Doc Math J DMV

On the Cyclic Homology

of Ringed Spaces and Schemes

Bernhard Keller

Received March

Revised September

Communicated by Peter Schneider

Abstract We prove that the cyclic homology of a scheme with an ample

line bundle coincides with the cyclic homology of its category of algebraic

vector bundles As a byproduct of the pro of we obtain a new construction

of the Chern character of a p erfect complex on a ringed space

Mathematics Sub ject Classication Primary E Secondary

E F

Keywords Cyclic homology Chern character Ringed space Scheme Perfect

complex Derived category

Introduction

The main theorem Let k b e a eld and X a scheme over k which admits an

ample line bundle eg a quasipro jective variety Let vecX denote the category

of algebraic vector bundles on X We view vecX as an exact category in the sense

of Quillen By denition a short sequence of vector bundles is admissible exact

i it is exact in the category of sheaves on X Moreover the category vecX is

k linear ie it is additive and its morphism sets are k vector spaces such that the

comp osition is bilinear In we have dened for each k linear exact category A a

der

A The sup erscript der indicates that the denition cyclic homology theory HC

is mo deled on that of the derived category of A In lo c cit it was denoted by

HC A As announced in lo c cit in this article we will show that the cyclic

homology of the scheme X coincides with the cyclic homology of the k linear exact

category vecX There is a canonical isomorphism cf Corollary



der

HC X HC vecX

The denition of the cyclic homology of a scheme is an imp ortant technical p oint

which will b e discussed b elow in Note that by denition Par there is an

analogous isomorphism in K theory

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Bernhard Keller

Motivation Our motivation for proving the isomorphism is twofold

Firstly it allows the computation of HC X for some nontrivial examples Indeed

supp ose that k is algebraically closed and that X is a smo oth pro jective algebraic va

riety Supp ose moreover that X admits a tilting bund le ie a vector bundle without

higher selfextensions whose direct summands generate the b ounded derived category

of the category of coherent sheaves on X Examples of varieties satisfying these hy

p otheses are pro jective spaces Grassmannians and smo oth quadrics

In we deduce from that for such a variety the Chern character induces

an isomorphism



K X HC k HC X

Z

Here the left hand side is explicitly known since the group K X is free and admits

a basis consisting of the classes of the pairwise nonisomorphic indecomp osable di

rect summands of the tilting bundle Cyclic homology of pro jective spaces was rst

computed by Beckmann using a dierent metho d

Our second motivation for proving the isomorphism is that it provides fur

der

ther justication for the denition of HC Indeed there is a comp eting and

previous denition of cyclic homology for k linear exact categories due to R Mc

McC

Carthy Let us denote by HC A the graded k mo dule which he asso ciates

McC

with A McCarthy proved in lo c cit a number of go o d prop erties for HC The

most fundamental of these is the existence of an agreement isomorphism



McC

HC A HC pro jA

where A is a k algebra and pro jA the category of nitely generated pro jective A

mo dules endowed with the split exact sequences In particular if we take A to b e

commutative we obtain the isomorphism



McC

vecX HC X HC

for all ane schemes X Sp ecA to identify the left hand side we use Weibels

isomorphism b etween the cyclic homology of an ane scheme and the cyclic

der

homology of its co ordinate algebra Whereas for HC this ismorphism extends

McC

to more general schemes this cannot b e the case for HC Indeed for n

n

the group H X O o ccurs as a direct factor of HC X However the group

X n

McC

HC vanishes for n by its very denition

n

Generalization Chern character Our pro of of the isomorphism

actually yields a more general statement Let X b e a quasicompact separated scheme

over k Denote by p er X the pair formed by the category of p erfect sheaves on

X and its full sub category of acyclic p erfect sheaves The pair p er X is a lo calization

pair in the sense of and its cyclic homology HC p er X has b een dened in

lo c cit We will show that there is a canonical isomorphism



HC X HC p er X

If X admits an ample line bundle we have an isomorphism



der

HC vecX HC p er X

so that the isomorphism results as a sp ecial case

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Cyclic homology

The rst step in the pro of of will b e to construct a map

HC p er X HC X

This construction will b e carried out in for an arbitrary top ological space X

endowed with a sheaf of p ossibly noncommutative k algebras As a byproduct we

therefore obtain a new construction of the Chern character of a p erfect complex P

Indeed the complex P yields a functor b etween lo calization pairs

P p er pt p er X

k

and hence a map

HC p er pt HC p er X HC X

The image of the class

chk HC p er pt HC k

under this map is the value of the Chern character at the class of P An analogous

construction works for the other variants of cyclic homology in particular for negative

cyclic homology The rst construction of a Chern character for p erfect complexes is

due to BresslerNestTsygan who needed it in their pro of of SchapiraSchneiders

conjecture They even construct a generalized Chern character dened on all

higher K groups Several other constructions of a classical Chern character are due

to B Tsygan unpublished

Cyclic homology of schemes Let k b e a commutative ring and X a scheme

over k The cyclic homology of X was rst dened by Lo day He sheaed the

classical bicomplex to obtain a complex of sheaves CC O He then dened the

X

cyclic homology of X to b e the hypercohomology of the total complex of CC O

X

Similarly for the dierent variants of cyclic homology There arise three problems

The complex CC O is unbounded to the left So there are at least two non

X

equivalent p ossibilities to dene its hypercohomology should one take Cartan

Eilenberg hypercohomology cf or derived functor cohomology in the sense

of Spaltenstein

Is the cyclic homology of an ane scheme isomorphic to the cyclic homology of

its co ordinate ring

If a morphism of schemes induces an isomorphism in Ho chschild homology do es

it always induce an isomorphism in cyclic homology

Problem is related to the fact that in a category of sheaves pro ducts are not exact

in general We refer to for a discussion of this issue

In the case of a no etherian scheme of nite dimension Beckmann and Weibel

Geller gave a p ositive answer to using CartanEilenberg hypercohomology

By proving the existence of an SBIsequence linking cyclic homology and Ho chschild

homology they also settled for this class of schemes whose Ho chschild homology

vanishes in all suciently negative degrees Again using CartanEilenberg hypercoho

mology Weibel gave a p ositive answer to in the general case in There he also

showed that cyclic homology is a homology theory on the category of quasicompact

quasiseparated schemes Problem remained op en

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Bernhard Keller

We will show in A that CartanEilenberg hypercohomology agrees with Spal

tensteins derived functor hypercohomology on all complexes with quasicoherent ho

mology if X is quasicompact and separated Since CC O has quasicoherent ho

X

mology this shows that problem do es not matter for such schemes As a

byproduct of A we deduce in B a partially new pro of of Bo ekstedtNeemans

theorem which states that for a quasicompact separated scheme X the unbounded

derived category of quasicoherent sheaves on X is equivalent to the full sub category

