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A Homology Theory for Etale Group oids

by Marius Crainic and Ieke Mo erdijk

UtrechtUniversity Department of Mathematics Netherlands

Abstract

Etale group oids arise naturally as mo dels for leaf spaces of foliations for orbifolds and for

orbit spaces of discrete group actions In this pap er weintro duce a homology theory for

etale group oids Weprove its invariance under Morita equivalence as well as Verdier duality

between Haeiger and this homologyWe also discuss the relation to the cyclic and

Ho chschild homologies of Connes convolution algebra of the group oid and derive some sp ectral

sequences whichserve as a to ol for the computation of these homologies

Keywords etale group oids homology duality sp ectral sequences cyclic homology

foliations

In this pap er weintro duce a homology theory for etale group oids Etale group oids serveas

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 generalized spaces

In the literature one nds roughly sp eaking two dierent approaches to the study of etale

group oids One approach is based on the construction of the convolution algebras asso ciated to

an etale group oid in the spirit of Connes noncommutative geometry and involves the

study of cyclic and Ho chschild homology and cohomology of these algebras 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

Our motivation in this pap er is twofold First wewanttogive a more complete picture of

the second approach by constructing a suitable homology theory which complements the existing

cohomology theory Secondlywe use this homology theory as the main to ol to relate the two ap

proaches

Let us b e more explicit In the second approach one denes for anyetale group oid G natural

cohomology groups with co ecients in an arbitrary G equivariant sheaf These were intro duced in

a direct wayby Haeiger As explained in they can b e viewed as a sp ecial instance of

the Grothendieck theory of cohomology of sites and agree with the cohomology groups of the

classifying space of G Moreover these cohomology groups are invariant under Morita equiva

lences of etale group oids This invariance is of crucial imp ortance b ecause the construction of the

etale group oid mo delling the leaf space of a given foliation involves some choices which determine

the group oid only up to Morita equivalence We complete this picture by constructing a homology

theory for etale group oids again invariant under Morita equivalence which is dual in the sense of

Verdier duality to the existing cohomology theoryThus one result of our work is the extension of

the six op erations of Grothendieck from spaces to leaf spaces of foliations

Our homology theory of the leaf space of a foliation reects some geometric prop erties of the



Research supp orted byNWO

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 and the references cited there It also

shows that the RuelleSullivan current of a measured foliation see lives in Haigers closed

cohomology The results in see also Prop osition imply that our homology is also the

natural target for the lo calized Chern character We plan to describ e some of these connections

more explicitly in a future pap er

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

oretic 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 connection explains several basic prop erties of the cyclic and p erio dic

homology groups and leads to explicit calculations The previous work on the BaumConnes

conjecture for discrete groups or for prop er actions of discrete groups on manifolds suggest that

this homology will play a role in the BaumConnes conjecture for etale group oids

From an algebraic p oint of view our homology theory is an extension of the homology 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 whichmodel

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 nonseparated nonparacompact manifolds For example as a sp e

cial case of our results one obtains the Verdier and Poincare duality for nonseparated manifolds

Our notion of compactly supp orted sections is also used in the construction of the convolution alge

bra of a nonseparated etale group oid We b elieve that this extension to nonseparated spaces has

amuch wider use that the one in this pap er and wehave tried to give an accessible presentation of

it in the app endix The results in the app endix also playacentral role in the calculation concerning

the cyclic homology of etale group oids in and make it p ossible to extend the results of for

separated group oids to the nonseparated case

We conclude this intro duction with a brief outline of the pap er

In the rst section we review the basic denitions and examples related to etale group oids and

in the second section we summarize the of etale group oids These two sections

serve as background and do not contain any new results Readers familiar with this background

should immediately go to section and consult the earlier sections for notational conventions

In section we present the denition of our homology theory and mention some of its imme

diate prop erties

In section a covariant op eration for anymap 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 homologyItalso

plays a crucial role in many calculations in

In section we prove that the op eration L has a right adjoint at the level of derived

categories thus establishing Verdier duality The Poincare dualitybetween Haeiger cohomology

and our homology of etale group oids is an immediate consequence

In section we summarize the main asp ects of the relation to cyclic homology This section

is based on to whichwe refer the reader for detailed pro ofs and further calculations

In an app endix weshowhow to adapt the denition of the functor X A assigning to a

c

space X and a sheaf A the group of compactly supp orted sections in suchaway that all the prop

erties as expressed in say can b e proved without using Hausdorness and paracompactness of

the space X This app endix can b e read indep endently from the rest of the pap er

Contents

Etale group oids

Sheaves and cohomology

Homology

Leray sp ectral sequence Morita invariance

Verdier duality

Relation to cyclic homology

App endix Compact supp orts in nonHausdor spaces

Etale group oids

In this section we review the denition of top ological group oids x the notations and mention

some of the main examples

Recall rst that a groupoid G is a small category in whichevery arrowisinvertible We will

 

 

for the set of ob jects and the set of arrows in G resp ectively and denote the and G write G

structure maps by

s

m i u

   

       

G G

 



G G G G

G

t

  

Here s and t are the source an target m denotes comp osition m g h g h i is the inverse

g

 

   

i g g and for any x G u x is the unit at xWe write g x y or x y to

x

 

   

indicate that g G is an arrow with s g x and t g y

   

A topological groupoid G is similarly given by top ological spaces G and G and bycontinuous

   

structure maps as in For a smo oth group oid G and G are smo oth manifolds and these

structure maps are smo oth moreover one requires s and t to b e submersions so that the b ered

   

G in is also a manifold pro duct G

 



G

Denition A top ological smo oth group oid G as ab oveiscalledetale if the source map

   

s G G is a lo cal homeomorphism lo cal dieomorphism This implies that all other

structure maps in are also lo cal homeomorphisms lo cal dieomorphisms

Germs Anyarrow g x y in an etale group oid induces a germg U xV yfrom

 



a neighborhood U of x in G to a neighborhood V of y Indeed we can deneg t where

       

 

x U G is so small that s G G has a section U G with x g

If U is so small that t j is also a homeomorphism resp dieomorphism theng U V is

U

also a homeomorphism resp dieomorphism We will also writeg for the germ at x of this map

g

 

g U V Note that is the identity germ and that hg hg if g x y and h y z

x

 

Examples of etale group oids Note that in examples and the space G is in general

not Hausdor

Any top ological space manifold X can b e viewed as an etale group oid X with identity

   

arrows only X X X etc We will often simply denote this group oid by X again

If a discrete group acts from the right on a space X one can form a group oid X

 

 

with X X and X X by taking as arrows x y those with

y x This group oid is called the translation group oid of the action

q q q

The Haeiger group oid has R for its space of ob jects An arrow x y in

q q q

is a germ of a dieomorphism R x R y This group oid and its classifying space B cf

b elow playacentral role in foliation theory

see for example For a foliation M F of co dimension q its holonomy

group oid HolM F can b e represented byanetale group oid Hol M F dep ending on the choice

T

of a complete transversal T ie a submanifold T M of dimension q whichistransversal to

the leaves and which meets every leaf at least once Two dierentsuch transversals T and T give



Morita equivalent see b elow etale group oids Hol M F and Hol M F

T T

Any orbifold gives rise to a smo oth etale group oid These group oids G coming from orbifolds

  

  

 

have the sp ecial prop erty that s t G G G is a prop er map see Group oids

   

with this prop erty are called prop er For a prop er group oid G is Hausdor whenever G is

 

Let G b e an etale group oid A right G space is a space X equipp ed with a map p X G

 

 

and an action X G X x g xg satisfying the usual identities If X is a right G

 



G

 

G space one can construct a group oid X G withX G X and X G X

 



G

g

   

an arrow x y in X G is an arrow p x p y with y xg A similar construction applies

of course to left G spaces

Homomorphisms Let G and K be etale group oids A homomorphism KG is given

       

bytwo continuous or smo oth maps K G and K G whichcommute

          

with all the structure maps in i e s g s g g h g h etc

Morita equivalence A homomorphism KG is called a Morita or weak or essential

equivalence if

         

