions are functorial in A.

Any resolution 2.4 maps into the truncated injective one. And, similarly, given

two resolutions 0 !A !S and 0 !A !I , there is a resolution R e.g. the

truncated injective one and a diagram:

2.5

I A

S R

which commutes up to homotopy.

Section 2.2: Homology/compactly supp orted cohomology 45

2.2.4 De nition of homology. Let A b e an arbitrary G -sheaf, and let 0 !A!

0 d

S ! :: : S ! 0 b e a c-soft resolution. Then S is a b ounded chain complex non-

zero in degrees b etween d and 0, and we de ne the homology groups H G ; Atobe

H G ; S . By 2.2.3 see 2.5 ab ove and Lemma 2.2.2, this de nition is indep endent

of the choice of the resolution. Observe that

H G ; A=0 for all n<d:

2.2.5 Extreme cases.

1: If G is a p oint, i.e. if G is a discrete group, then H G ; A is the usual group

homology of G .

2: If G is a discrete group oid, G is a simplicial set, and H G ; A is the usual

simplicial homology of G with twisted co ecients.

3: If G is a Hausdor space X viewed as a \trivial" group oid, 1.1.3.1 then

H G ; A= H X ; A is the usual cohomology with compact supp orts although

graded di erently. So the sp ectral sequence o ccurring in the pro of of lemma 2.2.2

could b e written as

H H G ; A= H G ; A :

p p+q

2.2.6 Long exact sequence. Any short exact sequence:

0 !A !B !C ! 0

of G -sheaves induces a long exact sequence in homology:

::: ! H G ; C ! H G ; A ! H G ; B ! H G ; C ! :::

n+1 n n n

The pro of is standard. The truncated Go dement resolutions give a short exact se-

quence of resolutions 0 !S A !S B !S C ! 0.

2.2.7 Compactly supp orted cohomology. It is sometimes more convenient to re-

index the homology groups and to see them as compactly supportedcohomology groups

see also 2.2.5.3 ab ove. Because of this, we de ne:

G ; =H G ; H

B G ; A".The same applies to the functors L ' which give a precise meaning to \H

n !

intro duced in the next paragraph: if ' : K!Gis a homomorphism, we de ne

R ' := L ' : Ab K ! AbG . With these notations, Leray sp ectral sequence of

! n !

q 2 p

G ; R ' A= 2.3.4 will b ecomes a cohomological sp ectral sequence with E -term H

p+q

H K ; A. If the b ers x=' are oriented k -dimensional manifolds, the transgression

of this sp ectral sequence will give \the integration along the b ers" map:

: H G ; R ! H K ; R:

c c

b er

G 2.2.8 Basic cohomology. Let G b e a smo oth etale group oid. The space

c;basic

of compactly supp orted basic forms is de ned as the Cokernel of:

d d

0 1

; ! G G

c c

46 Chapter 2: A homology theory for etale group oids

where d ;d are the maps coming from the nerveof G . In other words, G =

0 1

c;basic

. The basic compactly supportedcohomology of G , denoted H G , is de ned

c;basic

as the cohomology of the complex G with the di erential induced by DeRham

c;basic

di erential on G . There is an obvious pro jection from the reindexed homology see

2.2.7:

j : H G ; R ! H G ; 2.6

c c;basic

which is an isomorphism if G is prop er cf. 2.3.9. In this case we also have:

G f! 2 G : ! is Ginvariant; and supp ! is compact in M ]

c;basic

where : G! M is the pro jection into the quotient space of G ; see also 2.3.9. This

map asso ciates to ! 2 G the G -invariant form! ~ on G , given by:

c;basic

!~ x = ! y g:

x!y

2.2.9 Functoriality. Compactly supp orted cohomology of spaces 2.2.5.3 is co-

variant along lo cal homeomorphisms and contravariant along prop er maps. Anal-

ogous prop erties hold for homology of etale group oids. Consider a homomorphism

' : K!G between etale group oids.

n n

1: Supp ose that ' is prop er, in the sense that each ' : K !G is a prop er

map cf. 2.5.14. Then for any G -sheaf A one obtains homomorphisms:

n n

G ; A ! K ; ' A

c n c n

by pullback, and hence a homomorphism:

' : H G ; A ! H K ; ' A :

n n

In other words, homology is contravariant along prop er maps.

n n

2: Supp ose ' is etale, in the sense that each ' : K !G is a lo cal homeo-

morphism it is not dicult to see that the assumption is only ab out ' . Let S be a

c-soft G -sheaf. For the sheaf S = S onG summation along the b ers de nes a

homomorphism:

' ' S =' ' S ! S ;

n ! n !

n n n n

and hence a homomorphism :

n n

K ; ' S ! G ; S :

c n c n

These homomorphisms, for each n 0, commute with the face op erators 2.2. Since

the functor ' is always exact and preserves c-softness b ecause ' is etale, this gives

for each G -sheaf A a homomorphism:

H K ; ' A ! H G ; A :

n n

3: Supp ose that ' is etale, and moreover supp ose that for each n the square:

Section 2.2: Homology/compactly supp orted cohomology 47

n 0

2.7

K K

n 0

G G

is a pullback. Morphisms of this kind are exactly the pro jections X / G!G asso-

ciated to etale G -spaces X . For sucha ', there is an exact functor:

' : Ab K ! AbG

which preserves c-softness. at the level of underlying sheaves, it is simply the functor

0 0

' : AbK ! AbG of 2.5.9. For any c-soft K -sheaf B , there is a natural

0 !

isomorphism:

n n n

K ; B = K ; B = G ;' B = G ; ' B = G ; ' B ;

c n n c n c n ! c 0 ! c ! n

n n n

for any n 0. These yield an isomorphism

H K ; B = H G ; ' B ;

n n !

for any K -sheaf B .

Note that even if ' is not etale, a functor ' can b e de ned in this way but it is

no longer exact. See also 2.3.5.4.

2.2.10 Hyp erhomology. Consider any b ounded b elowchain complex A of G -

sheaves. Let A !R b e a q.i. into a b ounded b elowchain complex of c-soft

G -sheaves. Suchan R can b e constructed for example by considering a resolution

0 d

A !S ! :::S ! 0 as in 2.2.3 and then taking the total complex of the

double complex S p; q 2 Z; d p 0. De ne the hyp erhomology H G ; A

to b e the homology of the total complex asso ciated to the double complex B G ; R .

This de nition of H G ; A do es not dep end on the choice of the resolution R cf.

lemma 2.2.2.

Prop osition 2.2.11 Hyperhomology spectral sequences Let A beaboundedbelow

chain complex of G -sheaves as above, and consider for each q 2 Z the homology G -sheaf

H A . Therearespectral sequences:

E = H G ; H A = H G ; A ; 2.8

p q p+q

p;q

E = H H G ; A = H G ; A : 2.9

p q p+q

p;q

P ro of: Consider the truncated Go dement resolution 0 !A !S ! ::: !