of the unbounded derived category of all O mo dules whose ob jects are the com

X

plexes with quasicoherent homology A dierent pro of of this was given by Alonso

JeremasLipman in Prop

In order to get rid of problem we will slightly mo dify Lo days denition

Using sheaves of mixed complexes as introduced by Weibel we will show that the

image of the Ho chschild complex C O under the derived global section functor is

X

canonically a mixed complex M X The mixed cyclic homology of X will then b e

dened as the cyclic homology of M X For the mixed cyclic homology groups the

answer to is p ositive thanks to the corresp onding theorem in Ho chschild homology

due to WeibelGeller the answer to is p ositive thanks to the denition The

mixed cyclic homology groups coincide with Lo days groups if the derived global

section functor commutes with innite sums This is the case for quasicompact

separated schemes as we show in

Organization of the article In section we recall the mixed complex

of an algebra and dene the mixed complex M X A of a ringed space X A In

section we recall the denition of the mixed complex asso ciated with a lo calization

pair and give a sheaable description of the Chern character of a p erfect complex

over an algebra In section we construct a morphism from the mixed complex

asso ciated with the category of p erfect complexes on X A to the mixed complex

M X A We use it to construct the Chern character of a p erfect complex on X A

In section we state and prove the main theorem and apply it to the computation

of the cyclic homology of smo oth pro jective varieties admitting a tilting bundle In

app endix A we prove that CartanEilenberg hypercohomology coincides with derived

functor cohomology for unbounded complexes with quasicoherent homology on

quasicompact separated schemes In app endix B we apply this to give a partially

new pro of of a theorem of Bo ekstedtNeeman

Acknowledgment The author thanks the referee for his suggestions which

help ed to make this article more readable

Homology theories for ringed spaces

Let k b e a eld X a top ological space and A a sheaf of k algebras on X In

this section we consider the p ossible denitions of the cyclic homology of X A In

we recall the denition suggested by Lo day In we p oint out that with

this denition it is not clear that a morphism inducing isomorphisms in Ho chschild

homology also do es so in cyclic homology and its variants This is our main reason

for introducing the mixed homologies These also have the advantage of allowing

a unied and simultaneous treatment of all the dierent homology theories For the

sequel the two fundamental invariants are the mixed complex of sheaves M A and

its image M X A RX M A under the derived global section functor Both

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Cyclic homology

are canonical up to quasiisomorphism and are therefore viewed as ob jects of the

corresp onding mixed derived categories In the case of a p oint and a sheaf given

by an algebra A these complexes sp ecialize to the mixed complex M A asso ciated

with the algebra The mixed cyclic homology HC A is dened to b e the cyclic

mix

homology of the mixed complex M X A

Hochschild and cyclic homologies Following a suggestion by Lo day

per

the Hochschild complex C A and the bicomplexes CC A CC A and CC A

are dened in by comp osing the classical constructions cf for example

with sheacation The Hochschild homology cyclic homology of A are then

obtained as the homologies of the complexes

RX C A RX C C A

where RX is the total right derived functor in the sense of Spaltenstein of

the global section functor

Mixed cyclic homologies Supp ose that f X A Y B is a morphism

of spaces with sheaves of k algebras inducing isomorphisms in Ho chschild homology

With the ab ove denitions it do es not seem to follow that f also induces isomorphisms

in cyclic homology negative cyclic homology and p erio dic cyclic homology This is

one of the reasons why we need to replace the ab ove denitions by slightly dierent

variants dened in terms of the mixed complex asso ciated with A This complex was

introduced by C Weibel in However the mixed homologies we consider do not

always coincide with the ones of cf the end of this section

Let us rst recall the case of ordinary algebras For an algebra A we denote by

M A the mapping cone over the dierential t linking the rst two columns of

the bicomplex CC A We endow M A with the op erator B M A M A

induced by the norm map N from the rst to the second column of the bicomplex

Then endowed with its dierential d and with the op erator B the complex M A

b ecomes a mixed complex in the sense of Kassel ie we have

d B dB B d

The mixed complex M A completely determines the homology theories of A Indeed

we have a canonical quasiisomorphism

C A M A

which shows that Ho chschild homology is determined by M A We also have canon

ical quasiisomorphisms

 

L

CC A M A k CC A R Hom k M A

where the right hand sides are dened by viewing mixed complexes as ob jects of the

mixed derived category ie dierential graded dg mo dules over the dg algebra

generated by an indeterminate of chain degree with and d cf

Finally we have a quasiisomorphism

per

CC A R limP n M A

k

where P is a cobrant resolution closed resolution in the sense of

k

semifree resolution in the sense of of the dg mo dule k and the transition

map P n P n comes from a chosen morphism of mixed complexes

k k

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Bernhard Keller

P P which represents the canonical morphism k k in the mixed derived

k k

category For example one can take

M

P i

k

iN

as a mo dule endowed with the dierential mapping the generator of i to

i

The p erio dicity morphism then takes to and to Note that the

i i i

functor lim P n is actually exact so that R lim may b e replaced by lim in the

k

ab ove formula

Following Weibel Section we sheafy this construction to obtain a mixed

complex of sheaves M A We view it as an ob ject of the mixed derived category

DMix X of sheaves on X ie the derived category of dg sheaves over the constant

sheaf of dg algebras with value The global section functor induces a functor from

mixed complexes of sheaves to mixed complexes of k mo dules By abuse of notation

the total right derived functor of the induced functor will still b e denoted by RX

The mixed complex of the ringed space X A is dened as

M X A RX M A

The fact that the functor RX and the mixed derived category of sheaves is

well dened is proved by adapting Spaltensteins argument of section of Since

the underlying complex of k mo dules of M A is quasiisomorphic to C A we have

a canonical isomorphism



HH A H RX M A

We dene the mixed variants

per

HC A HC A HC A

mix

mix

mix

of the homologies asso ciated with A by applying the functors

L

k R Hom k resp R lim P n

k

to M X A and taking homology

These homology theories are slightly dierent from those of BresslerNest

Tsygan Weibel and Beckmann We prove in that mixed cyclic

homology coincides with the cyclic homology dened by Weibel if the global section

functor RX commutes with countable copro ducts and that this is the case if

X A is a quasicompact separated scheme

For a closed subset Z X we obtain versions with supp ort in Z by applying

the corresp onding functors to R X M A

Z

Now supp ose that a morphism X A Y B induces an isomorphism in HH

Then by denition it induces an isomorphism in the mixed derived category

RX M A RY M B

per

and thus in HC HC and HC

mix

mix

mix

Homology theories for categories

In this section we recall the denition of the cyclic homology or rather the

mixed complex of a lo calization pair from We apply this to give a description

of the Chern character of a p erfect complex over an algebra A sheaf of algebras

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Cyclic homology

over a p oint This description will later b e generalized to sheaves of algebras over a

general top ological space

A lo calization pair is a pair consisting of a small dierential graded k category

and a full sub category satisfying certain additional assumptions To dene its mixed

complex we pro ceed in three steps In the classical denition for algebras is

generalized to small k categories following an idea of Mitchells then in we

enrich our small k categories over the category of dierential complexes ie we dene

the mixed complex of a dierential graded small k category by making this denition

relative we arrive in at the denition of the mixed complex of a lo calization pair