 

The map s K G G dened on the space of pairs y g K G

 



G

   

with t g y isanetale surjection

The square



 

 

G

K

 

 

st

st

 

     

 

G G

K K

is a b ered pro duct

We often write K G to indicate that is such a Morita equivalence Two group oids G and

H are said to b e Morita equivalent if there are Morita equivalences HK G This is a transitive

relation One generally considers the category of etale group oids obtained by formally inverting the

Morita equivalences In this category an arrow HG is represented bytwo homomorphisms as

in

HK G

see for more details

Bundles see for example Let B b e a base space and G an etale

group oid A left G bundle over B consists of a space P amap P B andaleftactionofG

   

on P see which resp ects in the sense that ge e The action is called principal if the

canonical map b etween b ered pro ducts

 

   

G P P P g e ge e

 



B G

is a homeomorphism

 

If B K is the space of ob jects of another group oid K the bundle P is said to b e K

equivariantif P is also equipp ed with a right K action which commutes with the left action by G

   

   

ge h g eh in this case the maps P K and P G are denoted by s source and

P

t target resp ectivelyFor instance any homomorphism KG induces a K equivariant

P

principal G bundle

   

 

P K G

 



G

     

the space considered also in with s y g y t y g s g The isomorphism classes

   

P P

of K equivariant principal G bundles P can b e viewed as generalized or HilsumSkandalis mor

phisms

P KG

The category so obtained is equivalent to the category obtained byinverting the Morita equiv

alences see Thus showing that a certain construction is invariant under Morita equivalence

is the same as showing that it is functorial on generalized morphisms

 

n

Nerve and classifying space For an etale group oid G we write G for the space of

comp osable strings of arrows in G

g g g

  n

x x x

n

For n this agrees with the notation for the space of ob jects and arrows of G already

 

n

intro duced The spaces G n together form a simplicial space

     

G G G

   

n n

with the face maps d G G dened in the usual way

i

 

g g if i

n

   

d g g g g g g if i n

i n i i n

 

g g if i n

n

Its thick geometric realization is the classifying space of G denoted B G This space B G

classies homotopy classes of principal Gbundles A Morita equivalence HG

induces a weak homotopy equivalence B H B G

Overall assumptions It is imp ortanttoobserve that in many relevant examples the space

 

G of arrows of an etale group oid G is not Hausdor cf in However for any space X in

this pap er we do assume that X has an op en cover bysubsets U X which are each paracompact

Hausdor lo cally compact and of cohomological dimension b ounded byanumber d dep ending on

X but not on U These assumptions hold for any nonseparated manifold of dimension dandin

 

n

particular for each of the spaces G asso ciated to a smo oth etale group oid

Sheaves and cohomology

n

In this section we review the denition and main prop erties of the cohomology groups H G A

of an etale group oid G with co ecients in a G sheaf A These groups have b een studied by Haeiger

They can also b e viewed as cohomology groups of the top os of G sheaves Grothendieck

Verdier and were discussed from this p ointofviewin





G sheaves Let G b e an etale group oid A G sheaf is a sheaf S on the space G onwhich

 

G acts continuously from the right In other words S is a right G space for which the map



SG is etale a lo cal homeomorphism A morphism of G sheaves SS is a morphism of

sheaves whichcommutes with the action We will write ShG for the category of all G sheaves of

sets and Ab G for the category of ab elian G sheaves These categories haveconvenient exactness

prop erties it is well known that ShG is a top os and hence that AbG is an ab elian category

with enough injectives If R is a ring we write Mod G for the category of G sheaves of G mo dules

R

Thus AbG Mod G Later we will mostly work with the category Mod G ofG sheaves of

Z R

real vector spaces

Examples

 

For any set or ab elian group A the corresp onding constant sheaf on G can b e equipp ed

with the trivial G action We will refer to G sheaves of this form as constant G sheaves they are

simply denoted by A again

 

of germs of continuous realvalued functions on G has the natural The sheaf A C

 



G

 



structure of a G sheaf if g x y in G and A is a germ at g then g is dened as the

y

n



of dierential comp osition g cf Similarly if G is a smo oth etale group oid the sheaf

 



G

 

nforms on G has a structure of a G sheaf n

   

G Let E b e a sheaf on G no action To E we can asso ciate a G sheaf E G E

 



G

 

     

f e g g x y e E g The sheaf pro jection is the map E G G given by e g s g

y

    



h e g h Sheaves isomorphic to ones of this while the G action is given by comp osition e g

form are said to b e free G sheaves The freeness is expressed by the adjunction prop erty

Hom E G S Hom E S

 



G

G

for any G sheaf S

 

n

Each of the spaces G in the nerveofG cf has the structure of a G sheaf with sheaf

pro jection

g g g

n      

n

x x x G G x

n n n

   



h g g h ThisG sheaf is denoted F G and the G action given by comp osition g g

n n n

 

n

For n these sheaves are free in fact G G G The system of G sheaves

n

F G F G F G

 

has the structure of a simplicial G sheaf whose stalk at x G is the nerve of the comma category

xG This stalk is a contractible simplicial set

For any G sheaf of sets S one can form the free ab elian G sheaf ZS the stalk of ZS at

 

x G is the free ab elian group on the stalk S In particular from we obtain a resolution

x

ZF G ZF G Z

of the constant G sheaf Z where is dened by the alternating sums of the face maps in

 

If G is a top ological manifold of dimension d recall that its orientation sheaf or is given by

d

or U H U R see eg and the App endix for compactly supp orted cohomology in the

c





is nonHausdor It has a natural G action for any arrow g x y in G let U case where G

x

   

and U be neighborhoods of x and y so small that s G G has a section through g with

y

d d d d

 

t U U Then t induces a map H U H U so also a map H U H U

x y x y x y

c c c c

Hence by taking germs it gives an action or or

y x

   

Note that if G is oriented ie as a sheaf on G or is isomorphic to the constantsheafR





it is not necessarily constantasaG sheaf When it is ie when G is orientable and anyarrow

g x y gives an orientationpreserving germg cf wesaythatG is orientable

Morphisms Amorphismofetale group oids KG induces an evident functor

Sh G ShK

G AbK This functor has a rightadjoint by pullback and similarly an exact functor Ab

Sh G ShK

   

For an K sheaf S the sheaf S onG is dened for any op en set U G by

S U Hom U S

K

 

   

Here U f y g y K g y x x U g with K sheaf structure given byy gh

  

y g h The G action on this sheaf G is dened as follows for S and g x x

x



let U b e a neighborhood of x so that is represented by an element S U and let U be

x x

x

     



 

so small that s G G has a section U G through g with t U U Then

x x x



dene g S to b e the element represented by the morphism

x

   



S y f y z z f U

x

These adjoint functors and together constitute a top os morphism

Sh K ShG

If K G is a Morita equivalence then this morphism is an equivalence of categories ShK

ShG In fact top os morphisms ShK ShG corresp ond exactly to generalized morphisms

KGorequivalently to pairs of homomorphisms K HG cf

 

Invariant sections Let S be a G sheaf A section G S is called invariantif

   

y g x for any arrow g x y in G We write

G S

inv

for the set of invariant sections it is an ab elian group if S is an ab elian sheaf In fact G S

inv

S where G is the morphism into the trivial group oid

n

Cohomology For an ab elian G sheaf A the cohomology groups H G A are dened as the

cohomology groups of the complex

G T G T

inv inv

where AT T is any resolution of A by injective G sheaves In other words

n n

H G R G

inv

n

G with co ecients in A It is obvious Thus H G A is simply the cohomology of the top os Sh

that a homomorphism KG induces homomorphisms in cohomology

n n

 