S ! 0. It has the prop erty that for each q , it also yields c-soft resolutions of the

cycles Z , the b oundaries B and the homology H A . Write C for the triple complex:

q q q

p r

; ; S C = G

p;q ;r c

and let D b e the double complex:

D = C :

n;q p;q ;r

p+q =n

48 Chapter 2: A homology theory for etale group oids

The total complex of C , and hence also that of D , compute H G ; A . Furthermore, by

the prop erty of the resolution just mentioned and the fact that G ; preserves

exact sequences of c-soft sheaves wehave for xed p and r that:

p r

H C = G ; H S :

q p;;r c q

Hence, for a xed n,

p r

G ; H S : H D =

c q q n;

p+r =n

q 0 1 q

But H A !H S !H S ! :: : is a resolution of H A , so for a xed

q q

q the double complex G ; H S computes H G ; H A . Thus:

c q q

H H D =H G ; H A ;

n q ; n q

and 2.8 is simply the sp ectral sequence H H D = H TotD for the double

n q n+q

complex D . The second sp ectral sequence is obvious.

2.2.12 Cap pro duct. For an etale group oid G , the Ext-groups 1.2.6 act on the

homology byacap product:

H G ; B Ext B ; A ! H G ; A : 2.10

n np

This follows from the general considerations in 1.2.6. For example, for p =1 an

elementof Ext B ; A can b e represented by an exact sequence 0 B E

A 0, which yields a b oundary map H G ; B ! H G ; A for the long exact

n n1

sequence of 2.2.6. For p>1, the cap pro duct can b e constructed in the same wayby

decomp osing a longer extension 0 B E ::: E A 0into

1 n

short exact sequences.

In particular, when working over R, this yields a simple description of the cap

pro duct relating homology and cohomology of etale group oids:

H G ; B H G ; A ! H G ; B A :

n np R

The cap pro duct satis es the usual \pro jection formula" for a morphism :

C!A. Explicitly, induces : H G ; C ! H G ; A and : Ext B ; C !

p p

Ext B ; A, and wehave for any u 2 H G ; B and 2 Ext B ; C that:

u \ =u \ :

For p = 1 this is just the naturality of the exact sequence 2.2.6.

2.2.13 Remark. The d b oundary of the hyp erhomology sp ectral sequence 2.8:

: H G ; H A ! H G ; H A

p q p2 q +1

p;q

is given by the cap pro duct with an element u A 2 Ext H A ; H A : Let

q q q +1

: Z A !H A b e the quotient map from the sheaf of cycles Z A . Then

q q q q

the extension

Z A A Z A 0 0 H A

q q +1 q +1 q

de nes an element v 2 Ext H A ; Z A , and u A is A . This is

q q +1 q q +1

immediate from the construction of the sp ectral sequence pro of of 2.2.11, and the

general description of the b oundaries of the sp ectral sequence induced by a double complex.

Section 2.3: Leray sp ectral sequence; Morita invariance 49

2.2.14 Remark. Recall that a top ological category G is said to b e etale if all its

structure maps are lo cal homeomorphisms. Thus, such a category is given by maps

1 1

as in 1.1, except for the absence of an inverse i : G !G . The de nitions and

the results of this section hold equally well for the more general context of such etale

categories, and for this reason wehave tried to write the pro ofs in suchaway that they

apply verbatim to this general context. The same is true for the next section, provided

one takes sucient care to de ne Morita equivalence for categories in the appropriate

way. Because of this we need another technical to ol when we deal with etale categories;

it is a variantofawell known principle due to Segal see Prop.2:I in [93 ]:

2.2.15 Lemma and de nition: Let G and H be etale categories. Acontinuous

functor ' : G!H is cal led a strong deformation retract of H if thereisacontinuous

functor : H!G cal ledretraction and a continuous natural transformation of

functors F : ' ! Id cal led strong deformation retraction such that ' =

Id ;F' c =id for al l c 2G and the maps '; ; F are etale.

G ' c

In this case, for any H-sheaf A, ' induces an isomorphism:

H G ; ' A~!H H; A:

proof : Denote ' by l . Let the map induced by ':

:B G ; ' A ! B H; A; ajg ; :::; g =aj' g ; :::; ' g :

1 n 1 n

0 0

preserves c-softness [63] so it is enough to ! ShG Since ' is etale, ' : ShH

prove that is a homotopy equivalence of chain complexes, when A is c-soft. De ne a

chain map:

:B H; ' A ! B G ; A; ajh ; :::; h =aF t h j h ; :::; h :

1 n 1 1 n

Wehave =Id and is homotopic to Id by the following homotopy:

h : B H; ' A ! B H; ' A; h = 1 h ;

+1 i

i=0

h ajh ; :::; h = ajh ; :::; h ;Fs h ;l h ; :::; l h :

i 1 n 1 i i i+1 n

2.3 Leray sp ectral sequence; Morita invariance

In this section we construct for each morphism ' : K!Gbetween etale

group oids a functor ' from c-soft K -sheaves to c-soft G -sheaves. We derive a Leray

sp ectral sequence for this functor 2.3.4, of which the invariance of homology under

Morita equivalences will b e an immediate consequence 2.3.6.

2.3.1 Comma group oids of a homomorphism. Let ' : K!G b e a homomor-

phism of etale group oids. For each p oint x 2G consider the \comma group oid" x=',

0 1

whose ob jects are the pairs y; g : x ! ' y where y 2K and g 2G . An arrow

0 0 0 0

in x=' is an arrow k : y ! y in K such that ' k g = g . When k : y; g ! y ;g

equipp ed with the obvious b ered pro duct top ology, x=' is again an etale group oid.

50 Chapter 2: A homology theory for etale group oids

It should b e viewed as the b er of ' ab ove x; more exactly, there is a commutative

diagram see also 2.3.7:

x='

Note that an arrow g : x ! x in G induces a homomorphism:

g : x =' ! x=' 2.11

by comp osition. Thus the group oids x=' together form a right G -bundle of group oids.

0 0

If ' : K !G is a lo cal homeomorphism, then it is a G -sheaf of group oids.

More generally, for any A G the comma groupoid A=' is de ned by:

[

i i i 1

A=' = x=' K G ;i 2f0; 1g

x2A

with the induced top ology. The nerveof A=' consists of the spaces:

1 n

n 1 1

A=' = fy ::: y ;'y x:k 2K ;g 2G ;x 2 Ag :

0 n n i

When ' = id : G!G, these are simply denoted by x=G , A=G . Dually one de nes

the comma group oids '=x; '=A; G =x; G =A consisting on arrows \going into x".

2.3.2 The functors ' ;L' ; L' . Let ' : K!G b e as ab ove, and let A be a

! n ! !

K -sheaf. We de ne a simplicial G -sheaf B '; A in analogy with the de nition of the

n 1 0

bar-complex 2.2.1. On the spaces K G which form the nerveof G =', cf.

2.3.1 of strings of the form:

g ' k ' k

' y x ::: ' y

n 0

we de ne the maps:

1 0 n

G !K ; k ; :::k ;g 7! t k ; : K

1 n 1 n

1 0 n

G !G ; k ; :::k ;g 7! s g : : K

1 n n

Notice that any is etale. For any n 0we set:

B '; A= A :

n n !