For simplicity we work only with the Ho chschild complex at rst

We illustrate each of the three stages by considering the resp ective categories

asso ciated with a k algebra A the k category pro jA of nitely generated pro jec

b

tive Amo dules the dierential graded k category C pro jA of b ounded complexes

over pro jA and nally the lo calization pair formed by the category of all p erfect

complexes over A together with its full sub category of all acyclic p erfect complexes

The three resp ective mixed complexes are canonically quasiisomorphic Thanks to

this fact the mixed complex of an algebra is seen to b e functorial with resp ect to exact

functors b etween categories of p erfect complexes This is the basis for our description

of the Chern character in

k categories Let C b e a small k category ie a small category whose mor

phism spaces carry structures of k mo dules such that the comp osition maps are bi

linear Following Mitchell one denes the Ho chschild complex C C to b e the

complex whose nth comp onent is

a

C X X C X X C X X C X X

n n n n n

where the sum runs over all sequences X X of ob jects of C The dierential is

n

given by the alternating sum of the face maps

f f f f if i

n i i

d f f f f

i n i i

n

f f f if i

n

For example supp ose that A is a k algebra If we view A as a category C with

one ob ject the Ho chschild complex C C coincides with C A We have a canonical

functor

A pro j A

where pro j A denotes the category of nitely generated pro jective Amo dules By a

theorem of McCarthy this functor induces a quasiisomorphism

C A C pro j A

Differential graded categories Now supp ose that the category C is a

dierential graded k category This means that C is enriched over the category of

dierential Zgraded k mo dules dg k mo dules ie each space C X Y is a dg

k mo dule and the comp osition maps

C Y Z C X Y C X Z

k

are morphisms of dg k mo dules Then we obtain a double complex whose columns

are the direct sums of and whose horizontal dierential is the alternating sum

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Bernhard Keller

of the face maps

f f f f if i

n i i

d f f f f

i n i i

n

f f f if i

n

where deg f deg f deg f The Ho chschild complex C C of the dg

n

category C is by denition the sum total complex of this double complex The dg

categories we will encounter are all obtained as sub categories of a category CX of

dierential complexes over a k linear category X a k linear category is a k category

which admits all nite direct sums In this case the dg structure is given by the

complex H om X Y asso ciated with two dierential complexes X and Y

X

b

Hence if A is a k algebra the category C pro j A of b ounded complexes of

nitely generated pro jective Amo dules is a dg category and the functor

b

pro j A C pro j A

mapping a mo dule P to the complex concentrated in degree whose zero comp onent is

P b ecomes a dg functor if we consider pro j A as a dg category whose morphism spaces

b

are concentrated in degree By lemma the functor pro j A C pro j A

induces a quasiisomorphism

b

C pro j A C C pro j A

Pairs of dg categories Now supp ose that C C are full sub categories

of a category of complexes CX over a small k linear category X We dene the

Ho chschild complex C C of the pair C C C to b e the cone over the morphism

C C C C

induced by the inclusion here b oth C and C are viewed as dg categories For

example let A b e a k algebra Recall that a p erfect complex over A is a complex

of Amo dules which is quasiisomorphic to a b ounded complex of nitely generated

pro jective Amo dules Let p er A denote the pair of sub categories of the category of

complexes of Amo dules formed by the category p er A of p erfect Amo dules and

its full sub category p er A of acyclic p erfect Amo dules Clearly we have a functor

pro j A p er A ie a commutative diagram of dg categories

p er A

p er A

pro j A

This functor induces a quasiisomorphism

C pro j A C p er A

by theorem b of

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Cyclic homology

Mixed complexes and characteristic classes In the preceding para

graph we have worked with the Ho chschild complex but it is easy to check that

everything we said carries over to the mixed complex The conclusion is then

that if A is a k algebra we have the following isomorphisms in the mixed derived

category

 

M A M pro j A M p er A

This shows that M A is functorial with resp ect to morphisms of pairs p er A p er B

ie functors from p erfect complexes over A to p erfect complexes over B which resp ect

the dg structure and preserve acyclicity For example if P is a p erfect complex over

A we have the functor

P p er k p er A

k

which induces a morphism

M P M p er k M p er A

k

and hence a morphism

M P M k M A

If we apply the functors H resp H R Hom k to this morphism we obtain

morphisms

HH k HH A and HC k HC A

mix mix

which map the canonical classes in HH k resp HC k HC k to the Euler

mix

class resp the Chern character of the p erfect complex P

Characteristic Classes for Ringed spaces

Let k b e a eld X a top ological space and A a sheaf of k algebras on X In

this section we consider for each op en subset U of X the lo calization pair of p erfect

complexes on U denoted by p er Aj The mixed complexes M p er Aj asso ciated

U U

with these lo calization pairs are assembled into a sheaf of mixed complexes M er A

In we show that this sheaf is quasiisomorphic to the sheaf M A of mixed

complexes asso ciated with A In this isomorphism is used to construct the trace

morphism

M p er A RX M A

The construction of the characteristic classes of a p erfect complex is then achieved

using the functoriality of the mixed complex M p er A with resp ect to exact functors

b etween lo calization pairs

The main theorem will state that is invertible if X A is a quasicompact

separated scheme

The presheaf of categories of perfect complexes Recall that a strictly

perfect complex is a complex P of Amo dules such that each p oint x X admits an

op en neighbourho o d U such that P j is isomorphic to a b ounded complex of direct

U

summands of nitely generated free Aj mo dules note that such mo dules have no

U

reason to b e pro jective ob jects in the category of Aj mo dules A perfect complex is

U

a complex P of Amo dules such that each p oint x X admits an op en neighbourho o d

U such that P j is quasiisomorphic to a strictly p erfect complex

U

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Bernhard Keller

We denote by p er A the pair formed by the category of p erfect complexes and

its full sub category of acyclic p erfect complexes For each op en U X we de

note by p er Aj the corresp onding pair of categories of p erfect Aj mo dules Via