H G A H K A n

If is a Morita equivalence these are isomorphisms since ShG ShK

Leray sp ectral sequence For any morphism KG and any K sheaf A there is a

Leray sp ectral sequence

pq

p q pq

E H G R A H K A

q

The G sheaf R A can b e explicitly describ ed as the sheaf asso ciated to the presheaf U

q

H U Awhere U is the group oid asso ciated to the right action of K on the space U

used in cf See

Basic sp ectral sequence Let G b e an etale group oid and let A be a G sheaf By pullback

     

n n

along G G see A induces a sheaf AonG whichwe often simply denote

n

n

   

q p p

by A again Consider for each p and q the sheaf cohomology H G A of the space G For a

xed q these form a cosimplicial ab elian group and there is a basic sp ectral sequence

 



p q pq

H H G A H G A

 



p q

It arises from the double complex G T where AT is an injective resolution

p

q

It follows that if A A A is any resolution by G sheaves A with

 

p q

the prop erty that A is an acyclic sheaf on G thenH G A can b e computed bythedouble

p

complex

 

p q

G A

p

 

C ech sp ectral sequence An op en set U G is called saturated if for anyarrow

   

g x y in G one has s g U i t g U For suchaU there is an evident full subgroup oid

 

Gj G with U as space of ob jects If U is an op en cover of G by saturated op ens there is a

U

sp ectral sequence

p q pq

H U H A H G A

q q

where H A is the presheaf U H Gj Aj

U U



Hyp ercohomology For a co chain complex A of ab elian G sheaves the hyp ercohomology



n

groups H G A are dened in the usual way as the cohomology groups of the double complex

   

G T where A T is a quasiisomorphism into a co chain complex of injectives If A

inv

is concentrated in degree one recovers the ordinary cohomology dened in For each q Z

  

q

denote by H A the q th cohomology G sheaf of A IfA is b ounded b elow there is a sp ectral

sequence for hyp ercohomology analogous to the one in

 

  

p q p q

H H G A H G A

p

Ext functor Recall that for any G sheaf B the functor Ext B is dened as the pth

p p

right derived functor of the functor Hom B Thus H G AExt Z A For later purp oses

G

p

we recall Yonedas description of Ext B A as the group as equivalence classes of extensions

B E E A

n

see eg By comp osition of exact sequences one denes a cap pro duct



q p pq

Ext C B Ext B A Ext C A

The same applies of course to the category Mod G ofG sheaves of real vector spaces We use

R

p

the notation Ext B A here Recall also that over R the tensor pro duct denes a functor

R

p p

CB CA This gives an easy description of the cap pro duct in co B A Ext Ext

R R

homology

q p pq

H G A H G B H G A B

R

as



q p q p pq

Ext Z A Ext Z B Ext Z A Ext A A B Ext ZA B

R R





Internal hom For two G sheaves A and B the sheaf Hom A B onG carries a natural

A B or simply Hom G action hence gives a G sheaf Hom A B again We recall that

G

G Hom A B HomA B

inv

G

G The is the group of action preserving homomorphisms ie morphisms in the category Ab

derived functor of

HomA AbG AbG

p p

A orby Ext A will b e denoted by R Hom

Homology

In this section we will intro duce the homology groups H G A for anyetale group oid G and

n

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 book

c

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 pap er will denote this extended functor

c

     

n

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

are assumed to satisfy the general conditions of but we will not assume that G is Hausdor

   

We write d cdimG for the cohomological dimension of G Thus for any n and any

 

n

Hausdor op en set U G the usual cohomological dimension of U is at most d





Bar complex Let A be a G sheaf and assume that A is csoft as a sheaf on G we will

 

n

briey saythat A is a csoft G sheaf For each n consider the sheaf A AonG

n

n

   

n

constructed by pullbackalong G G x x x It is again a csoft

n n n

 

n

sheaf b ecause is etale The groups G A of compactly supp orted sections intro duced in

n c n

the App endix together form a simplicial ab elian group

     

B G A

G A G A G A



c c c

with face maps

   

n n

d G A G A

i c n c n

   

n n

dened as follows First for the face map d G G cf there is an evident map

i

g g

 n

g

x x is the identity map isomorphism in fact A d A whose stalk at

n n n

i

 

for i and the action by g A A A d A if i The map d in is

n x x n i

 

g g

now obtained from this by summation along the bres see

d

i

 

 

n n

A A G G

n n c c

O

P

 

n

G

c

d

i

   

n n

G d A G d d A

c n c i n

i i

The homology groups H G A are dened as the homology groups of the simplicial ab elian groups

n

or equivalently as those of the asso ciated chain complex given by the alternating sum

P

i

 

d

i

Similarlyany b ounded b elowchain complex S of csoft sheaves gives rise to a double complex



B G S

 

and we dene H G S to b e the homology of the asso ciated total complex



n

Lemma Any quasiisomorphism S T between boundedbelow chain complexes of csoft

 

G sheaves induces an isomorphism

H G S H G T

 

n n

P ro of The sp ectral sequence of the double complex takes the form E H H G S

 

p q

pq

   



p p

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

 

pq p c

pq

lemma thus follows from 

csoft resolutions Let A b e an arbitrary G sheaf There always exists a resolution

d

A S S

by csoft G sheaves For example since the category of G sheaves has enough injectives one can

  

takeany injective resolution A T and take S to b e the truncation T softness of

d



d

S then follows as in p Or one can use for T 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

A A A

 



obtained from the G sheaves of dierential forms on G Note that the last two resolutions are

functorial in A

Any resolution maps into the truncated injective one And similarlygiven two resolutions

  

AS and AT there is a resolution R eg the truncated injective one and

a diagram



A T

 

S R

which commutes up to homotopy

Denition of homology Let A b e an arbitrary G sheaf and let A S



d

S b e a csoft resolution Then S is a b ounded chain complex nonzero in degrees



between d and and we dene H G Atobe H G S By and lemma this

n n

denition is indep endent of the choice of the resolution Observethat

H G A for all n d

n

These groups can b e viewed also as compactly supp orted cohomology groups see and

b elow

Extreme cases





If G is a p oint ie if G is a discrete group then H G A is the usual group homology

n

of G

 



If G is a discrete group oid G is a simplicial set and H G A is the usual simplicial

n

 



homology of G with twisted co ecients

If G is a Hausdor space X viewed as a trivial group oid then H G A

n

n

H X A is the usual cohomology with compact supp orts although graded dierently So the

c

 



q

sp ectral sequence o ccurring in the pro of of lemma could b e written as H H G A

p

c

H G A

pq

Long exact sequence Any short exact sequence

A B C

of G sheaves induces a long exact sequence in homology

H G C H G A H G B H G C

n n n n

The pro of is standard The truncated Go dement resolutions give a short exact sequence of resolu

  

tions S A S B S C

Functoriality Compactly supp orted cohomology of spaces is covariant along lo cal

homeomorphisms and contravariant along prop er maps Analogous prop erties hold for homology of

etale group oids Consider a homomorphism KG between etale group oids

   

n n

Supp ose that is prop er in the sense that each K G is a prop er map cf

n

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

Supp ose is etale in the sense that each K G is a lo cal homeomorphism it

n

is not dicult to see that the assumption is only ab out Let S b e a csoft G sheaf For the sheaf

 

n

S S onG summation along the b ers denes a homomorphism

n

n

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 commute with the face op erators Since the functor

is always exact and preserves csoftness b ecause is etale this gives for each G sheaf A a

homomorphism

H K A H G A

n n

Supp ose that is etale and moreover supp ose that for each n the square

n

 

 

n

K K

n



n

   

n

G G

is a pullback Morphisms of this kind are exactly the pro jections X GG asso ciated to etale

G spaces X For sucha there is an exact functor

Ab K AbG

which preserves csoftness at the level of underlying sheaves it is simply the functor

   

