The Classification of Triangulated Subcategories 3

Home , Homotopy category, Triangulated category

Compositio Mathematica 105: 1±27, 1997. 1

c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

The classi®cation of triangulated subcategories

R. W. THOMASON ? CNRS URA212, U.F.R. de Mathematiques,Â UniversiteÂ Paris VII, 75251 Paris CEDEX 05, France email: thomason@@frmap711.mathp7.jussieu.fr

Received 2 August 1994; accepted in ®nal form 30 May 1995

Key words: triangulated category, Grothendieck group, Hopkins±Neeman classi®cation Mathematics Subject Classi®cations (1991): 18E30, 18F30.

1. Introduction The ®rst main result of this paper is a bijective correspondence between the strictly full triangulated subcategories dense in a given triangulated category and the subgroups of its Grothendieck group (Thm. 2.1). Since every strictly full triangulated subcategory is dense in a uniquely determined thick triangulated subcategory, this result re®nes any known classi®cation of thick subcategories to a classi®cation of all strictly full triangulated ones. For example, one can thus re®ne the famous classi®cation of the thick subcategories of the ®nite stable homotopy category given by the work of Devinatz±Hopkins±Smith ([Ho], [DHS], [HS] Thm. 7, [Ra] 3.4.3), which is responsible for most of the recent advances in stable homotopy theory.

One can likewise re®ne the analogous classi®cation given by Hopkins and Neeman

(R ) ([Ho] Sect. 4, [Ne] 1.5) of the thick subcategories of D parf, the chain homotopy

category of bounded complexes of ®nitely generated projective R -modules, where

R is a commutative noetherian ring. The second main result is a generalization of this classi®cation result of Hop-

kins and Neeman to schemes, and in particular to non-noetherian rings. Let X

be a quasi-compact and quasi-separated scheme, e.g. any commutative ring or

(X )

algebraic variety. Denote by D parf the derived category of perfect complexes, O

the homotopy category of those complexes of sheaves of X -modules which are O

locally quasi-isomorphic to a bounded complex of free X -modules of ®nite type.

D (X )

I say a thick triangulated subcategory A parf is a -subcategory if for

D (X ) A A E A

each object E in parf and each in , the derived tensor product

A X L is also in A. This is a mild condition on .If has an ample line bundle

(e.g. for X a classical, hence quasi-projective variety), it suf®ces by 3.11.1 below

A A X = (R ) R

to check that L .If Spec for a commutative ring, all

D (R ) thick subcategories A parf are -subcategories. The second main result

? Deceased October 1995.

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*INTERPRINT* PREPROOF CRCs Corr M/c 2 Pips: 87450 MATHKAP comp3750.tex; 8/05/1997; 8:14; v.5; p.1 2 R. W. THOMASON

(Thm. 3.15) gives for X a quasi-compact and quasi-separated scheme a bijective

D (X )

correspondence between the thick triangulated -subcategories of parf and

Y X Y X X Y

the subspaces which are unions of closed subspaces with

quasi-compact. To such a Y corresponds the thick -subcategory of those perfect

Y X complexes acyclic at each point of X .For noetherian, all subspaces are

quasi-compact, and the remaining condition that Y be a union of subspaces closed Y in X is usually expressed by saying that ` is closed under specialization'. For a detailed comparison with the previous work of Hopkins and Neeman see 3.17 below.

Re®ning the second main result by the ®rst, I obtain (Thm. 4.1) a classi®cation

A D (X )

of all strictly full triangulated -subcategories of parf : they correspond

Y; H ) Y X H K (X

bijectively to data ( where is a subspace as above and 0 on

) K (X )

Y is a 0 -submodule of the Grothendieck group of perfect complexes acyclic

A Y off Y . Thus is determined by a condition on the supports and a condition

H on the multiplicities. This classi®cation has been my personal motivation for developing the results of this paper. I seek to de®ne a good intersection ring of

`algebraic cycles' on schemes X where the classical construction of the Chow ring fails, for example on singular algebraic varieties or on regular schemes ¯at

and of ®nite type over Z. Inspired by the superiority of Cartier divisors over Weil divisors and by recent progress in local intersection theory, I believe the good

notion of `algebraic n-cycle' is that of those perfect complexes in some triangulated

D (X )

subcategory A parf which remains to be de®ned. Technological secrets

about `moving lemmas' demand that A should be a -subcategory, and show it cannot be thick in general. The classi®cation has proved to be very helpful here in clarifying the issues to be resolved. Other results presented here worth mentioning are the Tensor Nilpotence Theo- rem (Thms. 3.6 and 3.8), generalizing to schemes a result of Hopkins and Neeman for noetherian rings ([Ho] Thm. 10, [Ne] 1.1) analogous to the Nilpotence Theorem of Devinatz±Hopkins±Smith ([DHS] Thm. 1) in stable homotopy, and a useful but little-known necessary and suf®cient condition for the equality of two classes in the Grothendieck group of a triangulated category (Lemma 2.4), due to Landsburg inspired by Heller. I have tried to make each theorem, proposition, and lemma resistant to misinterpretation on being `zapped' out of its context by endlessly restating, or at least referencing, all hypotheses and de®nitions explicitly. However, I have followed a customary global convention that `ring' always means `commutative ring'.