U U

the restriction functors the assignment U M p erAj b ecomes a presheaf of

U

mixed complexes on X We denote by M er A the corresp onding sheaf of mixed

complexes

For each op en U X we have a canonical functor

pro j AU p er Aj

U

whence morphisms

M AU M pro j AU M p er Aj

U

and a morphism of sheaves

M A M er A

Key Lemma The above morphism is a quasiisomorphism

Remark This is the analog in cyclic homology of lemma of with the

same pro of as P Bressler has kindly informed me

Proof We will show that the morphism induces quasiisomorphisms in the stalks Let

x X Clearly we have an isomorphism



M er A M lim p er Aj

x U

where U runs through the system of op en neighbourho o ds of x We will show that

the canonical functor

lim p er Aj p er A

U x

induces a quasiisomorphism in the mixed complexes For this it is enough to show

that it induces equivalences in the asso ciated triangulated categories by b

Now we have a commutative square

lim p er Aj

p er A

U

x

lim strp er Aj

strp er A

U

x

Here we denote by strp er the pair formed by the category of strictly p erfect

complexes and its sub category of acyclic complexes For an algebra A we have

b

strp er A C pro j A by denition It is easy to see that the two vertical arrows

induce equivalences in the triangulated categories and the b ottom arrow is actually

itself an equivalence of categories Indeed we have the commutative square

lim strp er Aj

strp er A

U

x

lim strp er AU

strp er A

x

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Cyclic homology

Here the right vertical arrow is the identity and the left vertical arrow and the b ottom

arrow are clearly equivalences

The claim follows since the comp osition of the morphism

M A M lim p er Aj

x U

with the quasiisomorphism M lim p er Aj M p er A is the canonical quasi

U x

isomorphism M A M p er A

x x

p

Characteristic classes By denition of M er A we have a morphism of

mixed complexes M p er A X M er A By the key lemma the canon

ical morphism M A M er A is invertible in the mixed derived category Thus

we can dene the trace morphism

M p er A RX M A

by the following commutative diagram

M p er A X M er A

RX M A RX M er A

Now let P b e a p erfect complex It yields a functor

P p er k p er A

k

and hence a morphism in the mixed derived category

M P



M k M p er k M p er A RX M A M X A

If we apply the functor H resp R Hom k to this morphism we obtain morphisms

HH k HH A resp HC k HC k HC A

mix mix

mapping the canonical classes to the Euler class resp ectively to the Chern character

of the p erfect complex P

Remark The trace morphism M p er A M X A is a quasiisomorphism if

X is a p oint by or if X A is a quasicompact separated scheme by b elow

Remark B Tsygan Let P b e a p erfect complex and A H om P P the dg

X

algebra of endomorphisms of P So if P is brant cf A then the ith homology

of A identies with Hom P P i The dg category with one ob ject whose endo

D X

morphism algebra is A naturally embeds into p er A and we thus obtain a morphism

M A M p er A M p er A RX M A

whose comp osition with the canonical map M k M A coincides with the mor

phism constructed ab ove

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Bernhard Keller

Variant with supports Let Z X b e a closed subset Let p erA on X

b e the pair formed by the category of p erfect complexes acyclic o Z and its full

sub category of acyclic complexes For each op en U X denote by p erAj on Z the

U

corresp onding pair of categories of p erfect Aj mo dules Via the restriction functors

U

the assignment U M p erAj on Z b ecomes a presheaf of mixed complexes on

U

X We denote by M er A on Z the corresp onding sheaf of mixed complexes We

claim that M er A on Z is acyclic for x Z Indeed if U X n Z is an op en

x

neighbourho o d of x then by denition the inclusion

p er Aj on Z p er Aj on Z

U U

is the identity so that M p erAj on Z is nullhomotopic It follows that the canon

U

ical morphism M er A on Z M er A uniquely factors through

R M er A M er A

Z

in DMix X Using the quasiisomorphism M A M er A we thus obtain a

canonical morphism M er A on Z R M A making the following diagram

Z

commutative

M er A on Z R M er A M er A

Z

H

H

H

H

H

H

Hj

R M A M A

Z

We now dene the trace morphism M p erA on Z R X M A as the

Z Z

comp osition

M p erA on Z X M er A on Z R X M A

Z

We then have a commutative diagram

M p erA on Z M p er A

R X M A RX M A

Z

This yields a canonical lift of the classes constructed in section to the theories

supp orted in Z The trace morphism is invertible if X and U X n Z are quasi

Z

compact separated schemes by b elow

The main theorem examples proof

This section is devoted to the main theorem Let k b e a eld and X a

quasicompact separated scheme over k The mixed complex asso ciated with X is

dened as M X RX M O The main theorem states that the trace map

X

M p er X M X of is invertible in the mixed derived category

In we dene M p er X and examine its functoriality with resp ect to mor

phisms of schemes following In we state the theorem and as a corollary the

case of quasipro jective varieties As an application we compute in the cyclic

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Cyclic homology

homology of smo oth pro jective varieties admitting a tilting bundle as describ ed in the

introduction

The pro of of the main theorem o ccupies subsections to It pro ceeds by

induction on the number of op en anes needed to cover X The case of ane X is

treated in section The induction step uses a MayerVietoris theorem which

is based on the description of the b er of the morphism of mixed complexes induced

by the lo calization at a quasicompact op en subscheme This description is achieved

in It is based on ThomasonTrobaughs lo calization theorem which we recall in

section in a suitable form and on the lo calization theorem for cyclic homology of

lo calization pairs c which we adapt to our needs in

Definition and functoriality We adapt ideas of ThomasonTrobaugh

Let X b e a quasicompact separated scheme over a eld k We put p er X p er O

X

cf We claim that the assignment X M p er X is a functor of X Indeed

let atp er X b e the pair formed by the category of right b ounded p erfect complexes

with at comp onents and its sub category of acyclic complexes Then the inclusion

atp er X p er X

induces an equivalence in the asso ciated triangulated categories by and

hence an isomorphism

M atp er X M p er X

by b Now if f X Y is a morphism of schemes then f clearly induces a a

functor atp er Y atp er X and hence a morphism M p er Y M p er X Notice

that this morphism is compatible with the map M p er X RX M er X of

section

Now supp ose that X admits an ample family of line bundles Then the inclusion

strp er X p er X

induces an equivalence in the asso ciated triangulated categories and hence

an isomorphism M strp er X M p er X Note that strp er X is simply the category

of b ounded complexes over the category vec X of algebraic vector bundles on X

together with its sub category of acyclic complexes Hence we have the equality

M strp er X M vec X where M vec X denotes the mixed complex asso ciated

with the exact category vec X as dened in In particular if X Sp ec A is ane

we have canonical isomorphisms

  