AbK AbG of For any csoft K sheaf B there is a natural isomorphism

     

n n n

K B K B G B G B G B

c n n c n c n c c n

n n n

for any n 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 dened in this waybutitisnolonger

exact See also

Prop osition Let KG be two etale homomorphisms AAbG and a

continuous transformation of functors Denote by H K A H K A the map induced

by the sheaf map A Aa a andby the maps inducedby in homology cf

Then

H K A

N

N

N

N

 N

N

N

N

N

N

N



H G A

p

p

p

p

p

p

p

p

p



p

p

H K A

Moreover the construction of is functorial with respect to

Proof Wemay assume that A is csoft Then a homotopybetween the maps

B K A B G A

 

inducing and in homology is given by

n

X

i

 

H H B K A B G A

i n n

i

where the H s are dened as follows Consider

i

   

n n

h K G

i

      

t k k k if i

n

h k k

i n

          

k k s k k k if i n

i i i n

Using the obvious identity isomorphisms h A A and summation along the b er of

n n

i

the etale h s see in App endix we get the homomorphisms

i

H B K A B G A

 

i

The naturality with resp ect to is obvious 

Hyp erhomology Consider anyboundedbelowchain complex A of G sheaves Let A

 

R b e a qi into a b ounded b elowchain complex of csoft G sheaves SuchanR can b e constructed

 

d

for example by considering a resolution A S S as in and then taking the



 

p

total complex of the double complex S p q Z d p Dene the hyp erhomology

q

H G A to b e the homology of the total complex asso ciated to the double complex B G R

  

This denition of H G A do es not dep end on the choice of the resolution R cf lemma

 

Prop osition Hyperhomology Let A beaboundedbelow chain complex of



G sheaves as above and consider for each q Zthe homology G sheaf H A Thereisa spectral



q

sequence

H G H A H G A

 

p q p q

d

It S P ro of Consider the truncated Go dement resolution A S



 

has the prop erty that for each q it also yields csoft resolutions of the cycles Z the b oundaries B

q q

and the homology H A Write C for the triple complex



q

 

r p

C G S

pq r c

q

and let D b e the double complex

M

D C

nq pq r

pq n

The total complex of C and hence also that of D compute H G A Furthermore by the prop erty



 

p

of the resolution just mentioned and the fact that G preserves exact sequences of csoft

c

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

M

 

p r

H D G H S



q n c q



pr n

q q

is a resolution of H A so for a xed q the double H S But H A H S

 

q q

 

 

 

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 the desired sp ectral sequence is simply the sp ectral sequence H H D H TotD for

n q nq

the double complex D 

Cap pro duct For an etale group oid G theExtgroups act on the homology bya

cap pro duct

p

H G B Ext B A H G A

n np

For example for p an elementofExt B A can b e represented by an exact sequence

B E A which yields a b oundary map H G B H G A for the

n n

long exact sequence of For p the cap pro duct can b e constructed in the same wayby

decomp osing a longer extension B E E A into short exact

n

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

p

H G B H G A H G B A

n np R

The cap pro duct satises the usual pro jection formula for a morphism CA Explic

p p

itly induces H G C H G Aand Ext B C Ext B A and wehave for any

p

u H G B and Ext B C that

n

u u

For p this is just the naturality of the exact sequence

Remark The d b oundary of the hyp erhomology sp ectral sequence

H G H A H G H A

 

p q p q

pq

is given by the cap pro duct with an element u A Ext H A H A Let Z A

   

q q q q q

H A b e the quotient map from the sheaf of cycles Z A Then the extension

 

q q

q

d

H A Z A A Z A

  

q q q q

denes an element v Ext H A Z A and u A is A This is immediate

   

q q q q

from the construction of the sp ectral sequence pro of of and the general description of the

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

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 as in except for the absence of

   

an inverse i G G The denitions and the results of this section hold equally well for the

more general context of suchetale 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 dene Morita equivalence for categories in the appropriate way

In this pap er we will only use the homology for etale categories in Prop osition

Leray sp ectral sequence Morita invariance

In this section we construct for each morphism KGbetween etale group oids a functor

from csoft K sheaves to csoft G sheaves We deriveaLeray sp ectral sequence for this functor

of which the invariance of homology under Morita equivalences will b e an immediate consequence

Comma group oids of a homomorphism Let KG b e a homomorphism of etale





consider the comma group oid x whose ob jects are the pairs group oids For eachpoint x G

   

     

in x is an arrow y g x y where y K and g G An arrow k y g y g

  

k y y in K suchthat k g g When equipp ed with the obvious b ered pro duct top ology

x is again an etale group oid It should b e viewedastheberof ab ove x more exactly there

is a commutative diagram see also

x

x

K

x

G

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

g x x

by comp osition Thus the group oids x together form a right G bundle of group oids If

   

K G is a lo cal homeomorphismthenitisaG sheaf of group oids

 

More generally for any A G the comma group oid A is dened by

       

i i i

A x K G i f g

 



G

xA

with the induced top ology The nerveofA consists of the spaces

g

k k

  

  

n 

n

 

x Ag g G y y xk K fy A

n n i

When id GG these are simply denoted by xG AG Dually one denes the comma

group oids x A G x G A consisting on arrows going into x

The functors L L Let KG b e as ab ove and let A be a K sheaf We

n

dene a simplicial G sheaf B A in analogy with the denition of the barcomplex On the



     

n

spaces K G which form the nerveofG cf of strings of the form

 



G

   

g k k

n 

   

y x y

n

we dene the maps

     

n

   

K G K k k g t k

 

n  n

G

     

n

   

G G k k g s g K

 



n n

G

Notice that any is etale For any n we set

n

A B A

n n

n

 

By the stalk at x G is describ ed by

 

x A B x A B A

n n x c

n x n

This gives us the stalkwise denition of the simplicial structure on B A Tocheck the

n

continuity let us just remark that the b oundaries can b e describ ed globally Indeed using the maps

               

n n n n

d K G G K G G

   

 

i

G G

 

coming from the nerveofG see wehave d for all i n and there

n n i

are evidentmaps A d A compare to the denition of the b oundaries of B A



n

i n

are in fact

A d A d d A A

n n i n i n

n n i n n

To describ e the action of G on B A let g x x b e an arrowinG The homomorphism



induces an obvious map B x A B x Awhich via is the action by



 

x

x



g B A B A

 

x x

If S is a csoft K sheaf L S is dened as the chain complex of G sheaves asso ciated to

the simplicial complex B S If S is a b ounded b elowchain complex of csoft K sheaves

 

dene L S as the total complex of B S For an arbitrary K sheaf A L A is dened to

  

 

where S is a resolution of A as in More generallywe dene L A for any be B S

 

b ounded b elowchain complex of K sheaves using a resolution A R as in As in the case

 

of homology cf we see that L is well dened up to quasiisomorphism in particular the

derived functors

L H L Ab K AbG

n n

are well dened up to isomorphism For n we simply denote L AbK AbG by





Prop osition For any x G thereare isomorphisms

 

L A H x A for all x G

n x n

x

Proof This is an immediate consequence of relation and the fact that s preservec

n

softness since they are induced byetale maps 

Theorem LerayHochschildSerrespectral sequence For any homomorphism KG

between etale groupoids and any K sheaf A there is a natural spectral sequence

E H G L A H K A

p q pq

pq

Proof The sp ectral sequence follows from an isomorphism

H K A H G L A

and applied to L A

To prove we consider the double complex C AB G B A and weshow that

pq p q

there are maps C A B K A functorial in Asuch that the augmented complex

q q

C A C A C A B K A

q q q q

is acyclic for any csoft K sheaf A

Using the diagram

u

   

q p

 

p

K G

 

 G

G

v p

q

   

q

 

G K





G 

G

O

O

O

O

q

O

O

O w

O

O

O

O

q

   

q

K K

where are those dened b efore v w are the pro jections into the rst comp onents u

q q q p

is the pro jection into the last comp onents and wvwehaveby the general prop erties of the

q

App endix

 

p

C A G A

pq c q

p q

 

p

G u v A

c

q

   

q p

K G v A





c 

q

G

 

q

 

A K v

c q

q

 

q

   