By 2.5.9, the stalk at x 2G is describ ed by:

x ; A=B x='; A : 2.12 B '; A =

n n x c

n x n

This gives us the stalk-wise de nition of the simplicial structure on B '; A. To

check the continuity, let us just remark that the b oundaries can b e describ ed globally.

Indeed, using the maps:

n 1 1 n n1 1 1 n1

d : K G =G =' !K G =G ='

0 0

G G

Section 2.3: Leray sp ectral sequence; Morita invariance 51

coming from the nerveof G =' see 1.1.13, 2.3.1, wehave = d for all

n n1 i

0 i n and there are evident maps A! d A compare to the de nition

n i n1

of 2.2; the b oundaries of B '; A are in fact:

A : A!

A! d d A = d

n1 ! n1 ! i ! n1 ! i ! n !

n1 n1 i n n

To describ e the action of G on B '; A, let g : x ! x b e an arrowinG . The homo-

A which, via A ! B x='; morphism 2.11 induces an obvious map B x =';

x x

! B '; A : 2.12, is the action by g : B '; A

x x

If S is a c-soft K -sheaf, L' S is de ned as the chain complex of G -sheaves as-

so ciated to the simplicial complex B '; S . If S is a b ounded b elowchain complex

of c-soft K -sheaves, de ne L' S as the total complex of B '; S . For an arbitrary

K -sheaf A, L' A is de ned to b e B '; S where S is a resolution of A as in 2.2.4.

More generally,we de ne L' A for any b ounded b elowchain complex of K -sheaves

using a resolution A !R as in 2.2.10. As in the case of homology cf. 2.2.4 ,

we see that L' is well de ned up to quasi-isomorphism; in particular, the \derived

functors":

L ' =H L' : Ab K ! AbG

n ! n !

are well de ned up to isomorphism. For n =0we simply denote L ' : Ab K !

0 !

Ab G by ' .

Prop osition 2.3.3 For any x 2G , thereare isomorphisms:

L ' A H x='; A for all x 2G : 2.13

n ! x n

Proof: This is an immediate consequence of relation 2.12, and the fact that 's

preserve c-softness since they are induced by etale maps.

Theorem 2.3.4 \Leray-Hochschild-Serrespectral sequence" For any homomorphism

' : K!G between etale groupoids and any K -sheaf A there is a natural spectral

sequence:

E = H G ; L ' A= H K ; A :

p q ! p+q

p;q

Proof: The sp ectral sequence follows from an isomorphism:

H K ; A 2.14 H G ; L' A

and 2.2.11 applied to L' A.

To prove 2.14 we consider the double complex C A=B G ; B '; A and

p;q p q

we show that there are maps C A ! B K ; A, functorial in A, such that the

0;q q

augmented complex

::: !C A !C A !C A ! B K ; A ! 0 2.15

2;q 1;q 0;q q

is acyclic for any c-soft K -sheaf A.

Using the diagram:

p+1 q

/ p

G K

1 q

0 /

G K

q O

0 q

K K

52 Chapter 2: A homology theory for etale group oids

where ; ; ; are those de ned b efore, v; w are the pro jections into the rst com-

q q q p

p onents, u is the pro jection into the last comp onents and = wv ,wehaveby the

general prop erties of the App endix:

C A = G ; A

p;q c q !

p q

A = G ; u v

c !

q p+1

= K G ; v A

= K ; v A

c q !

A ; = K ;

c q ! q

B K ; A = K ; A :

q c

Via these equalities, the augmented chain complex 2.15 commies from an aug-

q k

y has the form: mented simplicial sheaf on K whose stalk at x :: :

M M M

::: ! A ! A ! A !A :

x x x x

g g g

f f f

1 2 1

' y x x x ' y x x ' y x

0 1 2 0 1 0

This is in fact the augmented bar complex computing the homology of the con-

tractible, discrete category G =' y with constant co ecients A . In particular it is

acyclic with the usual contraction f; g ; :::; g ; a 7! 1;f;g ; :::; g ; a.

1 n 1 n

2.3.5 Remarks and examples.

1. From the construction of the sp ectral sequence, and Remark 2.2.13, it follows

that the b oundary at the level of E is of typ e:

p;q

2 2 2

d = \u : E ! E with u 2 Ext L ' A;L ' A

q q G q ! q +1 !

p;q p;q p2;q +1

constructed as in 2.2.13 and replacing A by L' A.

2. The isomorphism 2.14 is actually a consequence of the quasi-isomorphism

L' pt = pt where pt is the map into the trivial group oid; this is a particular case

! ! !

of the naturality prop erty L' L = L' \up to quasi-isomorphism", which can

! ! !

b e proved in an analogous way. Compare to [97].

3. If ' : K!G is etale , S2Ab K , then there is no need of c-soft resolutions

to de ne L' S . Indeed, the condition on ' implies that the maps de ned in 2.3.2

! n

are etale, so there is a quasi-isomorphism L' S'B '; S .

4. Let ' : H!G b e a morphism for which all the squares in 2.7 are pullbacks.

0 0

K ! AbG \extends" to a functor Recall that in this case, the functor ' : Ab

0 !

' : AbK ! AbG , making the diagram:

forget

Ab K

AbK

forget

AbG

commute. This simple minded functor of 2.2.9 agrees up to quasi-isomorphism with

the functor L' , describ ed in 2.3.2. Indeed, for such a morphism ' and a p oint x 2G !

Section 2.3: Leray sp ectral sequence; Morita invariance 53

the comma group oid x=' is a space or more precisely, equivalent to the group oid cor-

resp onding to a space, cf. 1.1.3.1. In this case, the sp ectral sequence 2.3.4 degenerates

for c-soft sheaves B but not for arbitrary sheaves. If ' is moreover etale, it do es

always degenerate, and yields the isomorphism already proved in 2.2.9.3.

Corollary 2.3.6 \Morita invariance" For any Morita equivalence ' : K!G and

any G -sheaf A there is a natural isomorphism

H G ; A H K ; ' A:

p p

Proof : Theorem 2.3.4 gives a sp ectral sequence H G ; L ' ' A= H K ; ' A.

p q ! p+q

By 2.13 the stalk of L ' ' A at a p oint x 2G computes the homology of the nerve

q !

of x='.If' is a Morita equivalence, this nerve is a contractible simplicial set. Thus,

the sp ectral sequence degenerates to give an isomorphism:

H K ; ' A : H G ; L ' ' A

p p 0 !

It thus suces to observe that the G -sheaf L ' ' A is isomorphic to A itself.

0 !