2. `Rappels' For the convenience of the reader and to eliminate ambiguities, I will brie¯y recall some basic de®nitions and results of the theory of triangulated categories. Useful general references are the classic treatments [Ha] I and II, and [Ve], as well as

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[SGA4] XVII Section 1.2, [SGA5] VIII, [SGA6], and the more recent [BBD]

Section 1, [Ri], [Sp], and [BN]. T 1.1. Let T be a triangulated category. A full triangulated subcategory of is a full

subcategory A with the structure of a triangulated category such that the inclusion A!T functor i: is a triangulated (a.k.a. `exact') functor, i.e. such that it preserves distinguished (`exact') triangles and commutes with the suspension (`translation') endofunctors. Since the axioms of a triangulated category imply that any morphism is an edge of an exact triangle which is unique up to isomorphism of the opposite

vertex ([Ha] I Sect. 1 TR1, TR3, Prop. 1.1.c, or [Ve] I Sect. 1 no. 1), and since the

A T

functor i is fully faithful so two triangles in are isomorphic in if and only if A

theyareisomorphicin A, a triangle in is exact if and only if its image is exact A in T . Thus the triangulated category structure of is uniquely determined by that

on T .

Thus the de®nition above is equivalent to: a full triangulated subcategory of T

T T

is a full non-empty subcategory A of such that for every exact triangle of of A which two vertices are in A, then the third vertex is isomorphic to an object of

(c.f. [Ve] I Sect. 1 no. 2±3). T

1.2. A strictly full triangulated subcategory A of is a full triangulated subcate- T gory such that A contains every object of which is isomorphic to an object of

A. T

1.3. A thick (a.k.a epaisseÂ ) triangulated subcategory A of is a strictly full tri- A angulated subcategory such that every direct summand in T of an object of is

itself an object of A. Rickard ([Ri] Prop. 1.3) showed that this de®nition is equiv-

alent to the somewhat more complicated classic de®nition given by Verdier ([Ve] T I Sect. 2 1±1). Using his de®nition Verdier showed that a full subcategory A of

is a thick triangulated subcategory if and only if there exists a triangulated category

0 0

T !T A

T and a triangulated functor such that is the full subcategory of those 0

objects whose image in T is isomorphic to 0. (c.f. [Ve] I Sect. 2 nos. 1, 2, 3).

D D

1.4. A full triangulated subcategory A of a triangulated category is dense in A if each object of D is a direct summand of an object isomorphic to an object in .

REMARK 1.5. If A is a full triangulated subcategory of a triangulated category

T A

T , the intersection of all the thick triangulated subcategories of containing D

is a thick triangulated subcategory D .This is the smallest thick triangulated A

subcategory of T which contains . On the other hand, one checks easily that

e T

the strictly full subcategory A of all objects of which are summands of objects

T DA

isomorphic to objects in A is a thick triangulated subcategory of . Thus .

e D

But by Rickard's de®nition of thick one has ADsince is thick and contains

D = A A. Thus .

comp3750.tex; 8/05/1997; 8:14; v.5; p.3

4 R. W. THOMASON

D T D

Hence A is dense in a thick triangulated subcategory of if and only if is A the smallest thick triangulated subcategory of T which contains .

1.6. Let T be a triangulated category, which we suppose is essentially small (1.7)

and so replace with an equivalent triangulated category which has a set of objects.

(T )

The Grothendieck group K0 is the quotient group of the free abelian group

[B ] =

on the set of isomorphism classes of objects of T by the Euler relations:

T A]+[C]

[ whenever there is an exact triangle in

A B

( : : )

@ 1 6 1

@ C

See [SGA6] IV Section 1, [SGA5] VIII Section 2. The Grothendieck group

has the universal mapping property that any function from the set of isomorphism G

classes of objects of T to an abelian group such that the Euler relations hold

K (T ) ! G K ()

in G factors through a unique homomorphism 0 . 0 is covariant

A]+[B] = [AB]

for triangulated functors. One has [ from the exact triangle

! A B ! B [A]+[ ] = [A] [ ] =

A .Then 0, which implies 0 0in

A A T

the Grothendieck group. Note that if is the suspension of in ,sothere

! ! A [A]+[A]=[ ]=

exists an exact triangle A 0 , one has 00, so

A]=[A] K (T )

[ . From all this it follows that every element of 0 is of the form

C ] C T

[ for some object of the triangulated category . The analogous statement for

C ] the Grothendieck group of an abelian or exact category would not be true; the [ 's only generate the group in these cases.

1.7. A category T is essentially small if it is equivalent to a small category, i.e. if

there exists a set of objects of T (as opposed to a class of objects in the sense of

Godel±BernaysÈ set theory) such that every object of T is isomorphic to an object in this set.

Note that if T is an essentially small triangulated category then any full subcategory, any localization, and in particular any Verdier quotient by a thick triangulated subcategory ([Ve] I Sect. 2 no. 3) is essentially small. The stable homotopy cate-

gory of ®nite CW spectra is essentially small. For any quasi-compact and quasi-

separated scheme X (e.g. for any noetherian scheme) the triangulated category of

(X )

perfect complexes (3.1) on X; D parf , is essentially small ([TT] Appendix F).

D (A) If A is an essentially small abelian category then the derived category is

also essentially small. However, neither the category of all abelian groups, nor its

(Z ) derived category D -mod is essentially small: there are too many non-isomorphic abelian groups. Indeed these exist in every cardinality.

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3. Classi®cation of dense strictly full triangulated subcategories

As in 1.5, each strictly full triangulated subcategory A of a triangulated category

D T

T is dense in a uniquely determined thick subcategory of . Thus to classify

T D T

all such A in it suf®ces to classify: (1) the thick subcategories in ,and(2) D the strictly full dense triangulated subcategories A of . The second part of the classi®cation is given by:

THEOREM 2.1 Let D be an essentially small (1.7) triangulated category. Then

there is a one-to-one correspondence between the strictly full dense (1.2, 1.4)

D H

triangulated subcategories A in and the subgroups of the Grothendieck group

(D )

K0 .

K (A) K (D )

To A corresponds the subgroup which is the image of 0 in 0 .ToH

A A D

corresponds the full subcategory H whose objects are those in such that

A] 2 H K (D ) [ 0 .