K A

c q q

q

 

q

A B K A K

q c

q

Via these equalities the augmented chain complex commies from an augmented simplicial

k

k q



 

q

sheaf on K whose stalk at x y has the form

M M M

A A A A

x x x x

g g g

f f f

  

     

y x x x y x x y x

     

This is in fact the augmented bar complex computing the homology of the contractible

 

discrete category G y with constant co ecients A In particular it is acyclic with the usual

x

contraction f g g a fg g a 

n n

Remarks and examples

The isomorphism is actually a consequence of the quasiisomorphism 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 quasiisomorphism which can b e proved in an analogous way Com

pare to

If KG is etale SAb K then there is no need of csoft resolutions to dene

L S Indeed the condition on implies that the maps dened in are etale so there is a

n

quasiisomorphism L SB S



Let HG b e a morphism for which all the squares in are pullbacks Recall that

   

in this case the functor AbK AbG extends to a functor AbK AbG

making the diagram

forget

 

AbK

AbK



 

forget

 

AbG

AbG

commute This simple minded functor of agrees up to quasiisomorphism with the functor

 

L describ ed in Indeed for such a morphism and a p oint x G the comma group oid

x is a space or more precisely equivalent to the group oid corresp onding to a space cf In

this case the sp ectral sequence degenerates for csoft sheaves B but not for arbitrary sheaves

If is moreover etale it do es always degenerate and yields the isomorphism already proved in

Corollary Morita invariance For any Morita equivalence KG and any G sheaf A

there is a natural isomorphism

H G A H K A

p p

Proof Theorem gives a sp ectral sequence H G L A H K A By the

p q pq

 

stalk of L A at a p oint x G computes the homology of the nerveofxIf is a Morita

q

equivalence this nerveisacontractible simplicial set Thus the sp ectral sequence degenerates to

give an isomorphism

H G L A H K A

p p

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

Fib ered pro ducts of group oids For homomorphisms HG and KG their

b ered pro duct H K

G

q

H K

K

G

p

G

H

     

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

   

 

G G

   

         

is a y g z where y H z K and g y z in G An arrow y g z y g z

    

h k g The group oid pair of arrows h y y in H and k z z in K suchthatg

H K is again etale if G H K are This notion of b ered pro duct is the appropriate one for

G

group oids and generalized morphisms describ ed in and In particular if KG is a

Morita equivalence then so is p H KH G

Prop osition Changeofbase formula Consider a beredproduct of etale groupoids as in

For any csoft K sheaf S thereisacanonical quasiisomorphism

L S Lp q S

 

Proof For y H the comma group oid y p is Morita equivalent to the comma group oid

   

whichiscontinuous in y and which resp ects by a Morita equivalence y p y y

the action by H Using this observation the prop osition follows in a straightforward way from

and 

Compactly supp orted cohomology It is sometimes more convenient to reindex the

homology groups and to see them as compactly supp orted cohomology groups Because of this we

dene

n

H G H G

n

c

which give a precise meaning to H B G AThe same applies to the functors L intro duced

n

c

n

in this paragraph if KG is a homomorphismwe dene R L Ab K AbG

n

With these notations Leray sp ectral sequence b ecomes a cohomological sp ectral sequence with

p q pq

E term H G R A H K A If the b ers x are oriented k dimensional manifolds

c c

the transgression of this sp ectral sequence will give the integration along the b ers map

Z

k

H G R H K R

c c

b er

Orbifolds As wehave already mentioned in orbifolds are characterized byetale

group oids which are prop er Let G b e such a group oid The leaf space M of G ie the space

 

obtained from G dividing out by the equivalence relation x y 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 see

The obvious pro jection G M induces a sp ectral sequence

H M L A H G A

p q pq

G The stalk of L at x M is for any AAb

q

L H G A

q x q x x

   

   

wherex G is anyliftofxand G is the nite group f G s t xg this follows

x

from and the Morita equivalence x G

x

G the sp ectral sequence degenerates and gives an isomorphism In particular for AMod

R

H G A H M A

c

This also shows that the coinvariants functor

H G Mod G Mod

G

R R

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

G

c

Basic cohomology Let G b e a smo oth etale group oid The space G of compactly

cbasic

supp orted basic forms is dened as the Cokernel of

d d

   

 

G G

c c

G The where d d are the maps coming from the nerveofG In other words

G

cbasic

G is dened as the cohomology of the basic compactly supportedcohomology of G denoted H cbasic

complex G with the dierential induced by DeRham dierential on G There is an

c

cbasic

obvious pro jection from the reindexed homology see

H G R H G

c cbasic

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

 

G f G is Ginvariant and supp is compact in M

cbasic

where G M is the pro jection considered in This map asso ciates to G

cbasic

 

the G invariantform on G given by

X

   

x y g

g

xy

Verdier duality

In this section all sheaves are sheaves of R mo dules ie real vector spaces we can actually

use any eld of characteristic and Hom and are all over RWe will establish a Verdier typ e

duality for the functor L ie 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 toetale group oids But

our presentation is selfcontained As a sp ecial case we will obtain a Poincaredualitybetween the

Haeiger cohomology of etale group oids describ ed in Section and the homology theory Section

Tensor pro ducts As a preliminary remark we observe the following prop erties of tensor

pro ducts over R First if A is a csoft sheaf on a space Y and B is any other sheaf the tensor

pro duct AB is again csoft Moreover for the constant sheaf asso ciated to a vector space V we

haveY A V Y A V cf It follows by comparing the stalks that for a map

c c

f Y X also

f A B f A B

for any sheaf B on X see These prop erties extend to a morphism KG of etale

group oids for a csoft K sheaf A and any G sheaf B there is an isomorphism

A B A B

 

The sheaves RV Let us x an etale group oid K Anyopenset V K gives a free

 

K sheaf see of sets V given by the etale map s t V K and the K action dened

by comp osition Let RV b e the free Rmodule on this K sheaf V SoRV isaK sheaf of vector

spaces and for any other such K sheaf B wehave

Hom RV B Hom V B Hom V B V B

 

KR K 

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

V

of etale group oids of the kind describ ed in and RV can also b e obtained from the constant

sheaf R on K V as

RV e R

From this p oint of view the mapping prop erties 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

       

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 RV RW In this sense the

construction is functorial in V

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

M M

R V R V A

j i

j i

Proof It suces to provethatany K sheaf can b e covered by K sheaves of the form R V and

this is clear from 

The sheaves S Let S be any csoft K sheaf We write S for the sheaf SR V Note

V V

that

S Se R e e S R e e S

V

see In particular S is again csoft and has the following mapping prop erties

V

Hom S AHom R V Hom S A V H omS A Hom Sj Aj

K V K V V V

S

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

i

M M

S S S

V V V

i i i

  

is exact To see this it suces to prove that the sequence obtained byhomming into any injective

K sheaf T

M

Hom S T Hom S T

K V K V

i



is exact This is clear from the mapping prop erties

The sheaves S From nowonlet KG b e a homomorphism b etween etale

V

 

group oids For an op en set V K induces a map V G which ts into a commutative

V

diagram

K V

V

i

V

e

o

G

K

where i is the canonical Morita equivalence Thus for any csoft K sheaf S wehave

S e e S Sj

V V V

 

Notice that the group oid x is a space for anyobject x G this and the general

V

description of see give a simple description of the stalks of S It follows from this

V

description and the corresp onding fact for spaces that maps the exact sequence into an exact

sequence

M M

S S S

V V V

i i i

  

The K sheaves S T Again let KG be any homomorphism b etween etale

group oids let S b e a csoft K sheaf and let T b e an injective G sheaf Dene for each op en set

 

V K

S T V Hom S T

G V

 