2.3.7 Fib ered pro ducts of group oids. For homomorphisms ' : H!G and

: K!G, their b ered pro duct H K :

H K

0 1 0

is constructed as follows. The space of ob jects is the space H G K

0 0

G G

0 0

of triples y; g; z where y 2H ;z 2K and g : ' y ! z in G . An arrow

0 0 0 0 0

y; g; z ! y ;g ;z is a pair of arrows h : y ! y in H and k : z ! z in K such

that g ' h = k g . The group oid H K is again etale if G ; H; K are. This notion

of b ered pro duct is the appropriate one for group oids and generalized morphisms

describ ed in 1.1.6 and 1.1.9. In particular, if : K!G is a Morita equivalence, then

so is p : H K!H.

Prop osition 2.3.8 Change-of-base formula Consider a beredproduct H K of

etale groupoids as in 2.3.7. For any c-soft K -sheaf S , thereisacanonical quasi-

isomorphism:

' L' S 'Lp q S :

! !

Proof: For y 2H , the comma group oid y =p is Morita equivalent to the comma

group oid ' y = ,by a Morita equivalence y =p ! ' y = whichiscontinuous in y

0 0 0 0

and which resp ects the action by H. Using this observation, the prop osition follows

in a straightforward way from 2.3.6 and 2.5.11.

2.3.9 Orbifolds. As wehave already mentioned in 1.1.3.5, orbifolds are characterized

by etale group oids which are prop er. Let G b e such a group oid. The \leaf space" M

of G i.e. the space obtained from G dividing out by the equivalence relation x y

54 Chapter 2: A homology theory for etale group oids

i there is an arrowinG from x into y , will b e a Hausdor space; it is the underlying

space of the orbifold induced by G [75]. The obvious pro jection : G! M induces a

sp ectral sequence:

H M ; L A= H G ; A;

p q ! p+q

for any A2 Ab G . The stalk of L at x 2 M is:

q !

L H G ; A ;

q ! x q x~ x~

0 1

wherex ~ 2G is any lift of x, and G is the nite group f 2G : s = t =~xg

this follows from 2.3.3 and the Morita equivalence x= G .

In particular, for A2Sh G , the sp ectral sequence degenerates and gives an

isomorphism:

H G ; A H M ; A: 2.16

This also shows that the \co-invariants functor":

G ! Vs :=H G ; : Sh

G 0

is left exact and that H G ; see 2.2.7 are the right derived functors of .

2.3.10 Relations to cyclic cohomology. As an illustration, wenow brie y describ e

how the Ho chschild and the cyclic homology of the convolution algebra can b e describ ed

in terms of our sheaf homology.We refer to Chapter 4 for details.

Let G b e a not necessarily Hausdor smo oth group oid. Weintro duce the group oid

0 1 0

G , where B = f 2G : s = t g is the space of of lo ops G =B /

0 1

lo ops with the G -action given by conjugation ; g 7! g g. It has B as space of

ob jects, while the arrows b etween two lo ops and are all the arrows g from s

0 1 0

to s with the prop erty = g g . This group oid app ears in the computation of

the cyclic/Ho chschild cohomology of the convolution algebra C G asso ciated to the

group oid G where, again, in the non-Hausdor case one uses the adapted of the

G . The Ho chschild homology can b e expressed in terms of App endix to de ne C

our homology, as:

; HH C G = H G ; A

is the asso ciated simplicial sheaf where A is the sheaf of smo oth functions, and A

de ned in 4.5.6 of Chapter 4. Using Connes' category , and its version see

1.5.5, these homology groups are isomorphic to H G ; A . We remark that

G G as units, and we get, as \lo calization at units":

q 1

G = H G ; : HH C

[1]

p+q =n

In order to describ e the cyclic homology of the convolution algebra in terms of

our homology theory, one intro duces a new category ^ G which is the etale

n+1

;id = id ; , category obtained from G by imp osing the relations t

[n]

is a sheaf on ^ G ; Prop osition 4.6.1 and for all 2 B ;n 0. Then A

Prop osition 4.5.7 see also the de nition in 4.4.1 will imply:

G = H ^ G ; A : HC C

tw c

Section 2.4: Verdier duality 55

In cyclic homology, the previous \lo calization at units" b ecomes:

HP C G = H G ;i 2f0; 1g:

i i+2k

[1]

More generally,any G -invariant subspace OB de nes a group oid G =

O / G G , and the lo calized Ho chschild and cyclic homology indicated by the

subscript O . When O is elliptic i.e. or d < 1, for all 2O, it will b e shown in

Theorem 4.6.4 that:

G = H G ;i2f0; 1g: 2.17 HP C

O i+2k O i

2.4 Verdier duality

In this section all sheaves are sheaves of R- mo dules, i.e. real vector spaces we

can actually use any eld of characteristic 0, and Hom and are all over R.We will

establish a Verdier typ e duality for the functor L' i.e. ' viewed at the level of the

! !

derived categories and an asso ciated functor ' to b e describ ed, by extending one of the

standard treatments [63 ] to etale group oids. But our presentation is self-contained.

As a sp ecial case, we will obtain a Poincar e dualitybetween the Hae iger cohomology

of etale group oids describ ed in Section 2 and the homology theory Section 4.

2.4.1 Tensor pro ducts. As a preliminary remark, we observe the following prop-

erties of tensor pro ducts over R. First, if A is a c-soft sheaf on a space Y and B is any

other sheaf, the tensor pro duct A B is again c-soft. Moreover, for the constant sheaf

asso ciated to a vector space V wehave Y ; A V =Y ; A V cf. 2.5.6. It

c c

follows by comparing the stalks that for a map f : Y ! X , also:

f A ' B =f A B

! !

for any sheaf B on X see [16, 63]. These prop erties extend to a morphism ' : K!G

of etale group oids: for a c-soft K -sheaf A and any G -sheaf B , there is an isomorphism:

' A ' B =' A B :

! !

2.4.2 The sheaves R[V ]. Let us x an etale group oid K .Any op en set V K

0 1

gives a free K -sheaf see 1.2.2.3 of sets V , given by the etale map s : t V !K

and the K -action de ned by comp osition. Let R[V ] b e the free R-mo dule on this K -

sheaf V .SoR[V ]isaK -sheaf of vector spaces, and for any other such K -sheaf B we

have:

Hom R[V ]; B = Hom V ; B = Hom V; B = V ; B 2.18

K;R K

These four o ccurrences of B denote B as a K -sheaf of vector spaces, as a K -sheaf of

sets, and twice as a sheaf on K , resp ectively.

There is a natural morphism:

e = e : K =V !K 2.19 V

56 Chapter 2: A homology theory for etale group oids

of etale group oids of the kind describ ed in 2.2.9.3, and R[V ] can also b e obtained

from the constant sheaf R on K =V as:

R[V ]= e R : 2.20

From this p oint of view, the mapping prop erties 2.18 followby the adjunction b etween

e and e , together with the Morita equivalence K =V ' V where V is viewed as a trivial

group oid, 1.1.3.3.

0 1 1 0

If V; W K are op en sets and : V !K is a section of s : K !K

such that t V W , then comp osition with gives a morphism R[V ] ! R[W ]. In

this sense, the construction is functorial in V .