Proof. One checks that these formulas de®ne functions between the set of strictly

(A)

full dense triangulated subcategories and the set of subgroups: Trivially im K0

K (D ) K (D ) A

is a subgroup of 0 . The Euler relation in 0 gives readily that H is

D A D

a strictly full triangulated subcategory of .And H is dense in since for all

D 2D D D 2A [D D ]=[D]+[D]=[D][D]= 2H

one has H as 0.

It remains to be checked that the two functions are inverse to each other. For H

K (A ) H H K (D )

a subgroup, clearly im 0 H . But as in 1.6, any element of 0

[D ] D 2 D D 2 A [D ] 2 H

is of the form for some ,andthen H since . Thus

H K (A )

im 0 H , and the composition of the two functions gives the identity on the set of subgroups. To see that the reverse composition gives the identity on the set of

subcategories, completing the proof of the theorem, one must check that for each

2D D 2A [D ] 2 K (A) K (D) D one has if and only if im 0 0. But this is given by the following lemma.

LEMMA 2.2 Let A be a strictly full dense triangulated subcategory of the essen-

D D

tially small triangulated category D . Then for any object of , one has that

2A [D ]= K (D )= K (A) D if and only if 0in 0 im 0 .

Proof. Passing to an equivalent triangulated category, I may assume that D

has a set of objects. Consider the relation on the set of isomorphism classes of

0 0

D D A A A

objects of D de®ned by iff there exist and in such that there is

0 0

D A D A

an isomorphism = . One checks easily that the relation is an

equivalence relation. Denote by G the quotient by of the set of isomorphism

D i G D D

classes of objects. Denote by h the class in of the object of .

2 A hD i h i G D 2 A hD i h i

I claim D iff 0 in . For clearly implies 0 .

hD i h i A; A 2 A D A

Conversely if 0 ,thereare and an isomorphism =

0 0

A = A D A 2A

0 .Then , and as two of the three vertices of the exact triangle

! D A ! D A A

A are in the strictly full triangulated subcategory ,atthe 2A third vertex one has also D . This proves the claim.

comp3750.tex; 8/05/1997; 8:14; v.5; p.5 6 R. W. THOMASON

To complete the proof of the lemma, it remains to show that the set G is bijective

(D )= K (A) hD i$[D] with the group K0 im 0 via .

But G has a structure of an abelian monoid, with sum induced by the operation

h i G

of direct sum in D . 0 is the zero. In fact, has inverses and so is an abelian

D i2G A D D 2D

group. For given any element h ,as is dense in there is a such

0 0 0

D 2A hD i + hD i = hD D ih i=

that D .Then 00.

D G

Further the Euler relation holds in G, so that the surjection Obj induces

(D ) G [D ] hD i A ! B !

a surjection of abelian groups K0 sending to .Forlet

0 0

A D A C D

C be an exact triangle in . As above, there are objects , in such that

0 0 0 0

A C C A hA A i = = hC C i

A , are in so 0 . Taking the direct sum of the

0 0 0

! B ! C A A ! A ! A

exact triangle A with the exact triangles 0

0 0 0 0 0

C ! C A A ! B A C !

and 0 ! 0 gives an exact triangle

0 0 0 0

C (A A ) A A C C

C . The vertices and are in the strictly full

0 0

B A C

triangulated subcategory A, and hence so is the third vertex . Thus in

0 0 0 0

= hB A C i = hB i + hA i + hC i = hB ihAihCi G one has 0 , proving

the Euler relation holds there.

(D ) G

Thus one has a surjective homomorphism K0 . Since each element of

(D ) [D ] D 2 D

K0 is of the form for some by 1.6, the kernel of this surjection

D ] hD i D 2 A

consists of all [ such that 0, i.e. such that .Sothekernelis

K (A) G K (D )= K (A) 2 im 0 and = 0 im 0 .

COROLLARY 2.3 Let A be a full dense triangulated subcategory of the essentially

K ()

small triangulated category D . Then the homomorphism induced on 0 by the

(A) K (D )

inclusion of categories is a monomorphism of groups K0 0 .

e D

Proof. Let A be the strictly full triangulated subcategory of whose objects

A A A

are those objects of D isomorphic to objects in . The inclusion is an

K (A)

equivalence of triangulated categories and so induces an isomorphism 0 =

e e

(A) A A A K0 . Hence I may replace by and so may assume is a strictly full dense

triangulated subcategory of D .

= (K (A) ! K (D )) N

Let N ker 0 0 . By Theorem 2.1 the subgroups 0 and

(A) Z N

of K0 correspond to strictly full dense triangulated subcategories and

Z N

of A.Butthen and are also strictly full dense triangulated subcategories of

K (Z ) = = N = K (N ) K (D )

D .Butim 0 0 im im 0 in 0 . Then by Theorem 2.1,

= N D = K (Z )=K =(N)N 2 Z , whence 0 0 0.

The reader may ®nd the indirectness of the proof of this useful corollary psycho- logically uncomfortable. If so, rather than dosing himself with a benzodiazepine,

he may ®nd relief in deducing Corollary 2.3 from the following criterion for equal-

(D ) ity of two classes in K0 , whose proof is very similar to that of Lemma 2.2.

This criterion is due to Landsburg ([La]), inspired by an analog for K0 of exact categories due to Heller ([He] 2.1).