We claim that this denes a sheaf S T onK Indeed for an inclusion V W there is

an evidentmap S T W S T V induced by the map S S And for a covering

V W

S

V V the sheaf prop ertyfollows from the injectivityofT together with the exact sequence

i

Furthermore this sheaf S T carries a natural K action for any arrow k y z in K

   

let W and W be neighborhoods of y and z so small that s K K has a section through

y z



By S k with t W W Then gives a map R W R W and hence S

W y z y z W

z y

comp osition one obtains a map S T W S T W and hence by taking germs an

z y



action k S T S T

z y

Prop osition Duality formula Let KG be a morphism of etale groupoids For any

injective G sheaf T any csoft K sheaf S and any other K sheaf A there is a natural isomorphism

of abelian groups

Hom A S T Hom A S T

K G

In particular S T is again injective

Proof By and the fact that is right exact on sequences of csoft sheaves it suces to dene

a natural isomorphism

Hom R V S T Hom R V S T

G K

But using and the denitions

Hom R V S T V S T Hom S T Hom R V S T 

K G V G

As for spaces one can state and prove a somewhat stronger version of using the

internal hom see

Prop osition Duality formula strong form For any A S and T as in there is a natural

isomorphism of G sheaves

Hom A S T AS T Hom

K G

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

A S T Hom B Hom AS T Hom B Hom

G G

K G

natural in B This is immediate from and

Hom B Hom A S T Hom B Hom A S T

G K

K K

Hom B A S T

K

Hom B A S T

G

Hom B A S T 

G

Remark Let KG be an etale morphism suchthateach of the squares in is

a pullback Thus K E G for some etale G space E Then has a simple description as in

and is left adjointto Thus

T Hom AHom Hom AS T Hom A S S T

K G K K

Since this holds for any A prop osition implies that for sucha

S T Hom S T Hom S T

K G

Duality for complexes Wenow extend these isomorphisms to co chain complexes It

will b e convenienttowork with chain complexes for A and S and cochain complexes for T in

Thus we will use the following convention if A is a chain complex and B is a co chain complex

HomA B is the cochain complex dened by

Y

n q

HomA B HomA B

p

pq n



Recall for later use that if B is injective and b ounded b elow then for any quasiisomorphism

 

of chain complexes A C the map HomA B HomC B is again a quasiisomorphism

   

by a standard mapping cone argument it is enough to prove the assertion for C in this case





p

remark that HomA B is the total complex of a double co chain complex whose rows HomA B

 

p

are acyclic by the injectivityof B

Similarly for a b ounded b elowchain complex S of sheaves as in we dene the co chain





complex S T by



Y



q

S T S T



p

pq n

With these conventions gives an isomorphism of co chain complexes

 

Hom A S T Hom A S T

   

G K



for anycochain complex T of injective G sheaves and any b ounded b elowchain complexes A and



S of K sheaves with S csoft There is also an obvious strong version of

 

 

Hom Hom A S T A S T

   

K G

 



The functor T Nowlet d cohdimK and x a resolution

d

R S S



of the constant sheaf R by csoft K sheaves cf For a co chain complex T of injective G sheaves

dene

  

T T S

Then is adjointto in the

Theorem Adjointness Let KG be a morphism between etale groupoids For any



boundedbelow chain complex A of csoft K sheaves and any boundedbelow cochain complex T of



injective G sheaves there is a natural quasiisomorphism

 

Hom A T Hom L A T

 

K G

there is also a strong version derived from

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



instead of G by R Since A F A is a quasiisomorphism using with A F instead

    

 

of A and S S the fact that T are injective cf and the general remark in we

 

get a quasiisomorphism

 

S T Hom A F S T Hom A

    

G K

Remark that for any b ounded b elowchain complex B of csoft K sheaves wehave a quasi



isomorphism followed by an isomorphism L B L B F B F This for

    

B A S and the quasiisomorphism A A S give L A A F S Us

         



ing this the fact that T s are injective and the statement of the theorem follows easily 

Poincareduality follows in the usual way

 

Corollary Poincare duality Let K beanetale groupoid and suppose that K is a topological

manifold of dimension dLet or be the orientation K sheaf There is a natural isomorphism

pd

H K or H K R p Z

p



Proof Let G b e the trivial group oid In let T b e the complex R concentrated in

i

degree p and let A b e the complex A S as in As A is quasiisomorphic to R the

 

i

complex on the right of the quasiisomorphism in has



H Hom A T H K R



p



Now consider the left hand side of the quasiisomorphism of HomA T in this



sp ecial case Note rst that R is quasiisomorphic to the orientation sheaf concentrated in degree

d

R or d

 



R see is the complex Indeed for any op en set V K RV Hom V S

c



V R and the argumentforisjustlike the one for spaces Thus for which computes H

c

 

T R pand A S R



 

pd

H Hom A T H K T H K or 



K inv

Relation to cyclic homology

In the homology of etale group oids and the Leray sp ectral sequence paragraph are the main

to ols in dealing with cyclic homology of etale group oids We shall give here an overview of the

main results in expressing the cyclic homology of the convolution algebra of an etale group oid

in terms of the homology of etale categories For instance generalizes the previous results of

Connes for G a space Burghelea and Karoubi for G a group Brylinski and Nistor for G

a separated etale group oid Feigin and Tsygan Nistor In this section all sheaves

are sheaves of vector spaces

Mixed complexes of sheaves Let G b e an etale group oid By a mixed complex of G sheaves

we mean a mixed complex A bB in the category Ab G This means a family fA n g of



n

G sheaves equipp ed with maps of degree b A A and maps of degree B A

n n n

A such that b B b B For the general notions and constructions concerning

n

mixed complexes in any ab elian category see Recall that anysuch mixed complex A bB



gives rise to a double complex B AinAb G hence the Ho chschild and cyclic homology sheaves

are dened see also in

g g

HH A HC A Ab G

 

The Ho chschild and cyclic hyp er homology of the mixed complex A bB denoted HH G A



HC G A are dened as the hyp erhomology of the complexes of G sheaves A bandTotB A



resp ectively compare to From wegettwo sp ectral sequences with E terms

g g

H G HH A HH G A andH G HC A HC G A

   

p q pq p q pq

The sp ectral sequences of the double complex B G T where T A bor TotB A givetwo

   

sp ectral sequences with E terms

HH H G A HH G A andHC H G A HC G A

   

p q pq p q pq

Also from we get the SBI sequence relating HH G A and HC G A The p erio dic cyclic

 

homology is dened as usual as lim HC G A



S

Cyclic G sheaves It is well known and it is a motivating example that any cyclic ob ject

in an ab elian category gives rise to a mixed complex In particular any cyclic G sheaf A



ie a contravariant functor Ab G from Connes category induces a mixed complex

g g

of G sheaves The corresp onding homologies are still denoted HH A HC A HH G A

  

HC G A



 

A basic example of a cyclic G sheaf for a smo oth etale group oid is C dened by C n the

   

n

pullback of the sheaf of smo oth functions on G along the diagonal emb edding G

n

   

n

 

n is the vector G with the cyclic structure describ ed as follow At c G the stalk of C

 

 

of smo oth functions dened for x x G around cand space of germs f x x

n n

 

f x xx x if i n

i i n

 

d f x x

i n

 

f x x x if i n

n n

   

t f x x f x x x

n n n n

Using the quasiisomorphism C b which app ears in the work of Connes see also lemma

in weget

M

q

HH G C H G

n p

pq n

Y

HP G C H G i f g

i ik

k

Cyclic homology of the convolution algebra The convolution algebra of a smo oth etale

group oid G was used by Connes as a noncommutative mo del for the leaf space of G When G

is Hausdor C G is the lo cally convex algebra of compactly supp orted smo oth functions on

c

P





     

G with the convolution pro duct uv g u g v g Its continuous Ho chschild and

g g g

 





n

cyclic homology are computed by the cyclic vector space C G with C G C G

c

 

n

C G here denotes the pro jective tensor pro duct and the last isomorphism is an algebraic

c

one see

Using the functor describ ed in the App endix the denition of the convolution algebra

c

C G extends to the nonHausdor case see and in It also b ecomes clear that the

c

continuous version of the Ho chschild cyclic and p erio dic cyclic homology of this algebra should