M A M pro j A M vec X M p er X

The main theorem Let X b e a quasicompact separated scheme over a eld

k The mixed complex asso ciated with X is dened as M X RX M O

X

Note that by denition we have

HC X HC M X HC X HC M X

mix

mix mix

Theorem The trace morphism

M p er X M X

is invertible More generally if Z is a closed subset of X such that U X n Z is

quasicompact then the trace morphism

M p erX on Z R X M O

Z Z X

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is invertible

Corollary Let X be a quasicompact separated scheme over a eld k Then there

is a canonical isomorphism



HC p er X HC X

In particular if X admits an ample line bund le eg if X is a quasiprojective vari

ety there is a canonical isomorphism



der

HC vec X HC X

The corollary was announced in where we wrote HC vec X instead

der

of HC vec X It is immediate from the theorem once we prove that for quasi

compact separated schemes there is an isomorphism



HC X HC X

mix

This will b e done in

The theorem will b e proved in The plan of the pro of is describ ed in the

introduction to this section

The example of varieties with tilting bundles Supp ose that k is an

algebraically closed eld and that X is a smo oth pro jective algebraic variety Sup

p ose moreover that X admits a tilting bund le ie a vector bundle T without higher

selfextensions whose direct summands generate the b ounded derived category of the

category of coherent sheaves on X as a triangulated category Examples of vari

eties satisfying these hypotheses are pro jective spaces Grassmannians and smo oth

quadrics

Proposition The Chern character induces an isomorphism

K X HC k HC X

Z

Here the left hand side is explicitly known since the group K X is free and

admits a basis consisting of the classes of the pairwise nonisomorphic indecomp osable

direct summands of the tilting bundle For example if X is the Grassmannian of k

dimensional subspaces of an ndimensional space the indecomp osables are indexed by

all Young diagrams with at most k rows and at most n k columns Cyclic homology

of pro jective spaces was rst computed by Beckmann using a dierent metho d

The prop osition shows that if X is a smo oth pro jective variety such that

n

H X O for some n then X cannot admit a tilting bundle Indeed

X

n

the group H X O o ccurs as a direct factor of HC X and therefore has to

X n

vanish if the assumptions of the prop osition are satised

Proof Let A b e the endomorphism algebra of the tilting bundle T and r the Jacobson

radical of A We assume without restriction of generality that T is a direct sum of

pairwise nonisomorphic indecomp osable bundles Then Ar is a pro duct of copies

of k since k is algebraically closed We will show that the mixed complex M X is

canonically isomorphic to M Ar For this consider the exact functor

pro jA vecX

A

It induces an equivalence in the b ounded derived categories

b b

D pro jA D vecX

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Indeed we have a commutative square

T

A

b b

D pro jA D vecX

L T

A

b b

D mo d A D cohX

where mo dA denotes the ab elian category of all nitely generated right Amo dules

and cohX the ab elian category of all coherent sheaves on X Since T is a tilting

bundle the b ottom arrow is an equivalence Since X is smo oth pro jective it follows

that A is of nite global dimension Hence the left vertical arrow is an equivalence

Again b ecause X is smo oth pro jective the right vertical arrow is an equivalence

Hence the top arrow is an equivalence So the functor

T p erA p erX

A

induces an equivalence in the asso ciated triangulated categories and hence an isomor

phism



M p erA M p erX



by b Of course it also induces an isomorphism K pro jA K vecX

and the Chern character is compatible with these isomorphisms by its description in

So we are reduced to proving that the Chern character induces an isomorphism



K A HC k HC A

Z

For this let E A b e a semisimple subalgebra such that E identies with the quo

tient Ar The algebra E is a pro duct of copies of k and of course the inclusion E A

induces an isomorphism in K It also induces an isomorphism in HC by

since A is nitedimensional and of nite global dimension These isomorphisms are

clearly compatible with the Chern character and we are reduced to the corresp onding

p

assertion for HC E This is clear since E is a pro duct of copies of k

Proof of the main theorem in the affine case Supp ose that X

Sp ec A Then we know by section that the canonical morphism M A

M p er X is invertible Now WeibelGeller have shown in that the canonical

morphism

M A R X M O

ce X

is invertible where M O is viewed as a complex of sheaves on X and R X

X ce

denotes CartanEilenberg hypercohomology cf section A Moreover Weibel

Geller have shown in that the complex M O has quasicoherent homology

X

By section A it follows that the canonical morphism

RX M O R X M O

X ce X

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is invertible Using the commutative diagram

M A R X M O

ce X

M p er X RX M O

X

we conclude that M p er X RX M O is invertible for ane X

X

ThomasonTrobaughs localization theorem Let X b e a quasicompact

quasiseparated scheme We denote by T p er X the full sub category of the un

b ounded derived category of the category of O mo dules whose ob jects are the

X

p erfect complexes This category identies with the triangulated category asso ciated

with the lo calization pair p er X as dened in Recall that a triangle functor

S T is an equivalence up to factors if it is an equivalence onto a full sub category

whose closure under forming direct summands is all of T A sequence of triangulated

categories

R S T

is exact up to factors if the rst functor is an equivalence up to factors onto the kernel

of the second functor and the induced functor S R T is an equivalence up to

factors

Theorem

a Let U X be a quasicompact open subscheme and let Z X n U Then the

sequence

T p erX on Z T p er X T p er U

is exact up to factors

b Suppose that X V W where V and W are quasicompact open subschemes

and put Z X n W Then the lines of the diagram

T p erX on Z T p er X T p er W



j

T p erV on Z T p er V T p erV W

are exact up to factors and the functor j is an equivalence up to factors

The theorem was proved in section of Note that the rst assertion of part

b follows from a The second assertion of b is a sp ecial case of the main assertion

in take U V Z X n W in lo ccit A new pro of of the theorem is due

to A Neeman

Localization in cyclic homology of DG categories In this section we

adapt the lo calization theorem to our needs Let

F G

A B C

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b e a sequence of small at exact DG categories such that F is fully faithful GF

and the induced sequence of stable categories

B C A

is exact up to factors

Theorem The morphism

M F

ConeM A M B M C

induced by M G is a quasiisomorphism

Proof The pro of consists in extracting the relevant information from Indeed

F

since F is fully faithful we may consider A B as a lo calization pair and since

GF the square

F

A B

G

C

b

of as a morphism of lo calization pairs ie a morphism of the category L

str

By applying the completion functor of lo c cit we obtain a morphism

F

A B

C

of the category L Applying the functor C m to this morphism yields the morphism

M A M B

M G

M C

of DMor Mix by the remarks following prop osition of On the other hand

applying the functor I of to the morphism yields the identity of C

in M and applying M denoted by C in yields the identity of M C in DMix

By the naturality of the isomorphism of functors in a call it we obtain a

commutative square in DMix

ConeM A M B M C

M G

Cone M C M C

So the left vertical arrow of the square is invertible in DMix which is what we had

p

to prove

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Perfect complexes with support and local cohomology Let X b e a