LEMMA 2.4. (Landsburg) Let D be an essentially small triangulated category,

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THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 7

0 0

D D [D ]=[D] K (D )

and let D , be two objects of .Then in 0 if and only if there 2D

are objects A; B ; C and two exact triangles

- -

A B D A B D

J] J]

J J

0 0

(2.4.1)

J J

C C

Proof. First I replace D by an equivalent triangulated category with a set of

D D

objects. Now let be the relation on the set of objects de®ned by if there exist A; B ; C and two exact triangles as in (2.4.1). The relation is an

equivalence relation: symmetry and re¯exivity are clear; and as for transitivity,

0 1

D A ! B D ! C ;

note that if D because of exact triangles 1 1 1 and

0 0

0 00

1 0 1

! B D ! C ; D D

A1 1 1 , while because of the exact triangles

0 0

00 00

2 0 2 2

! B D ! C ; A ! B D ! C ; D D

A2 2 2 and 2 2 2 ,then

1 2 1 2

- -

A B D B D C C ;

because of the exact triangles A1 2 1 2 1 2

0 0 0 0

1 2 0 1 2

- -

A B D B D C C ;

and A1 2 1 2 1 2 , conjugated by the

0 0 0 00

(B B D ) D B D B D (B B D ) D =

isomorphisms 1 2 = 1 2 and 1 2

0 00

D B D G D

B1 2 .Let be the quotient of the set of objects of by the

G K (D ) hD i$[D] equivalence relation . It remains to show that = 0 via .

G has a structure of an abelian monoid, the sum being induced by direct sum

G hD i G

in D . In fact, is an abelian group. For given an element of ,ithas

D i hD D i h i =

an inverse h since 0 0 because of the exact triangles

! D D ! D D ! ! D

D and 0 . The Euler relation holds in

A ! D ! C

G.Forif is an exact triangle, considering it alongside the exact

! A C ! C hD ihACi=hAi+hCi triangle A gives .

Thus by the universal property of the Grothendieck group one has a homo-

(D ) G [D ] hD i morphism, clearly surjective, K0 sending to . This surjection is

injective, and so is an isomorphism of abelian groups. For given two elements in

0 0

(D ) [D ] [D ] D; D D

K0 , as in 1.6 they have the form , for objects of .Iftheygoto

D D

thesameelementofG,then and there exist two exact triangles (2.4.1).

(D ) [B D ] = [A]+[C] = [B D]

Then in K0 the Euler relations give ,so

0 0

B ]+[D]=[B]+[D ] [D ]=[D] 2 [ ,and in this group.

EXAMPLE 2.5 The proof of Lemma 2.2, the backbone of Theorem 2.1, was

already implicit in the proof ([TT] 5.5.4) of a key special case of the K0-criterion

of Thomason±Trobaugh for the extension of perfect complexes (3.1) on an open X subscheme U to perfect complexes on the ambient scheme .Asanexampleand since it will be needed in Section 3, I state this criterion:

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EXTENSION LEMMA. Let X be a quasi-compact and quasi-separated scheme,

X X Y U X

Y a closed subspace such that is quasi-compact, and a

(X Y )

quasi-compact open subscheme. Denote by K0 on the Grothendieck group

Y E

of the triangulated category of perfect complexes on X acyclic off .Let be a

U \ Y X

perfect complex on U acyclic off . Then there exists on a perfect complex

Y F jU E

F acyclic off and such that the restriction is quasi-isomorphic to ,if

E ] K (U U \ Y ) K (X Y ) and only if the class [ in 0 on is in the image of 0 on .

Proof. This statement is [TT] 5.2.2, whose proof is spread out through [TT] Sec-

= X X tion 5. In essence, one reduces to the case Y and where has an ample family

of line bundles by an inductive argument. In this case, a Koszul complex trick for

U X extending to X morphisms de®ned on between perfect complexes on shows

that the perfect complexes on U which extend are the objects of a strictly full triangulated subcategory. Trobaugh's revelation ([TT] 5.5.1) is that this subcategory is dense. Now Lemma 2.2 ®nishes the proof.

4. Classi®cation of thick subcategories of perfect complexes

3.1. `Rappels':Let X be a quasi-compact and quasi-separated scheme. Recall that

any classical algebraic variety, more generally any noetherian scheme, and the af®ne

R ) scheme Spec( for any commutative ring is quasi-compact and quasi-separated

([EGA] I).

X O

A strict perfect complex on is a bounded complex of locally free X -modules

of ®nite type ([SGA6] I 2.1, or [TT] 2.2.2). A perfect complex on X is a complex

E O X U

of sheaves of X -modules such that there is an open cover of by 's such

jU U that E is quasi-isomorphic to a strict perfect complex on ([SGA6] I 4.7, or

[TT] 2.2.10).

(X )

One denotes by D the derived category of the abelian category of sheaves

O D (X )

of X -modules, and by parf its strictly full subcategory whose objects are

(X ) D (X ) the perfect complexes on X: D parf is a thick triangulated subcategory of

by ([SGA6] I 4.9, 4.10, 4.17 or [TT] 2.2.13). It is contained in the thick subcate-

D (X )

gory qc of complexes cohomologically bounded below with quasi-coherent

cohomology sheaves.

(X )

The objects of D parf are characterized by the following `®nite presentation'

E D (X )

condition ([ThLG] Prop. 1.1, c.f. [TT] 2.4.1): an object of qc is a perfect

(E ) D (X )

(X )

complex iff the functor MorD takes all direct sums in ,even those with in®nitely many factors, to direct sums of abelian groups. Note there

is a similar characterization of the homotopy ®nite CW spectra in the full stable homotopy category (e.g., [ThSH] 2.5, with the dual to 2.6). This fact is part of the parallel between the classi®cation Theorem 3.15 below and the classi®cation theorem [HS] Theorem 7 in the ®nite stable homotopy category.

If the scheme X has an ample family of line bundles ([SGA6] II 2.2.4 or

(R )

[TT] 2.1.1); for example, if X is an af®ne scheme Spec , or is quasi-projective

R ) over a Spec( , or is a separated regular noetherian scheme, ([TT] 2.1.2), then

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any perfect complex on X is globally quasi-isomorphic to a strict perfect complex

( )(R )) ([SGA6] II 2.2.8, or [TT] 2.3.1). In particular D Spec parf is equivalent to the triangulated category obtained from the category of bounded chain complexes of

®nitely generated projective R -modules by inverting the quasi-isomorphisms. This

(R ) is the category abusively denoted D in [Ne], in ¯agrant incompatibility with the standard use of this symbol as in [Ha], [Ve], and all the works of Grothendieck.