 

n

b e dened using the cyclic vector space C G with C G C G and the usual cyclic

c

structure In this way the ChernConnesKaroubi character Ch K C G HP C G i

i i

c c

f g extends to the nonHausdor case

The Ho chschild homology of C G For G as b efore not necessarily Hausdor we

c

  

  

 

G see where B intro duce the group oid of lo ops G B f G s

 



G

 

t g is the space of lo ops with the G action given by conjugation g g g

There is a simplicial complex C of csoft G sheaves which is obtained bytwisting C

tw

 

 

   

see by lo ops More precisely C n s C n where s B G denotes the restriction

tw

   

tw

dened as follows At B the to B of the source map with the twisted b oundaries d i





     

stalk of C n is the vector space of germs f x x dened for x x G around s

n n

tw

and

 

f x xx x if i n

i i n

tw

 

d f x x

n

i

   

f x x x if i n

n n

The following is a reformulation of in

Prop osition For any smooth etale groupoid G there is a natural isomorphism

HH C G H G C

c tw

Recall that is the category dened in the same way as except that the cyclic relation

n

t is omitted Remark that C actually has a structure given by the b oundaries

n

tw

tw

d just describ ed and the cyclic action

i

     

t f x x f x x x

n n n n

In other words C can b e viewed as a contravariant functor ShG or equivalentlyas

tw

a G sheaf The previous prop osition b ecomes

HH C G H G C

tw

The cyclic homology of C G We relate now HC C G to the homology of an etale

c c

category Remark that

n

        

t f x x f x x x

n n n n

which shows that C is in fact a G sheaf where G istheetale category obtained

tw





n

n A from G by imp osing the relations t id id for all B

n

n

reformulation of and in is

Prop osition For any smooth etale groupoid G there is a natural isomorphism

HC C G H G C

c tw

We remark that the SBI sequence can b e describ ed via the isomorphisms and the one of

as the Gysintyp e long exact sequence arising from the Leray sp ectral sequence applied to the

obvious pro jection map G G Note that wehave tacitly made use of the

extension of homology for etale group oids to etale categories cf

Remark A Morita equivalence G K induces Morita a equivalence G K hence

the Morita invariance for homology implies the Morita invariance of the Ho chschild and cyclic

homology of the smo oth convolution algebras

Lo calization Remark that GG as units so we recover HH G C as lo calization

at units

HH G C HH C G

The isomorphisms describ ed in will give

M

q

G H G HH C

p n

c

pq n

q

where is the G sheaf of q forms and also Theorem in

Prop osition For any smooth etale groupoid G there is a natural isomorphism

Y

HP C G H G i f g

i ik

c

k

   

More generallyany G invariant subspace OB denes a group oid G O G G

O

and the lo calized Ho chschild and cyclic homology indicated by the subscript O When O is elliptic

ie or d for all O it is shown in Theorem that

Y

HP C G H G if g

i O ik O

c

k

The case of orbifolds Let M M U b e an orbifold M is the underlying top ological

space U an orbifold atlas Due to remark the Ho chschild cyclic and p erio dic cyclic homologies

do not dep end on the representation of the orbifold M by a smo oth prop er etale group oid We

simply denote these homologies by HH MHC MHP M

Note that for any representation of M by a prop er etale group oid G the lo op group oid G

is again a prop er etale group oid Denote byM the underlying space of the orbifold induced by

G ie M is the leaf space of G This space can b e constructed directly by using an

orbifold atlas for Mitwas intro duced in this wayin it serves there for the denition of a

geometric Chern character needed in the formulation of the index theorem for orbifold from this

p oint of view the next prop osition explains this choice Alternatively representing M as a quotient

NL where L is a compact Lie group acting on M with nite stabilizers see then

b

MN L

b

     

where N f x M L x xg is Brylinskis space with the Laction x g xg g g

Then applied to G and give the following result which also makes the connection with

Kawasakis denition of the Chern character for orbifolds

Prop osition For any orbifold M

Y

ik

HP M H M i f g

i

c

k

App endix Compact supp orts in nonHausdor spaces

In this app endix we explain how the usual notions concerning compactness and sheaves on

Hausdor spaces extend to our more general context see For basic denitions and facts for

sheaves on Hausdor spaces we refer the reader to any of the standard sources

csoft sheaves Let X b e a space satisfying the general assumptions in An ab elian sheaf

A on X is said to b e csoft if for any Hausdor op en U X its restriction Aj is a csoft sheaf on

U

U in the usual sense By the same prop erty for Hausdor spaces it follows that csoftness is a lo cal

S

prop erty ie a sheaf A is csoft i there is an op en cover X U suchthateach Aj is a csoft

i U

sheaf on A

The functor Let A b e a csoft sheaf on X and let A be its Godement resolution ie

c

A U U A is the set of all not necessarily continuous sections for anyopenU X

discr

For any Hausdor op en set W X let W A b e the usual set of compactly supp orted sections c

If W U there is an evident homomorphism extension by W A U A U A

c c

For any not necessarily Hausdor op en set U X we dene U A to b e the image of the map

c

M

W A U A

c

W

where W ranges over all Hausdor op en subsets W U

Observe that U A so dened is evidently functorial in A and that for any inclusion U U

c

wehave an extension by zero monomorphism

U A U A

c c

The following lemma shows that in the denition of U A itisenoughtoletW range over

c

a Hausdor op en cover of U in particular it shows that the denition agrees with the usual one if

U itself is Hausdor

S

Lemma Let A be a csoft sheaf on X For any open cover U W whereeach W is

i i

Hausdor the sequence

M

W A U A

c i c

i

is exact

T L

W W A Proof It suces to showthatforany Hausdor op en W U the map

c i

i

W A is surjective This is well known see eg 

c

This lemma is in fact a sp ecial case of the following Prop osition MayerVietoris

S

Prop osition Let X U beanopen cover indexedbyanorderedsetI andletA be a csoft

i

i

sheaf on X Then there is a long exact sequence

M M

U A X A U A

c i c c i i

  

i i i

  

Here U U U as usual Thereisofcourse a similar exact sequenceifI is not

i i i i

 n  n

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 eachoftheU is Hausdor as follows Let

i

S

X W be a cover by Hausdor op en sets and consider the double complex

j

j J

M

W C U A

c j j pq i i

 p  q

where the sum is over all j ji i For a xed p the column

p q

C is a sum of exact MayerVietoris sequences for the Hausdor op en sets W augmented by



p j j

 p

L

W A Keeping the notation U X W if q pwe C

c j j i i j j p

 p  q  p

j j

 p

observe that for a xed q the row C is a sum of MayerVietoris sequences for the spaces



q

U with resp ect to the op en covers fW U g So if the prop osition would hold for covers

i i j i i

 q  q

by Hausdor sets eachrow C q is also exact By a standard double complex argumentit



q

follows that the augumentation 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

i

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

X Anowfollows by Lemma Toshow exactness elsewhere consider for each nite subset

c

S

I



U and the subsequence I I the space U

i

iI



M M

I



A U A U A U

c i i c c i

  

i i in I i in I

    

of Clearly is the directed union of the sequences of the form where I I ranges

over all nite subsets of I So exactness of follows from exactness of eachsuch sequence of the

form Thus it remains to prove the prop osition in the sp ecial case of a nite cover fU g of X

i

by Haudor op en sets

So assume X U U where each U is Hausdor For n there is nothing to prove

n i

For n the sequence has the form

M

U U A U A U A U U A

c c c c

This sequence is exact at X Aby and evidently exact at other places Exactness for n

c

can b e proved using exactness for n Indeed consider the following diagram whose upp er two

rows are the sequences for n and whose third row is constructed by taking vertical cokernels

so that all columns are exact we delete the sheaf A from the notationcompare to pp in

U U U U U

c c c c

U U U U U U U U

c ij c ij c c c c

U U U U

C

c c c c

To show that the middle rowisexactitthus suces to prove that the lower rowisexactThisrow

can b e decomp osed intoaMayerVietoris sequence for the case n already shown to b e exact