Lemma 2.4.3 For any K -sheaf of vector spaces A there is an exact sequence of the

form:

M M

R [V ] ! R [V ] !A ! 0 :

j i

Proof: It suces to prove that any K -sheaf can b e covered by K -sheaves of the form

R [V ], and this is clear from 2.18.

2.4.4 The sheaves S . Let S be any c-soft K -sheaf. We write S for the sheaf

V V

S R [V ]. Note that:

S = S e R= e e S R= e e S

V ! ! !

see 2.4.1. In particular, S is again c-soft, and has the following mapping prop erties:

S ; A = V; H omS ; A = Hom Sj ; Aj : Hom S ; A=Hom R [V ];Hom

V V V K V K

2.21

Now supp ose V = V is an op en cover. We claim that the asso ciated sequence

M M

:: : ! S ! S !S ! 0 2.22

V V V

i i i

0 1 0

is exact. To see this, it suces to prove that the sequence obtained by homming into

any injective K -sheaf I ,

0 ! Hom S ; I ! Hom S ; I ! :::

K V K V

is exact. This is clear from the mapping prop erties 2.21.

2.4.5 The sheaves ' S . From nowonlet' : K!G b e a homomorphism

! V

between etale group oids. For an op en set V K , ' induces a map ' : V !G,

which ts into a commutative diagram:

K =V

K :

Section 2.4: Verdier duality 57

where i is the canonical Morita equivalence. Thus, for any c-soft K -sheaf S ,wehave:

' S =' e e S =' Sj : 2.23

! V ! ! V ! V

Notice that the group oid x=' is a space 1.1.3.3 for any ob ject x 2G ; this

and the general description of ' see 2.13 give a simple description of the stalks of

' S . It follows from this description and the corresp onding fact for spaces that '

! V !

maps the exact sequence 2.22 into an exact sequence:

M M

::: ! ' S ! ' S ! ' S ! 0 : 2.24

! V ! V ! V

i i i

0 1 0

2.4.6 The K -sheaves ' S ; I . Again, let ' : K!G be any homomorphism

between etale group oids, let S b e a c-soft K -sheaf, and let I b e an injective G -sheaf.

De ne for each op en set V K :

' S ; I V = Hom ' S ; I :

G ! V

! 0

. Indeed, for an inclusion V W S ; I on K We claim that this de nes a sheaf '

! !

there is an evident map ' S ; I W ! ' S ; I V induced by the map S !S .

V W

And for a covering V = V , the sheaf prop erty follows from the injectivityof I

together with the exact sequence 2.24. Furthermore, this sheaf ' S ; I carries a

natural K -action: for any arrow k : y ! z in K , let W and W b e neighborhoods of y

y z

1 0

and z so small that s : K !K has a section through k with t : W ! W .

y z

Then gives a map R [W ] ! R [W ] and hence S !S . By comp osition, one

y z W W

y z

! !

obtains a map ' S ; I W ! ' S ; I W , and hence by taking germs an action

z y

! !

k : ' S ; I ! ' S ; I :

z y

Prop osition 2.4.7 Duality formula Let ' : K!G be a morphism of etale groupoids.

For any injective G -sheaf I , any c-soft K -sheaf S and any other K -sheaf A, thereisa

natural isomorphism of abelian groups:

Hom ' A S; I : Hom A;' S ; I

G ! K

In particular, ' S ; I is again injective.

Proof: By 2.4.3 and the fact that ' is right exact on sequences of c-soft sheaves, it

suces to de ne a natural isomorphism:

Hom ' R [V ] S; I : Hom R [V ];' S ; I

G ! K

But, using 2.18 and the de nitions:

! !

Hom R [V ];' S ; I V; ' S ; I Hom ' S ; I Hom ' R [V ] S ; I

= = =

K G ! V G !

As for spaces [63 ], one can state and prove a somewhat stronger version of 2.4.7,

using the \internal hom" see 1.2.12:

Prop osition 2.4.8 Duality formula, strong form For any '; A; S and I as in 2.4.7

there is a natural isomorphism of G -sheaves

A;' S ; I ' A S; I : ' Hom Hom

K G

58 Chapter 2: A homology theory for etale group oids

Proof: It suces to prove that for any G -sheaf B there is an isomorphism:

Hom B ;Hom A;' S ; I Hom B ;Hom ' A S; I ;

G G !

K G

natural in B . This is immediate from 2.4.7 and 2.4.1:

! !

A;' S ; I = Hom ' B ;Hom A;' S ; I Hom B ;' Hom

K G

K K

= Hom ' B A;' S ; I

= Hom ' ' B A S; I

G !

= Hom B ' A S; I :

G !

2.4.9 Remark. Let ' : K!G b e an etale morphism such that each of the squares

in 2.7 is a pull-back. Thus K = E / G for some etale G -space E . Then ' has a

simple description as in 2.2.9.3, and is left adjointto' .Thus:

S ;' I : Hom ' A S; I =Hom A S;' I =Hom A;Hom

G ! K K

Since this holds for any A, prop osition 2.4.7 implies that for sucha ',

' S ; I =Hom S ;' I = Hom ' S ; I :

K G

2.4.10 Duality for complexes. Wenow extend these isomorphisms to cochain

complexes. It will b e convenienttowork with chain complexes for A and S and cochain

complexes for I in 2.4.7, 2.4.8. Thus, we will use the following convention: if A is a

chain complex and B is a co chain complex, HomA; B isthe cochain complex de ned

by:

n q

HomA; B = HomA ; B :

p+q =n

Recall for later use that if B is injective and b ounded b elow, then for any quasi-

isomorphism of chain complexes A !C the map HomA ; B ! HomC ; B

is again a quasi-isomorphism by a standard \mapping cone" argument it is enough

to prove the assertion for C = 0; in this case remark that HomA ; B is the total

complex of a double co chain complex whose rows HomA ; B are acyclic by the

injectivityof B .

Similarly, for a b ounded b elowchain complex S of sheaves as in 2.4.7 we de ne

the co chain complex ' S ; I by:

! ! q

' S ; I = ' S ; I :

p+q =n

With these conventions, 2.4.7 gives an isomorphism of co chain complexes

Hom A ;' S ; I Hom ' A S ; I ; 2.25

K G !

for anycochain complex I of injective G -sheaves, and any b ounded b elowchain com-

plexes A and S of K -sheaves with S c-soft. There is also an obvious \strong" version

of 2.25:

' Hom A ;' S ; I ' A S ; I : Hom

K G

Section 2.4: Verdier duality 59

! 0

2.4.11 The functor ' I . Now let d = cohdimK , and x a resolution:

0 d

0 ! R !S ! ::: !S ! 0

of the constant sheaf R by c-soft K -sheaves cf. 2.2.3. Foracochain complex I of

injective G -sheaves de ne:

! !

' I = ' S ; I :

Then ' is adjointto ' in the derived category:

Theorem 2.4.12 Adjointness Let ' : K!G be a morphism between etale groupoids.