I refer to [Ha] II, [SGA6] I, and [Sp] for the standard notations and results on L

operations like the total derived tensor product and the total derived inverse

image Lf . Note that generally in [Ha] and [SGA6], and were de®ned

(X )

only on D , i.e. on cohomologically bounded above complexes, but they have

(X )

been extended to all D by Spaltenstein ([Sp] 6.5, 6.7). See also [BN]. I will

Lf f X

sometimes abbreviate and simply as and , leaving the understood

O X

and suppressing the `L' when by the context these clearly refer to functors between derived categories rather than the inducing functors on categories of complexes.

Finally I note that I usually follow the customs of the tribe of algebraic geome-

n 1

E ! E

ters, for whom the differentials in a complex increase degree, @ : ,and @

who call ker @=im cohomology. For topologists, differentials decrease degree @

and ker @=im is homology. To translate between the languages of these antipodal

E = E n

peoples one reindexes the algebraic geometer's complex by setting ,so

H (E )=H (E:) E n

reindexing . For example, I say a complex is cohomologi-

H (E ) n cally bounded above if = 0for 0. Among the topologists one would

say this complex E: is homologically bounded below.

X E O

DEFINITION 3.2 Let be a scheme and a complex of sheaves of X -modules.

(E ) X

The cohomological support of E is the subspace Supph of those points

x 2 X O (E )

at which the stalk complex of X;x -modules is not acyclic.

E ) H (E )

Thus Supph( Supp is the union of the supports in the classic

n2Z

sense ([EGA] OI 3.1.5) of the cohomology sheaves of .

E LEMMA 3.3 Let X be a quasi-compact and quasi-separated scheme. Let be a

perfect complex on X.

x 2 X E O

(a) For any , is an acyclic complex of X;x -modules if and only if

k (x) k (x)

E is an acyclic complex of -modules.

f Y ! X (b) If Y is a quasi-compact and quasi-separated scheme and : a

morphism of schemes, then

Lf E )=f ( (E ))

Supph( Supph

E ) X X (E ) (c) Supph( is closed in , and Supph is quasi-compact. Proof. (a) Consider the strongly converging KunnethÈ spectral sequence (e.g.

[EGA] III 6.3.2)

k (x)): E = (H (E ); k (x)) =) H (E

p+q

Tor q

p;q x O

X X;x

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10 R. W. THOMASON

E H (E )= E k (x)

This shows at once that if is acyclic, i.e. if 0, then is

x x

acyclic. Conversely, suppose E is not acyclic. As the perfect is cohomologically

N H (E ) 6=

bounded there is a least such that N 0. Then in the spectral sequence

2 x

= q

E 0for as well as for 0. Then for the spectral sequence

p;q

p = q = N H (E ) k (x)

= O

gives in the corner 0, an isomorphism N

X;x

H (E k (x)) E H (E ) = q < N H (E )

q N

N .As is perfect and 0for , is

x x x

X O

a ®nitely generated X;x -module ([SGA6] I 2.10b, 5.8.1, or [TT] 2.2.3, 2.2.12).

H (E ) 6= H (E ) k (x) 6=

q O

Then by Nakayama's lemma N 0 implies 0, so

x x

X;x

k (x)

E is not acyclic. This proves (a).

2 (Lf E ) (Lf E ) k (y )

To prove (b), note by (a) that y Supph iff is

L L

x = f (y ) 2 X (Lf E ) k (y ) (E

not acyclic. But for one has =

O O

Y X

k (y ) k (y ) k (x) k (x) (x))

k .As is an extension ®eld of it is faithfully ¯at over ,

k (x)

L L L

k (y ) E k (x)) E k (x)

and ( is not acyclic iff is not acyclic, that

O O

k (x)

X X

2 (E ) is, by (a) again, iff x Supph . This proves (b).

It remains to prove (c). Suppose ®rst that X is a noetherian scheme. The coho-

E O

mology sheaves of the perfect complex are coherent X -modules ([SGA6] I

(E )) H X

3.5, or [TT] 2.2.8, 2.2.12). Thus each Supp( is closed in ([EGA] I

E ) n

6.8.5), and as ( is cohomologically bounded, for all but ®nitely many one has

n n

H (E )) = = ; (E )= (H (E ))

Supp( Supp 0 . Thus Supph Supp is closed in

X X (E ) X .For a noetherian scheme, any subspace, e.g. Supph is quasi-compact

([EGA] I 2.7.1, [B-AC] II Sect. 4 no. 2 Prop. 9). This proves (c) for X noetherian.

Now suppose only that X is quasi-compact and quasi-separated. By absolute 0

noetherian approximation ([TT] C.9, 3.20) there exist a noetherian scheme X ,an

0 0

g X ! X F X E Lg F

af®ne map : , and a perfect complex on such that = .By

E )=g ( (F ) X (F )

(b), Supph( Supph , and so is closed in as Supph is closed

0 g

in the noetherian X . As the af®ne map is a quasi-compact map ([EGA] I 6.1.1),

(E ) = g (X (F )) X Supph Supph is quasi-compact. This proves (c) and

completes the proof of the lemma.

Y X

LEMMA 3.4 Let X be a quasi-compact and quasi-separated scheme. Let

be a closed subspace such that X is quasi-compact. Then there exists a perfect

X (E )Y complex E on such that Supph .