quasicompact quasiseparated scheme U X a quasicompact op en subscheme and

Z X n U Let j U X b e the inclusion

Proposition The sequence

M er X on Z M er X j M er U

embeds into a triangle of DMix X This triangle is canonically isomorphic to the Z

local cohomology triangle associated with M er X In particular there is a canonical

isomorphism



M er X on Z R X M er X

Z

Moreover the canonical morphisms t into a morphism of triangles

M p erX on Z M p er X M p er U M p erX on Z

M er X M er X M er U M er X

Z Z

in the mixed derived category where and are short for RX and R X

Z Z

Proof Let V X b e op en Consider the sequence

M p erV on Z M p er V M p erV U

If we let V vary it b ecomes a sequence of presheaves on X We will show that there

is a sequence of mixed complexes of presheaves

f g

A B C

such that

we have g f in the category of mixed complexes of presheaves

in the derived category of mixed complexes of presheaves the sequence

b ecomes isomorphic to the sequence

for each quasicompact op en subscheme V X the canonical morphism from

the cone over the morphism AV B V to C V induced by g is a quasi

isomorphism

This implies that rstly the sequence of sheaves asso ciated with the sequence

embeds canonically into a triangle

A B C A

where the tilde denotes sheacation and the connecting morphism is constructed as

the comp osition



C ConeA B A

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Cyclic homology

and secondly we have a morphism of triangles

AV B V C V AV

RV A RV B RV C RV A

for each quasicompact op en subscheme V X to prove this last assertion we use

that RV lifts to a derived functor dened on the category of all sequences

0 0

f g

A B C

with g f

To construct the sequence we have to pre sheafy a part of the pro of of

For this let ip er X denote the category of all brant A p erfect complexes

Then the inclusion ip er X p er X induces an equivalence in the asso ciated triangu



lated categories and thus we have an isomorphism M ip er X M p er X in DMix

Note that this even holds if X is an arbitrary ringed space In particular it holds

for each op en subscheme V X instead of X Hence the presheaf V M p er V

is isomorphic in the derived category of presheaves to V M ip er V Similarly for

the other terms of the sequence so that we are reduced to proving the assertion for

the sequence of presheaves whose value at V is

M ip erV on Z M ip er V M ip erU V

For this let I V b e the exact dg category of brant A complexes on V

and let I V b e the category whose ob jects are the exact sequences

p

i

K L M

of I V such that i has split monomorphic comp onents K is acyclic o Z and i

x

is a quasiisomorphism for each x Z Then I V is equivalent to a full exact dg

sub category of the category of ltered ob jects of I V cf example d of

Let I V on Z b e the full sub category of I X whose ob jects are the sequences



K L

and I U V the full sub category whose ob jects are the sequences



M L

Let G I V I V U b e the functor

K L M M M

and F I V on Z I V the inclusion Then the sequence

F G

I V on Z I V I V U

is an exact sequence of the category M of and in particular we have

str

g

GF We take the subsequence of p erfect ob jects Let ip erV on Z b e the full



sub category of I V on Z whose ob jects are the K L with K ip erV on Z

g

let ip erV b e the full sub category of the K L M with M p er V and let

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Bernhard Keller

g

ip erV U b e the full sub category of the L M with M j p erV U

U

Consider the diagram

F G

g g g

ip erV on Z ip erV ip erV U

ip erV on Z ip erV ip erV U

where the three vertical functors are given by



K L K

K L M M

L M M j

U

Its left hand square is commutative up to isomorphism and its right hand square is

commutative up to the homotopy

pj

U

M j Lj

U U

The vertical arrows clearly induce equivalences in the asso ciated triangulated cate

gories By applying the functor M to the diagram and letting V vary we obtain a

commutative diagram in the derived category of presheaves of mixed complexes on

X The vertical arrows b ecome invertible and the top row b ecomes

g g g

M ip erV on Z M ip erV M ip erV U

where V runs through the op en subsets of X This is the sequence of presheaves

A B C announced at the b eginning of the pro of Using theorem a and

p

theorem one sees that it has the required prop erties

MayerVietoris sequences Let X b e a quasicompact quasiseparated

scheme and V W X quasicompact op en subschemes such that X V W

Proposition There is a canonical morphism of triangles in the mixed derived cat

egory

M p er X M p er V M p er W M p erV W

M er X M er V M er W M er V W

where is short for RX

Proof Put Z X n W The rst line of the diagram is deduced from theorem b

using Clearly the two squares app earing in the diagram are commutative

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Cyclic homology

We have to show that the square involving the arrows of degree

M p erV W M p er X

M er V W M er X

is commutative as well By lo ccit the connecting morphism is the comp osition

M p erX on Z M p er X

M p erV W M p erV on Z

Here the vertical morphism is invertible by theorem b and b The

second line of the diagram is the MayerVietoris triangle for hypercohomology So

the connecting morphism of the second line is obtained as the comp osition



M er V W M er V M er X M er X

Z Z

where and are short for RX and R X Now it follows from prop osi

Z Z

tion that the rightmost square of the diagram of the assertion is commutative as

p

well

Proof of theorem Let V V b e op en anes covering X If n

n

S

V theorem holds by section If n we cover X by V V and W

i

in

The intersection V W is then covered by the n sets V V i n These

i

are ane since X is separated So theorem holds for V W and V W by the

induction hypothesis Thus it holds for X V W by prop osition The assertion

for now follows by prop osition

Z

Proof of corollary In cf also C Weibel dened HC X

as the homology of the complex of k mo dules

R X C C O

ce X

where R denotes CartanEilenberg hypercohomology cf section A and

ce

CC O is the sheacation of the classical bicomplex Now WeibelGeller have

X

shown in that the Ho chschild complex C O has quasicoherent homology Thus

X

each column of CC O has quasicoherent homology and hence the sum total com

X

plex of CC O has itself quasicoherent homology Hence by theorem A the

X

ab ove complex is isomorphic to

RX C C O

X

Now as in the case of an algebra cf CC O may also b e viewed as

X

the sum total complex of the bicomplex B C M O asso ciated with the mixed

X

complex of sheaves M O cf Section What remains to b e proved then is

X

that the canonical map

B C RX M O RX B C M O

X X

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Bernhard Keller

is invertible in the derived category of k vector spaces Now indeed more generally

we claim that we have



B C RX M RX B C M

for any mixed complex of sheaves M with quasicoherent homology As the reader

will easily check this is immediate once we know that the functor RX commutes

with countable direct sums when restricted to the category of complexes with quasi

coherent homology This follows from Corollary in It may also b e proved

by the argument of For completeness we include a pro of Let K i I b e a

i

family of complexes with quasicoherent homology It is enough to prove that H X

L

takes K K to the sum of the H X K Now X is of nite cohomological

i i

dimension on the category of quasicoherent mo dules Indeed for an ane X this

follows from Serres theorem I I I and for arbitrary X it is proved by induction

on the size of an ane cover of X here we use that X is quasicompact and separated