For any boundedbelow chain complex A of c-soft K -sheaves and any boundedbelow

cochain complex I of injective G -sheaves there is a natural quasi-isomorphism:

Hom A ;' I ' Hom L' A ; I :

K G !

there is also a \ strong" version derived from 2.4.8.

Proof: Denote by F the free resolution of the constant sheaf obtained by tensoring

1.4 for K instead of G by R. Since A F !A is a quasi-isomorphism, using

2.25 with A F instead of A and S = S , the fact that ' I are injective cf.

2.4.7 and the general remark in 2.4.10 we get a quasi-isomorphism:

Hom A ;' S ; I ' Hom ' A F S ; I : 2.26

K G !

Remark that for any b ounded b elowchain complex B of c-soft K -sheaves we

have a quasi-isomorphism followed by an isomorphism: L' B 'L' B F =

! !

' B F . This for B = A S and the quasi-isomorphism A 'A S give

L' A ' ' A F S . Using this, the fact that I 's are injective and 2.26, the

! !

statement of the theorem follows easily.

Poincar e duality follows in the usual way:

Theorem 2.4.13 Poincar e duality Let K bean etale groupoid, and suppose that K

is a topological manifold of dimension d.Let or be the orientation K -sheaf 1.2.2.6.

There is a natural isomorphism:

_ p+d

H K ; R p 2 Z: H K ; or

Proof: Let G = 1 b e the trivial group oid. In 2.4.12, let I b e the complex R

concentrated in degree p, and let A b e the complex A = S as in 2.4.11. As A

is quasi-isomorphic to R, the complex on the right of the quasi-isomorphism in 2.4.12

has:

0 _

H Hom' A ; I =H K ; R :

! p

Now consider the left hand side of the quasi-isomorphism of 2.4.12, HomA ;' I

in this sp ecial case. Note rst that ' R is quasi-isomorphic to the orientation sheaf

concentrated in degree d:

' R ' or [d] : 2.27

! 0

Indeed, for any op en set V K , ' RV =Hom V ; S ; R see 2.23 is the

complex which computes H V ; R , and the argument for 2.27 is just like the one

for spaces. Thus, for I = R [p] and A = S ' R [0],

0 ! 0 ! p+d

H Hom A ;' I = H K ;' I = H K ;or :

K inv

60 Chapter 2: A homology theory for etale group oids

2.5 App endix: Compact supp orts in non-Hausdor

spaces

In this section we explain how the usual notions concerning compactness and

sheaves on Hausdor spaces extend to our more general context see 1.1.14. For basic

de nitions and facts for sheaves on Hausdor spaces, we refer the reader to any of the

standard sources [49, 63 , 16 ].

2.5.1 c-soft sheaves. Let X b e a space satisfying the general assumptions in 1.1.14.

An ab elian sheaf A on X is said to b e c-soft if for any Hausdor op en U X its

restriction Aj is a c-soft sheaf on U in the usual sense see e.g. [16, 49, 63 ]. By the

same prop erty for Hausdor spaces, it follows that c-softness is a lo cal prop erty, i.e.,

a sheaf A is c-soft i there is an op en cover X = U such that each Aj is a c-soft

i U

sheaf on A.

2.5.2 The functor . Let A b e a c-soft sheaf on X and let A b e its Go dement

resolution i.e. A U =U ; A is the set of all not necessarily continuous

discr

sections, for anyopen U X . For any Hausdor op en set W X , let W; A

b e the usual set of compactly supp orted sections. If W U , there is an evident

homomorphism, \extension by0"W; A ! U; A U; A . For any not

c c

necessarily Hausdor op en set U X ,we de ne U; A to b e the image of the map:

W; A ! U; A ;

where W ranges over all Hausdor op en subsets W U .

Observe that U; A so de ned is evidently functorial in A, and that for any

inclusion U U wehave an \extension by zero" monomorphism:

U; A ! U ; A :

c c

The following lemma shows that in the de nition of U; A it is enough to let

W range over a Hausdor op en cover of U ; in particular, it shows that the de nition

agrees with the usual one if U itself is Hausdor .

Lemma 2.5.3 Let A be a c-soft sheaf on X .For any open cover U = W whereeach

W is Hausdor , the sequence:

W ; A ! U; A ! 0

c i c

is exact.

Proof: It suces to show that for any Hausdor op en W U , the map

\ M

W W ; A ! W; A

c i c

is surjective. This is well known see e.g. [49].

This lemma is in fact a sp ecial case of the following Prop osition \Mayer- Vietoris":

2.5 App endix: Compact supp orts in non-Hausdor spaces 61

Prop osition 2.5.4 Let X = U beanopen cover indexedbyanordered set I , and

let A be a c-soft sheaf on X . Then there is a long exact sequence:

M M

::: ! U ; A ! U ; A ! X; A ! 0 2.28

c i i c i c

0 1 0

0 1 0

, as usual. Thereisofcourse a similar exact sequence \ ::: \ U = U Here U

i i i :::i

n n

0 0

if I is not ordered.

Proof: The prop osition is of course well known in the case where X is a paracompact

Hausdor space. We rst reduce the pro of to the case where each of the U is Hausdor ,

W beacover by Hausdor op en sets, and consider the double as follows. Let X =

j 2J

complex:

C = W \ U ; A ;

p;q c j :::j i ::i

p q

0 0

where the sum is over all j < :::

0 p 0 q

the column C is a sum of exact Mayer-Vietoris sequences for the Hausdor op en

W ; A. Keeping the notation sets W , augmented by C =

c j :::j j :::j p;1

j <:::

0 0

U = X = W if q = 1=p,we observe that for a xed q 1, the row C is

i ::i j :::j ;q

q p

0 0

a sum of Mayer-Vietoris sequences for the spaces U with resp ect to the op en covers

i ::i

fW \ U g. So, if the prop osition would hold for covers by Hausdor sets, eachrow

j i ::i

C q 1 is also exact. By a standard double complex argument it follows that

the augmentation column C is also exact, and this column is precisely the sequence

in the statement of the prop osition. This shows that it suces to show the prop osition

in the sp ecial case where each U is Hausdor .

So assume each U X is Hausdor . Observe rst that exactness of the sequence

2.28 at X; Anow follows by Lemma 2.5.3. To show exactness elsewhere, consider

for each nite subset I I the space U = U and the subsequence:

0 i

i2I

M M

::: ! U ; A ! U ; A ! U ; A ! 0 2.29

c i i c i c

0 1 0

0 1 0 0 0

of 2.28. Clearly 2.28 is the directed union of the sequences of the form 2.29, where

I I ranges over all nite subsets of I . So exactness of 2.28 follows from exactness

of each such sequence of the form 2.29. Thus, it remains to prove the prop osition in

the sp ecial case of a nite cover fU g of X by Hausdor op en sets.