Proof. By absolute noetherian approximation ([TT] C.9, [EGA] IV 8.3.11)

(Z)

there exist a noetherian scheme X of ®nite type over Spec , a closed subspace

0 0 0

0 1

X f X ! X Y = f (Y )

Y , and an af®ne map : such that .Ifthereisa

0 0 0 0 0

E X Y (E E ) = Lf E

perfect complex on such that = Supph ,then will be

Y = (E ) a perfect complex on X with Supph by 3.3b. Thus it suf®ces to prove the

result when X is a noetherian scheme. Y

For X noetherian, the closed has ®nitely many irreducible components, with

y ;::: ;y

generic points 1 k ([B-AC] II Sect. 4 no. 2, [EGA] I 2.1.5, 2.7). If there

E X (E )= y E = E

are perfect complexes on such that Supph i,then is

i i= i i 1

comp3750.tex; 8/05/1997; 8:14; v.5; p.10

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 11

(E )= y =Y

perfect with Supph i . Thus it suf®ces to prove the case where =

i 1

= y y

Y is irreducible closed with generic point .

= (A) y X

Then let U Spec be an af®ne open neighborhood of in .Let

ff ;::: ;f g A

1 n be a ®nite set of generators for the ideal in the noetherian ring

\ U (A)

corresponding to the reduced closed subscheme Y Spec ([EGA] I 4.6.1,

K = K (f ;::: ;f ) = (A ! A)

4.2). Consider the Koszul complex n ,the

1 i1

A ! A) tensor product of chain complexes ( which are 0 except in degrees 1and

0, and have A in those two degrees between which the differential is given by multi-

f K (f ;::: ;f ) (A) n

plication by i . 1 is a strict perfect complex on Spec . On one hand

(K (f ;::: ;f )) (f = ) \ \ (f = )= (A=(f ;::: ;f )) =

n n Supph 1 n 1 0 0Supp 1

Y \ U (K (f ;::: ;f )) (H (K (f ;::: ;f )) ) = n

. On the other hand, Supph 1 n Supp 1

(A=(f ;::: ;f ))Y \ U (K )=Y \U

Supp 1 n . So Supph .

K K K

Let be the suspension of the Koszul complex ,so is also perfect with

K )= (K )=Y \U K K

Supph( Supph .Thesum is perfect with cohomo-

\ U K (U Y \ U ) [K K ]=[K ]+[K ]=

logical support Y .In 0 on one has

K ][K ] = K

[ 0. By the 0-extension criterion of Thomason±Trobaugh (2.5, or

X (E ) Y

[TT] 5.2.2) there is a perfect complex E on with Supph andsuchthat

jU ' K K y 2 U \ (E ) y = Y (E )

E .Then Supph ,giving Supph since

E ) Y = (E ) 2

Supph( is closed (3.3). Thus Supph .

i U ! X j V ! X

LEMMA 3.5 (Mayer-Vietoris). Let X be a scheme, : and :

U \ V ! X

two open immersions, and denote by k : the open immersion of the

;F 2 D(U [ V )

intersection. Then for E there is a natural long exact sequence:

(k E ;k F )

(U \V )

MorD

(E ;F )

(U [V )

MorD

(i E ;i F ) (j E ;j F ) ( : : )

(U ) D (V )

MorD Mor 351

(k E ;k F )

(U \V )

MorD

( E ;F )

(U [V )

MorD

U [ V (Note by a standard abuse one has replaced X by as the target of the

immersions i; j; k .)

comp3750.tex; 8/05/1997; 8:14; v.5; p.11

12 R. W. THOMASON

l P ! X i O !O X

Proof. For : an open immersion let !: P -Mod -Mod be the

O F

functor `extension by 0' which sends a sheaf of P -modules to the sheaf of

O W X

X -modules given by, for open

(W ) W P

F if

l F )(W ) :

( !

* P

0ifW l

Extend l! to complexes by applying ! in each degree of the complex. The

l l a l

adjunction between the functors l! and on sheaves induces an adjunction !

l on complexes. As l! and are exact functors on sheaves, these functors preserve quasi-isomorphisms of complexes, so on passing to the derived category one still

has an adjoint pair, and in particular an adjunction isomorphism

(l l E ;F ) (l E ;l F ): ( : : )

(X ) D (P )

MorD ! Mor 352

[ V

Consider the sequence of complexes on U , where the morphisms are induced

0 0

" l l ! l "l

by the various adjunction maps : ! 1orbythe ! for the immersions

\ V U V U [ V

between U , , ,and

k k E ! i i E j j E ! E ! : ( : : )

0 ! ! ! ! 0 353

[ V U V

This sequence is exact on U , being locally split exact on and on since the

k k E jU j j E jU

relevant adjunction maps restrict to natural isomorphisms ! = ! ,

i i E jU E jU D (U [ V )

! = , etc. Thus the sequence gives an exact triangle in .

( ;F )

(U [V ) Applying to this exact triangle the contravariant functor MorD yields

a canonical long exact `Puppe' sequence ([Ha] I 6.1, or [Ve] I Sect. 1 1±2), which

= i; j; k 2 conjugated by the isomorphisms (3.5.2) for l yields (3.5.1).

THEOREM 3.6 (Tensor nilpotence). Let X be a quasi-compact and quasi-separated

X F

scheme. Let E be a perfect complex on , and a complex of sheaves of

O F 2 D (X ) qc

X -modules which has quasi-coherent cohomology (i.e., ). Let

E ! F D (X )

f : be a morphism in .

2 X f k (x)= D (k (x))

Suppose for all x that 0in . Then there exists a

n n n

f E ! F D (X )

positive integer n such that : is 0 in .