It therefore follows from by theorem A b lemma A and Serres theorem I I I



n

that we have an isomorphism H X K H X K and similarly for K

i i

for some xed n cf the pro of of theorem A for the denition of the truncation

n

functor So we may assume that the K and K are uniformly b ounded b elow

i

But then we may compute the H X K using resolutions K F by uniformly

i i i

b ounded b elow complexes of asque sheaves The sum of the F is again b ounded

i

b elow with asque comp onents and is clearly quasiisomorphic to K Now X

commutes with innite sums since X is quasicompact so the claim follows

Appendix A On CartanEilenberg resolutions

We prove that CartanEilenberg hypercohomology coincides with derived func

tor hypercohomology on all unbounded complexes of sheaves with quasicoherent

homology on a quasicompact separated scheme More precisely we prove that in

this situation CartanEilenberg resolutions are actually K injective resolutions in

the sense of

A Terminology Let A b e a Grothendieck category Spaltenstein dened a

complex I over A to b e K injective if in the homotopy category there are no non zero

morphisms from an acyclic complex to I This is the case i each morphism M I

in the derived category is represented by a unique homotopy class of morphisms of

complexes

In A C Weibel prop osed the use of the term brant for K injective

Indeed one can show that a complex is K injective i it is homotopy equivalent to

a complex which is brant for the global closed mo del structure on the category

of complexes in which cobrations are the comp onentwise monomorphisms This

structure is an additive analogue of the global closed mo del structure on the category

of simplicial sheaves on a Grothendieck site The existence of the global structure in

the case of simplicial sheaves was proved by Joyal cf We have not

b een able to nd a published pro of of the fact that the category of complexes over

a Grothendieck category admits the global structure an unpublished pro of is due to

F Morel However the key step may b e found in Prop

Whereas in the homotopy category the notions of brant for the global structure

and K injective b ecome essentially equivalent there is a slight dierence at the level

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of complexes brant ob jects for the global structure are exactly the K injective

complexes with injective comp onents

We will adopt the terminology prop osed by Weibel We call a complex brant i

it is K injective in the sense of Spaltenstein This will not lead to ambiguities since

we will not use the global closed mo del structure

A Sheaves with quasicoherent cohomology Let X b e a scheme and K

a complex of O mo dules unbounded to the right and to the left Let I b e a

X

CartanEilenberg resolution of K ie

a I is a Z Zgraded O mo dule endowed with dierentials d of bidegree

X I

and d of bidegree such that d d

II I II

pq

b I vanishes for q and

c I is endowed with an augmentation K I ie a morphism of dierential

Z Zgraded O mo dules where K is viewed as concentrated on the paxis

X

p

p p p

such that for each p the induced morphisms K I and H K H I are

I

injective resolutions

p p

p p

It follows that for each p the induced morphisms B K B K and Z K Z I are

I I

injective resolutions and that the rows of I are pro ducts of complexes of the form

M or M M

where M is injective

d

Let J Tot I denote the pro duct total complex of I and K J the mor

phism of complexes induced by The morphism is called a total CartanEilenberg

resolution of K The CartanEilenberg hypercohomology of K is the cohomology of

the complex

R X K X J

ce

The morphism is usually not a quasiisomorphism

Theorem a The complex J is brant A

b If K has quasicoherent homology the morphism K J is a quasi

isomorphism Hence CartanEilenberg hypercohomology of K coincides with

derived functor hypercohomology of K in the sense of Spaltenstein

Part a holds more generally whenever K is a complex of ob jects over an ab elian

category having enough injectives and admitting all countable pro ducts This was

proved by C Weibel in A For completeness we include a pro of of a b elow

Part b was proved by C Weibel in lo c cit for the case of complete ab elian

categories with enough injectives and exact products for example mo dule categories

The case we consider here is implicit in Nevertheless we thought it useful

to include the explicit statement and a complete pro of

In preparation of the pro of let us recall the notion of a homotopy limit cf

for example If T is a triangulated category admitting all countable pro ducts and

f

p

X X p N X

p p

is a sequence in T its homotopy limit holim X is dened by the Milnor triangle

p

Y Y

X holim X holim X X

q p p p

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Bernhard Keller

where the morphism has the comp onents

Y

f

q

can

X X X X

q q q p

p

Note that the homotopy limit is unique only up to non unique isomorphism We will

encounter the following situation Consider a sequence of complexes

f

p

K K K

p p

over an additive category admitting all countable pro ducts such that the f are com

p

n n

X p onentwise split epi or more generally for each n and p the morphism X

p

pk

is split epi for some k Then we have a comp onentwise split short exact sequence

of complexes

Y Y

lim K K K

p p q

p q

and hence the inverse limit lim K is then isomorphic to holim K in the homotopy

p p

category

Proof of the theorem a Note that the bicomplex I is the inverse limit of its quotient

q

complexes I obtained by killing all rows of index greater than q Let J b e the

q

q

d

pro duct total complex of I Then the sequence of the J has inverse limit Tot I

q

and its structure maps are split epi in each comp onent Hence I is isomorphic to

the homotopy limit of the sequence of the J Since the class of brant complexes is

q

stable under extensions and pro ducts it is stable under homotopy limits Therefore

it is enough to show that the J are brant Clearly the J are iterated extensions of

q q

rows of I suitably shifted So it is enough to show that the rows of I are brant

But each row of I is homotopy equivalent to a complex with vanishing dierential

and injective comp onents Such a complex is the pro duct of its comp onents placed

in their resp ective degrees and is thus brant

p

b For p Z dene K to b e the quotient complex of K given by

p p p p

K B K K K

p

and K to b e the sub complex of K given by

p p p

K K B K

p p

Dene J and J by applying the resp ective functor to each row of J Then

p p

the morphism K J is a CartanEilenberg resolution for each p Z Since

p p p

d

K is left b ounded it follows that the induced morphism K Tot J is a

quasiisomorphism for each p Z Now x n Z and consider the diagram

n

n p

H K

H K

n n p

d d

H Tot J H Tot J

For p n the top morphism is invertible It now suces to show that for p the

p n

d

Tot J b ottom morphism is invertible Equivalently it is enough to show that H

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p n

d

Tot J vanishes for p For this let x X We have to show that H

x

vanishes Since taking the stalk is an exact functor this reduces to showing that

p

d

the complex Tot J is acyclic in degree n For this it is enough to show that

x

p p

d

Tot J U is acyclic in degree n for each ane neighbourho o d of x Now J is

p

a CartanEilenberg resolution of K Therefore if we apply prop osition A b elow

p

d

to the functor F U we see that Tot J U is acyclic in all degrees n p

i p

Indeed we have R F H K for all p and all i by Serres theorem I I I

p

since H K is quasicoherent

p

A Unbounded complexes with uniformly bounded cohomology Let A

b e an ab elian category with enough injectives which admits all countable pro ducts