So assume X = U [ ::: [ U where each U is Hausdor . For n = 1, there is

1 n i

nothing to prove. For n = 2, the sequence has the form

0 ! U \ U ; A ! U ; A U ; A ! U \ U ; A ! 0 :

c 1 2 c 1 c 2 c 1 2

This sequence is exact at X; Aby 2.5.3, and evidently exact at other places. Exact-

ness for n = 3 can b e proved using exactness for n = 2. Indeed, consider the following

diagram, whose upp er tworows are the sequences for n =2; 3 and whose third rowis

constructed by taking vertical Cokernels, so that all columns are exact we delete the

62 Chapter 2: A homology theory for etale group oids

sheaf A from the notationcompare to pp. 187 in [15]:

0 0 0 0

 

/ / / /

U U U U [ U 

0 0

c 12 c 1 c 2 c 1 2

 

/ / / /

 U 

U U U U U [ U [ U 

1i

c 123 c 1 c 2 c 3 c 1 2 3

 

/ / / /

U U U U 

c 123 c 13 c 23 c 3

 

0 0 0 0

To show that the middle row is exact, it thus suces to prove that the lower rowis

exact. This row can b e decomp osed into a Mayer-Vietoris sequence for the case n =2,

already shown to b e exact,

0 ! U ! U U ! U \ U [ U ! 0

c 123 c 13 c 23 c 3 1 2

and the sequence:

0 ! U \ U [ U ! U ! C ! 0 :

c 3 1 2 c 3

The exactness of the latter sequence is easily proved by a diagram chase, using

exactness of the right-hand column.

An identical argument will show that the exactness for a cover by n + 1 op ens

follows from exactness for one by n op ens, so the pro of is completed by induction.

Prop osition 2.5.4 is our main to ol for transfering standard facts from sheaf theory

on Hausdor spaces to the non-Hausdor case, as illustrated by the following corollaries.

Corollary 2.5.5 Let Y X be a closed subspace, and let A be a c-soft sheaf on X .

There is an exact sequence

i r

0 ! X Y; A ! X; A ! Y; A ! 0

c c c

i is extension by zero, r is the restriction.

Proof: This including the fact that the map r is well de ned follows by elementary

homological algebra from the fact that the Corollary holds for Hausdor spaces, by

using 2.5.4 for a cover of X by Hausdor op en sets U and for the induced covers of Y

by fU \ Y g and X Y by fU Y g.

i i

Corollary 2.5.6 For a family A of c-soft sheaves on X the direct sum A is again

i i

c-soft, and:

 X; A : X; A 

c i c i

2.5 App endix: Compact supp orts in non-Hausdor spaces 63

In particular, when working over R, we have for any c-soft sheaf S of R-vector

spaces and any vector space V that the tensor product S V here V is the constant

sheaf is again c-soft, and the familiar formula:

X; S V X; S V: 2.30

c R c R

Corollary 2.5.7 Let A !B be a quasi-isomorphism between chain complexes of

c-soft sheaves on X . Then:

X; A ! X; B 

c c

is again a quasi-isomorphism.

Proof: By a \mapping cone argument" wemay assume that B =0. In other

words, wehave to show that X; A is acyclic whenever A is. This follows from

the Mayer-Vietoris sequence 2.5.4 together with the Hausdor case.

We remark that it is necessary to assume that the chain complexes are b ounded

b elowifX do es not have lo cally nite cohomological dimension, as in 1.1.14.

The following Corollary is included for application in chapter 4.

Corollary 2.5.8 Let Y X be a closed subspace, and let : X ! R beacontinuous

map such that 0 = Y .Let A be a c-soft sheaf on X . Then for any 2 X; A,

j =0 i 9 "> 0: j =0

";"

here j is the restriction r as in 2.5.5.

 

Proof: For " 0, write Y = fx 2 X : j x j"g, and for eachopen U X write

U; A= f 2 U; A: j =0g :

c U \Y

It suces to show that:

" 0

X; A ! X; A

c c

">0

is epi. Let fU g be a cover of X by Hausdor op en sets, and consider the diagram:

u 

/ /

U ; A

U Y; A

U ; A

c i

i;">0 i

 

v 

/ /

X Y; A X; A

X; A

">0

where the isomorphisms on the right come from 2.5.5. We wish to show that v is epi.

Since u is epi by the Hausdor case, it suces to show that is epi, or, equivalently,

that is epi. This is indeed the case by 2.5.4.

It is quite clear that using c-soft resolutions one can de ne compactly supp orted

cohomology H X; A for any A2Ab X . In particular, we get an extension H X; 

c c

of X; to all sheaves; this extension is still denoted by X; .

c c

64 Chapter 2: A homology theory for etale group oids

Prop osition 2.5.9 Let f : Y ! X bea continuous map. There is a functor f :

Ab Y ! AbX with the fol lowing properties:

 

i For any open U X and any B2Ab Y , U; f B = f U ; B .

c ! c

 

Y , f B = f x ; B . ii For any point x 2 X and any B2Ab

! x c

 

iii f is left exact and maps c-soft sheaves into c-soft sheaves.

 

iv For any beredproduct

Z Y

 

Z X

along an etale map e and for any c-soft B2AbY , there isacanonical isomorphism

q p B e f B :

! !

see 2.5.11 below for the case where e is not etale.

Proof: Of course the prop osition is well known in the Hausdor case. For the more

general case, recall rst from [18] the corresp ondence for any Hausdor space Z between

c-soft sheaves S on Z and abby cosheaves C on Z , given by:

W; S = C W 2.31

natural with resp ect to the op ens W Z : Given the cosheaf C , the stalk of the

corresp onding sheaf S at a p oint z 2 Z is given by the exact sequence:

0 !CZ z !CZ !S ! 0 : 2.32

We use this corresp ondence in the construction of f . However, see remark 2.5.10

b elow for a description of f which do esn't use this corresp ondence.

We discuss rst the construction of f on c-soft sheaves. Let B2Ab Y bec-

soft. First, assume X is Hausdor . Let B b e a c-soft sheaf on Y , and de ne a cosheaf

C = cB by C U =f U ; B . Note that C is indeed a abby cosheaf, by 2.5.4.

By the corresp ondence 2.31, there is a c-soft sheaf S on X , uniquely determined up

to isomorphism by the identity U; S =C U for anyopen U X .Thus, if X is

Hausdor , we can de ne f B to b e this sheaf S .

In the general case, cover X by Hausdor op ens U , and de ne in this way for each

i a c-soft sheaf S on U by:

i i

V; S = f V ; B : 2.33

c i c

Then again by the equivalence b etween sheaves and cosheaves there is a canonical

satisfying the co cycle condition. Therefore the !Sj isomorphism : S j

i U ij j U

ij ij

sheaves S patch together to a sheaf S on X , uniquely determined up to isomorphism

by the condition that Sj = S by an isomorphism compatible with . Thus we can

U i ij

de ne f B to b e S .

 

We prove the prop erties i iv in the statement of the prop osition for B2Ab Y 

 

c-soft. Prop erty i clearly holds for an op en set U contained in some U ,by 2.33. i

2.5 App endix: Compact supp orts in non-Hausdor spaces 65

 

For general U , prop erty i then follows by the Mayer-Vietoris sequence. Next, identity

2.32 yields for any p oint x 2 X an exact sequence:

 

0 ! Y f x ; B ! Y; B ! f B ! 0 ;

c c ! x

 

and hence, by 2.5.5 the isomorphism ii . of the Prop osition. Finally, iv is clear from

the lo cal nature of the construction of f .