O O

X X

n L n

F F

Proof. (c.f. [Ho] Thm. 10, [Ne] 1.1) Recall that exists for

2 D (X ) F 2 D (X )

any F , not just for , by [Sp] 6.5, 5.9.

n f ' X

3.6.1. I claim the conclusion `9 such that 0' is a local question on .

X U 9n

For suppose is covered by opens such that for each of these opens

f jU = D (U )

with 0in . Passing to a re®nement of this cover, I may assume

U X

the are af®ne and hence quasi-compact. As is quasi-compact, there is a

X = U k n

®nite subcover i . I will show by induction on that there is an =

i 1

f = D (X ) k = such that 0in . To start the induction at 1, note then that

= U f = k

X 1 so 0. To do the induction step assuming the result for 1, set

0 n

V = U 9n f = D (V )

i . By the induction hypothesis, such that 0in .Set = i 2

comp3750.tex; 8/05/1997; 8:14; v.5; p.12

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 13

0 m m

= fn ;n g f jU = D (U ) f jV = D (V )

m max 1 so 1 0in 1 and 0in ,aswell

f jU \ V = jU \ V = D (U \ V ) X = U [ V

as 1 0 1 0in 1 .Since 1 , the Mayer±

m m m

f 2 ( E ; F )

(X )

Vietoris sequence (3.5.1) then implies that Mor D is

m m

t 2 (k E ;k F )

(U \V )

of some morphism MorD 1 .Thatistosay,on

m m m

f E ! F D (X )

unfolding the proof of 3.5.1, that : factors in as

N N

m m

E ! F

? x

? ?

m m

( E ) "( F );

y ?

( : : )

3:6 1 1

N N

m m

E ! F k k k

k! !

k t)

! (

! k k k k where :Id ! = ! is the composite of the obvious isomorphism

and the natural third edge of the exact triangle induced by the exact sequence

k k ! n = m

(3.5.3), and where : ! Id is the adjunction map. But then for 2 ,

n m m

f ( f ) ( f )= D (X )

= 0in , as it has an induced factorization through

m m

f ) (k t) k ((k f ) t)

( ! , which identi®es under natural isomorphisms to !

f = D (U \ V ) and hence to 0 since k 0in 1 . This proves the induction step,

and hence the claim.

n f = 3.6.2. The desired conclusion 9 0 being local, and the hypotheses over

X implying the hypotheses over any quasi-compact open subscheme, to prove

= (R )

the theorem I may and do restrict to the af®ne case, X Spec .ThenI

may (3.1) assume E is a strict perfect complex. Let be the dual complex

_ _ _

E H (E ; O ) ' R H (E ; O ) E F ' E F

X O

om X om .Then is a

X O

complex with quasi-coherent cohomology. There is a natural isomorphismX

(E ;F ) (O ;RH (E ;F ))

(X ) D (X )

MorD Mor om

(O ;E F ); ( : : : )

(X )

MorD 3621

( ) a R H (E ; )

induced by the adjunction E om and the natural iso-

R H (E ;F ) ' RH (E ; O ) F ' E F E

morphism om om X for strict

perfect ([SGA6] I 7.4, 7.7, boosted by [TT] 2.4.1.a,b and [Ha] I Sect. 7). Under

[

f E ! F f O ! E F

the isomorphism (3.6.2.1) : corresponds to a : X in

D (X ) f

qc . Under other instances of this natural isomorphism corresponds to

n [ n n

f O O ! (E F ) x 2 X f k (x)

= X

: X and for all , corresponds

[

f k (x) F E F E O

to . Thus replacing by and by X , I reduce to the case

E O

X .

F 2 D (X ) F

By [BN] 5.1 (c.f. [SGA6] II Sect. 3 or [TT] B.16, B.17 for qc ) O

is quasi-isomorphic to a complex of quasi-coherent X -modules, so I may assume

F is such a complex. Under the equivalence of the category of quasi-coherent

O X = (R ) R F

X -modules on Spec and the category of -modules, is identi®ed to

comp3750.tex; 8/05/1997; 8:14; v.5; p.13

14 R. W. THOMASON

R f O ! F f R ! F

a complex of -modules, and : X is identi®ed to a map : in

(R ) f 2 H (F ) D -Mod , i.e. to a class .

Thus it remains to show that:

R f 2 H (F ) 3.6.2.2. If F is a complex of -modules, and is such that for all

2 (R )f k (x) = H (F k (x)) n

x Spec 0in , then there is a positive integer

n 0

f = H ( F )

such that 0in .

F 2 D (R ) 3.6.3. I next reduce to the case where F is strict perfect. For -Mod is

quasi-isomorphic to a complex of free R -modules, which in turn is the direct colimit

of its strict perfect subcomplexes ([TT] 2.3.2). So I may assume that F lim for

F g

f a directed system of strict perfect complexes. For such a ®ltering system

0 0 0 0

H (F ) H (F ) f 2 H (F ) f 2 H (F ) =

lim ,so is the image of a for some

. Passing to the co®nal directed subsystem of and setting equal to the

f f 2 H (F )

image of , I may assume that there is a family of compatible under

H (F )

the structure maps of the system and inducing f in the colimit . Note that

F R

now all F and so also are complexes of ¯at -modules, whence the derived

F k (x)

tensor product over R , , is represented by the tensor product of chain

k (x) complexes F .