and let F A Ab b e an additive functor commuting with all countable pro ducts

Let K b e a complex over A and let K J a CartanEilenberg resolution

p

Supp ose that K for all p and that there is an integer n with

i p

R F H K

for all i n and all p Z

p

d

Lemma We have H F Tot J for al l p n

Note that this assertion is clear if K is homologically left b ounded The p oint

is that it remains true without this hypothesis

p p

Proof Dene K and J as in the pro of of prop osition A The canonical

p p

morphisms J J are split epi in each bidegree and J identies with the

p p

d d

inverse limit of the J Hence we have Tot J lim Tot J and the morphisms

p p

d d

Tot J Tot J

are comp onentwise split epi Since F commutes with countable pro ducts we therefore

p

d d

have F Tot J lim F Tot J By lemma A b elow it is therefore enough to show

i

d

that the groups H F Tot L vanish for all i n and all p where L is the kernel

p p

p p

of the canonical morphism J J Now L is in fact a CartanEilenberg

p

p p

resolution of the kernel of the morphism K K which is isomorphic to

the complex

p p p

K B K Z K

p

d

This complex is quasiisomorphic to H K placed in degree p So Tot L is homotopy

p

p

equivalent to an injective resolution of H K shifted by p degrees Hence

i i p ip p

d

F Tot L H RF H K p R F H K H

p

p

By assumption this vanishes for i p n

A A MittagLeffler lemma Let n b e an integer and let

p+1

0

K p N K K K

p p

b e an inverse system of complexes of ab elian groups such that the are surjective in

p

i

is the kernel of for all i n and all p where K each comp onent and H K

p

p p

i

Lemma We have H lim K for al l i n

p

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Bernhard Keller

i

Proof By induction we nd that H K for all i n Now we have exact

p

sequences

i i i

Z K K Z K

p p

p



i i i i

for all i n Since B K Z K the maps Z K Z K are surjective for

p p p p

n n n

i n The fact that H K implies that the maps Z K Z K are

p p

p

surjective as well By the MittagLeer lemma the sequence

III

i i i

lim Z K lim K lim Z K

p p

p



i i i

is still exact for i n Since lim Z K Z lim K this means that H lim K

p p p

p

for i n

Appendix B A comparison of derived categories

B BoekstedtNeemans theorem Let X b e a quasicompact separated sche

me D Qcoh X the derived category of the category Qcoh X of quasicoherent sheaves

on X D X the derived category of all sheaves of O mo dules on X and D X its full

X q c

sub category whose ob jects are the complexes with quasicoherent homology

As an application of theorem A we give a partially new pro of of the following

result of Bo ekstedtNeeman We refer to Prop for yet another pro of

Theorem The canonical functor D Qcoh X D X is an equivalence of

q c

categories

The pro of pro ceeds by induction on the size of an ane cover of X The crucial

step is the case where X is ane Our pro of for this case is new For completeness

we have included the full induction argument

Proof In a rst step supp ose that X is ane X Sp ec A We identify Qcoh X with

Mo d A and then have to show that the sheacation functor F D Mo d A D X

induces an equivalence D Mo d A D X Clearly the image of A viewed as a

q c

complex of Amo dules concentrated in degree is O By the lemma b elow it

X

suces therefore to show that



a We have A Hom O O and Hom O O n for each n

D X X X D X X X

b The ob ject O is compact in D X ie the asso ciated functor

X q c

Hom O

D X X

q c

commutes with innite direct sums

c An ob ject K D X vanishes if Hom O K n vanishes for all n Z

q c D X X

The three assertions a b and c all follow easily from the fact that we have an

isomorphism



Hom O X H

D X X

q c

which we will now prove Indeed let K D X By denition we have

q c

Hom O K H RX K

D X X

Now we have morphisms

H RX K H RX K H RX H K X H K

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The morphism is invertible b ecause RX is a right derived functor The mor

phism is invertible by theorem A b lemma A and Serres theorem I I I

Now supp ose that X is the union of n op en ane sets U U By induction

n

on n and the ane case we may assume that the claim is proved for U U and

S

U Let j U X and j V X b e the inclusions Let Y X n U V

i

in

and let i Y X b e the inclusion For any ob ject K D X we have a triangle

q c

R K K j j K R K

Y Y

Here the second morphism is the adjunction morphism and R K is dened up to

Y

unique isomorphism by the triangle The ob ject j K is a complex of sheaves on U

n n

and H j K j H K is quasicoherent So j K is in the faithful image of D Qcoh U

Because X is separated j preserves quasicoherence cf So the triangle

lies in D X The subset Y X is a closed subset of V and i j i where i is the

q c

inclusion of Y into V This implies that R K j R K The ab ove triangle

Y Y V

thus shows that D X is generated by the j K and the j K where K b elongs to

q c

D Qcoh U and K to D Qcoh V It remains to b e checked that morphisms b etween

j K and j K in D Mo d O are in bijection with those in D Qcoh X Indeed we

X

have

Hom j K j K Hom j j K K

D X D V

By the induction hypothesis the latter group identies with

Hom j j K K Hom j K j K

D Qcoh V D Qcoh X

The same argument applies to morphisms from j K to j K This ends the pro of

p

B Derived categories of modules Let A b e a ring and T a triangulated

category admitting all innite direct sums Supp ose that F D Mo d A T is a

triangle functor commuting with all direct sums For the convenience of the reader

we include a pro of of the following more and more wellknown

Lemma The functor F is an equivalence if and only if



a We have A Hom F A F A and Hom F A F An for al l n

T T

b The object FA is compact in T ie Hom F A commutes with innite direct

T

sums

c An object X of T vanishes i Hom F A X n for al l n Z

T

Proof Let S T b e the smallest triangulated sub category of T containing FA and

stable under forming innite direct sums Then since FA is compact the inclusion

S T admits a right adjoint R by Browns representability theorem cf also

Now if X T and RX X X RX is a triangle over

the adjunction morphism then Hom F A X n vanishes for all n Z by the long

T

exact sequence asso ciated with the triangle So X vanishes by assumption c and S

coincides with T So FA is a compact generator for T Now the claim follows from

p

Documenta Mathematica

Bernhard Keller

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Comment Math Helv

Bernhard Keller

UFR de Mathematiques

Universite Paris

Institut Mathematique de Jussieu

UMR du CNRS

Case

place Jussieu

Paris Cedex

France

kellermathjussieufr

wwwmathjussieufr keller

Documenta Mathematica

Documenta Mathematica