Y we de ne f A 2 AbX as the kernel of the map For general A2 Ab

0 1 0 1

f S ! f S where 0 !A!S !S ! ::: is a c-soft resolution of A

! !

from the rst part it follows that it is well de ned up to isomorphisms. The prop er-

 

ties i and ii are now immediate consequences of the de nition and of the previous

 

case. Using 2.5.7 and ii it easily follows that f transforms acyclic complexes of c-soft

sheaves on Ab Y into acyclic complexes on AbX . This immediately implies that '

is left exact.

2.5.10 Remark. We outline an alternative construction and pro of of Prop osition

2.5.9, which do es not use the corresp ondence b etween sheaves and cosheaves. This

construction will b e used in the pro of of 2.5.11 b elow. We will assume that B is c-soft

and X is Hausdor . As in the pro of of 2.5.9, the construction of f for general X is

then obtained by glueing the constructions over a cover by Hausdor op ens U X .

So, let B b e a c-soft sheaf on Y .For any op en set V Y , denote by B the sheaf

on Y obtained by extending Bj by zero. Thus B is evidently c-soft, and Y; B =

V V c V

V; B . Moreover, an inclusion V W induces an evident map B ,!B .

c V W

Now let Y = W beacover by Hausdor op en sets. This cover induces a long

exact sequence:

M M

::: ! ! !B ! 0 B B

W W

i i i

0 1 0

of c-soft sheaves on Y . By Corollary 2.5.7, the functor Y; applied to this long

exact sequence again yields an exact sequence, and this is precisely the Mayer-Vietoris

sequence of 2.5.4. For each i ; :::; i let f : W ! X b e the restriction of

0 n i ;:::;i i ;:::;i

n n

0 0

f ; this is a map b etween Hausdor spaces, so wehavef B de ned as

i ;:::;i ! W

0 i ;:::;i

usual. De ne f B as the cokernel tting into a long exact sequence:

M M

::: ! f B ! f B ! f B ! 0 : 2.34

i i ! W i ! W !

0 1 i i 0 i

0 1 0

 

 = f For x 2 X ,wehavef B x \ W ; B by the Hausdor case. So

x c i ! W i

0 i 0

taking stalks of the long exact sequence in 2.34 at x and using the Mayer-Vietoris

1 1

 

x ; B as in 2.5.9 ii . Prop- x we nd f B = f sequence 2.5.4 for the space f

! x c

 

erty 2.5.9 i is proved in a similar way using 2.5.5.

The functor f can b e extended to the derived category D Y by taking a c-

0 1

soft resolution 0 !A!S !S ! ::: and de ning Rf A as the complex

f S . Up to quasi-isomorphisms, this complex is well de ned and do es not dep end

on the resolution S ,by 2.5.5. In this way,we obtain in fact a well de ned functor

Rf : D Y ! D X at the level of derived categories, which is sometimes simply

denoted by f again. In particular, H R' A gives in fact the right derived functors

! !

R f of f .

! !

66 Chapter 2: A homology theory for etale group oids

Lemma 2.5.11 For any pul lback diagram:

Z Y

 

Z X

and any sheaf B on Y , thereisacanonical quasi-isomorphism:

Rq p B'e Rf B :

! !

Proof: Using Mayer-Vietoris for covers of X and Z by Hausdor op en sets, it suces

to consider the case where X and Z are b oth Hausdor . Clearly it also suces to prove

the lemma in the sp ecial case where B is c-soft.

Let Y = W as in 2.5.10, so that f B ts into a long exact sequence 2.34

i !

of c-soft sheaves on X . Applying the exact functor e to this sequence and using the

lemma in the Hausdor case, one obtains a long exact sequence of the form:

M M

::: ! q p Bj ! q p Bj ! e f B ! 0 : 2.35

! W ! W !

i i i

0 1 0

Now let p B !S b e a c-soft resolution over the pullback Z Y . Then for any

 is a c-soft resolution is a c-soft resolution of p B , so q S op en U Y , S

1 1

W !

p W p W 

of q p B . The lemmanow follows by comparing the sequence 2.35 to the de ning

sequence

M M

1 1

 ! q S ! 0 q S ! q S ::: !

! ! !

p W p W

i i i

0 0 1

i i

0 0 1

def

for q p B = q S .

! !

2.5.12 f on etale maps. Let f : Y ! X b e an etale map, i.e. a lo cal homeomor-

phism. It is well known that the pullback functor f : Ab X ! AbY has an exact

left-adjoint f : Ab Y ! AbX , describ ed on the stalks by f B = B .

 

! ! x y

y 2f x

This construction agrees with the one in 2.5.9. In particular, for etale f , the counit of

the adjunction de nes a map:

 : f f A !A ;

f !

\summation along the b er", for any sheaf A on X .

2.5.13 Notation: Let A2ShX ; B2ShY b e c-soft sheaves. Most of the maps

we are going to deal with are of typ e:

 

u x 2B y 2 Y ; ; f : X ; A ! Y; B ; ; f u y =

x y c c

 

y x2f

B .In for some etale map f : X ! Y , and some morphism of sheaves : A! f

other words, ; f is the comp osition:

2:5:9:3

int

! Y; B ; B ' Y; f f ! X ; f B X ; A

c c ! c c

2.5 App endix: Compact supp orts in non-Hausdor spaces 67

where int := : f f B!B is, at the stalk at y 2 Y see also 2.5.9:

f !

M X X

B !B ; b x 7! b :

y y x x

1 1

 

x2f y x2f y

 

Rather than writing the explicit formula for ; f ,we prefer to brie y indicate the

maps f and using the notation:

 

X; A ! Y; B ; ajx 7! a jf x 

c c

x 2 X; a 2A .

2.5.14 f on prop er maps. De ne a map f : Y ! X between non-necessarily

Hausdor spaces to b e proper if:

i the diagonal Y ! Y Y is closed.

ii for any Hausdor op en U X and any compact K U , the set f K is

compact.

It is easy to see that if f is prop er then f = f , as in the Hausdor case. Furthermore,

for any c-soft sheaf A on X , there is a natural map X; A ! Y; f A de ned

c c

by pullback, as in the Hausdor case.

2.5.15 Remark: Although this do es not simplify matters, one could theoretically

interpret some of the constructions and results of this App endix as follows. First,

observe that for Hausdor group oids, the results in Sections 1-6 of the pap er can b e

based on the usual de nition of and are indep endent of the App endix. Now, any

non-separated manifold or suciently nice space, cf. 1.1.14 X can b e viewed as a

trivial group oid 1.1.3.1, which is Morita equivalent to the Hausdor etale group oid G

 

de ned from an op en cover fU g of X by Hausdor op en sets, by taking G to b e the

 

0 0 1

. G = G disjoint sum of the U , and G

X i