U U U U U U

c c c c

and the sequence

U U U U C

c c

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

the righthand column

An identical argumentwillshow that the exactness for a cover by n op ens follows from

exactness for one by n op ens so the pro of is completed by induction 

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

dor spaces to the nonHausdor case as illustrated by the following corollaries

Corollary Let Y X be a closed subspace and let A be a csoft sheaf on X There is an exact

sequence

i r

X Y A X A Y A

c c c

i is extension by zero r is the restriction

Proof This including the fact that the map r is well dened follows by elementary homological

algebra from the fact that the Corollary holds for Hausdor spaces by using 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 i

Corollary For a family A of csoft sheaves on X the direct sum A is again csoft and

i i

X A X A

c i c i

In particular when working over R we have for any csoft sheaf S of Rvector spaces and any

vector space V that the tensor product S V here V is the constant sheaf is again csoft and

R

the familiar formula

X S V X S V

c R c R

Corollary Let A B be a quasiisomorphism between chain complexes of csoft sheaves on

 

X Then

X A X B

 

c c

is again a quasiisomorphism

Proof By a mapping cone argument wemay assume that B In other words wehaveto



show that X A is acyclic whenever A is This follows from the MayerVietoris sequence

 

c

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 

The following Corollary is included for application in

Corollary Let Y X be a closed subspace and let X R beacontinuous map such that

Y Let A be a csoft sheaf on X Then for any X A

c

j i j 

Y

here j is the restriction r as in

Y

 

Proof For write Y fx X j x j g and for eachopenU X write

U Af U Aj g

c U Y

c

It suces to show that

M

X A X A

c c

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

i

L

L

L

u

U A

U Y A

U A

i

c i

i

c

i

c

i

i



L

v

X A

X Y A

X A

c

c

c

where the isomorphisms on the right come from We wish to show that v is epi Since u is epi

by the Hausdor case it suces to show that is epi or equivalentlythat is epi This is indeed

the case by 

It is quite clear that using csoft resolutions one can dene compactly supp orted cohomol

ogy H X Aforany AAb X In particular we get an extension H X of X toall

c

c c

sheaves this extension is still denoted by X

c

Prop osition Let f Y X beacontinuous map There is a functor f AbY AbX

with the fol lowing properties

 

i For any open U X and any BAbY U f B f U B

c c

   

x B Y f B f ii For any point x X and any BAb

x c

 

iii f is left exact and maps csoft sheaves into csoft sheaves

 

iv For any beredproduct

p

Z Y

Y

X

q

f

e

Z X

along an etale map e and for any csoft BAb Y thereisacanonical isomorphism

e f B q p B

see 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 the corresp ondence for any Hausdor space Z between csoft sheaves S on Z and

abby cosheaves C on Z given by

W S C W

c

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 Z is given by the exact sequence

CZ z CZ S

z

We use this corresp ondence in the construction of f However see remark b elow for a

description of f which do esnt use this corresp ondence

We discuss rst the construction of f on csoft sheaves Let BAb Y b e csoft First assume

X is Hausdor Let B b e a csoft sheaf on Y and dene a cosheaf C cB by C U f U B

c

Note that C is indeed a abby cosheaf by By the corresp ondence there is a csoft sheaf S

on X uniquely determined up to isomorphism by the identity U S C U for anyopen U X

c

Thus if X is Hausdor we can dene f B to b e this sheaf S

In the general case cover X by Hausdor op ens U and dene in this way for each i a csoft

i

sheaf S on U by

i i

V S f V B

c i c

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

S j S j satisfying the co cycle condition Therefore the sheaves S patch together to

ij j U i U i

ij ij

a sheaf S on X uniquely determined up to isomorphism by the condition that Sj S byan

U i

i

isomorphism compatible with Thus we can dene f B to b e S

ij

   

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

 

Prop erty i clearly holds for an op en set U contained in some U by For general U prop erty i

 

i then follows by the MayerVietoris sequence Next identity yields for anypoint x X an

exact sequence

 

Y f x B Y B f B

c c x

   

and hence by the isomorphism ii of the Prop osition Finally iv is clear from the lo cal nature

of the construction of f

For general AAbY wedene f A AbX asthekernel of the map f S f S

where A S S is a csoft resolution of A from the rst part it follows that it

   

is well dened up to isomorphisms The prop erties i and ii are now immediate consequences of

 

the denition and of the previous case Using and ii it easily follows that f transforms acyclic

complexes of csoft sheaves on AbY into acyclic complexes on AbX This immediately implies

that is left exact 

Remark We outline an alternative construction and pro of of Prop osition whichdoes

not use the corresp ondence b etween sheaves and cosheaves This construction will b e used in the

pro of of b elow We will assume that B is csoft and X is Hausdor As in the pro of of

the construction of f for general X is then obtained by glueing the constructions over a cover by

Hausdor op ens U X

i

So let B b e a csoft sheaf on Y For anyopenset V Y denote by B the sheaf on Y obtained

V

by extending Bj by zero Thus B is evidently csoft and Y B V B Moreover an

V V c V c

inclusion V W induces an evidentmapB B

V W

S

Nowlet Y W be a cover by Hausdor op en sets This cover induces a long exact sequence

i

M M

B B B

W W

i i i

  

i i i

  

of csoft sheaves on Y By Corollary the functor Y applied to this long exact sequence

c

again yields an exact sequence and this is precisely the MayerVietoris sequence of For each

i i let f W X b e the restriction of f thisisamapbetween Hausdor spaces

n i i i i

 n  n

so wehavef B dened as usual Dene f B as the cokernel tting into a long

i i W

 n i i

n



exact sequence

M M

f B f f B B

i W i i W

 i   i i

  

i i i

  

 

x W B by the Hausdor case So taking stalks For x X wehavef W f

i i i x c

  

of the long exact sequence in at x and using the MayerVietoris sequence for the space

       

x B asin ii Prop erty i is proved in a similar way using x we nd f B f f

x c

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



tion AS S and dening Rf A as the complex f S Up to quasi



isomorphisms this complex is well dened and do es not dep end on the resolution S by In

this waywe obtaine in fact a well dened 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

Lemma For any pul lback diagram

p

Z Y

Y

X

q

f

e

Z X

and any sheaf B on Y thereisa canonical quasiisomorphism

Rq p Be Rf B

Proof Using MayerVietoris 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 csoft

S

Let Y W as in so that f B tsinto a long exact sequence of csoft sheaves

i

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

W W

i i i

  

i i i

  



Now let p B S b e a csoft resolution over the pullback Z Y Then for anyopen U Y

X

 

is a csoft resolution of q p B The lemma is a csoft resolution of p B so q S S

  W

p W p W

now follows by comparing the sequence to the dening sequence

M M

q S  q S  q S

p W p W

i i i

  

i i i

  

def

for q p B q S 

f on etale maps Let f Y X be an etale map ie a lo cal homeomorphismItiswell

known that the pullback functor f Ab X AbY has an exact leftadjoint f AbY

AbX describ ed on the stalks by f B  B This construction agrees with the one in

 

x y

y f x

In particular for etale f the counit of the adjunction denes a map

f f A A

f

summation along the b er for any sheaf A on X

f on prop er maps Dene a map f Y X between nonnecessarily Hausdor spaces

to b e proper if

i the diagonal Y Y Y is closed

X

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

csoft sheaf A on X there is a natural map X A Y f A dened by pullback as in the

c c

Hausdor case

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 of the pap er can b e based on the usual denition of and

c

are indep endent of the App endix Now any nonseparated manifold or suciently nice space cf

X can b e viewed as a trivial group oid which is Morita equivalent to the Hausdor etale





to b e the group oid G dened from an op en cover fU g of X by Hausdor op en sets by taking G

i

     

disjoint sum of the U and G G G

i X

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Marius Crainic

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POBox TAUtrecht The Netherlands

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