T = fx 2 (R ) j f k (x)= H (F T T k (x))g

Set Spec 0in Note

0 0

T = fx 2 (R ) j f k (x) = H (F k (x))g =

if .Also Spec 0in

(R ) T (R )

Spec . I claim that each is a constructible set in Spec . For, representing

0 e 0

f 2 H (F f k (x)= ) f 2 F

the homology class by an element , one has 0in

0 e

H (F f k (x) F = f k (x) R ! k (x)) k (x)

iff is a boundary in ,i.e.iff0 :

0 1 0 1

(F F R ) k (x) =@ F

=@ F .As is a ®nitely presented -module, by [EGA] IV

X = S = (R ) T

9.4.5 on setting Spec one gets that is a locally constructible subset

(R ) (R ) T

of Spec .AsSpec is quasi-compact and separated, is then constructible

([EGA] 0III 9.1.12), proving the claim. But then the two remarks preceding the

T = (R )

claim imply that there exists an such that Spec by [EGA] IV 1.9.9,

f 2 H (F f k (x) = x 2 (R ) )

1.9.4. For this , satis®es 0forall Spec .If

n n

F n f = H ( F )

for the strict perfect this implies that for some 0in , one

n n

f = H ( F ) F F

gets 0in by taking the image under ! . This completes

R the reduction to proving (3.6.2.2) for F strict perfect over the ring .

3.6.4. Next I will reduce to the case where R is a noetherian ring of ®nite Krull

R R

dimension. Any commutative ring is the direct colimit lim of its subrings of

Z R

®nite type over .These are noetherian rings of ®nite dimension ([EGA] IV

5.5.4, 0IV 14.1.4). As the strict perfect complex F consists of a ®nite number of nonzero direct summands of ®nitely generated free modules and ®nitely many

maps @ between them, the projections on the summands and the maps between

them being represented by ®nitely many matrices with values in R , there exists

F R F R F

= R

a and a strict perfect complex over such that . Passing

comp3750.tex; 8/05/1997; 8:14; v.5; p.14

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 15

F = R F R

to the co®nal system of , I get a family of strict perfect

complexes over the , compatible under the structure maps of the system and

0 0

F F H (F ) H (F ) =

such that = lim .Thenas lim ,thereisa such that

! !

0 0

f 2 H (F ) f 2 H (F ) 0

is the image of a . Restricting to the co®nal system of

f f

and setting to be the image of , I may assume I have a compatible

f 2 H (F ) F

family of .Noteas is strict perfect, the tensor product of complexes

( ) F ( )

F represents the total derived functor .

T (R ) x 2 (R ) f

Let Spec be the subset of those Spec such that

H (F k (x)) T k (x) =

0in . As in 3.6.3 above, is constructible. Denot-

(R ) ! (R )

ing by :Spec Spec the structure maps of the system and by

(R )= (R ) ! (R )

:Spec lim Spec Spec the projections from the inverse limit

1 1

(T ) T (T ) = fx j f k (x) =

of schemes, one clearly has and

(F k (x))g = (R )

0in H Spec . Then by [EGA] IV 8.3.4 there is an such that

T = (R ) x 2 (R ) f k (x)= f

Spec ; i.e., such that for all Spec , 0. If in

n n n

0 n 0

H ( F ) R f = ( f )= H ( F )

for the noetherian ,then 0in .This

completes the reduction to the case where R is noetherian of ®nite dimension.

3.6.5. Now it remains to show: if R is a noetherian ring of ®nite Krull dimension,

R f 2 H (F ) F a bounded complex of ®nitely generated projective -modules, and

x 2 (R )f k (x)= H (F k (x)) 9n>

is such that 8 Spec 0in ,then 0such that

n 0

f = H ( F ) 0 in . To prove this, following the strategy of [Ne], I will induct

on the dimension of R .

3.6.6. To prepare the induction, I claim that if N is the ideal of all nilpotent

R=N R

elements of R , and if the statement 3.6.5 holds for , then it holds for .Forthen

n1 0 1

9n > f = H (( F ) R=N )

1 0suchthat 0in R . That is, on choosing a repre-

n n n

n1 1 0 1 1 1 0

f ( F ) 9x 2 ( F ) 9y = r f 2 N ( F ) i

sentative of in , and 1 i

n1 0 1

r 2 N R f 2 ( F ) f = @x + y r 2 N

i i

with i , such that . As each

9n > (r ;::: ;r ) N

is nilpotent, 2 0 such that for the ®nitely generated ideal 1 k ,

n n n n

n22 2210

(r ;::: ;r ) = y 2 (r ;::: r ) ( F ) = k

1k 0. Then 1 0. The element

n n n

n2 1 2 2

( f ) = (@x + y) y =

is a sum of terms 0 and of terms of the

@x b @x a @x b a b @x y

form a ,and where and are products of and . n

n1 1

= @( f @x) = @ f @@x = @a = @b =

As @y 0, in any case 0and

+n +n +n n n

n1 2 1 2 1 1 2

@x b = @(a x b) f 2 @ ( F ) f =

a . Thus ,so 0in

0 n1 2

( F )

H . This proves the claim.

= R=N (R ) (R=N ) = (R ) Note dim R dim as Spec and Spec Spec red have homeomorphic underlying spaces ([EGA] I 4.5, IV 5.1).

3.6.7. Now I ®nish the proof of 3.6.5 with the inductive argument. To start the

= R

induction with the case dim R 0, I may by the claim 3.6.6 replace by the

N R

reduced noetherian ring of dimension 0 which is R= . But thereafter is the

(x) product of its ®nitely many residue ®elds k (e.g., [B-AC] VII Sect. 1 no. 3

comp3750.tex; 8/05/1997; 8:14; v.5; p.15

16 R. W. THOMASON

H (F )

Exemple 1, IV Sect. 2 no. 5 Prop. 9, II Sect. 3 no. 5 Prop. 16). Thus = L

(F k (x)) f k (x)= x f =

H and as 0forall , one has 0.

= d >

To do the induction step, suppose dim R 0 and that 3.6.5 is already

d R

demonstrated for noetherian rings of dimension 1. Again replacing by its

R=N R R ! k ( )

quotient by the nilradical, I may suppose is reduced. Let i = i 1

be the map of R to the product of the residue ®elds at the ®nitely many minimal

R 2 (R ) R k ( ) i

primes of , i Spec .As is reduced, each such residue ®eld is in fact

Q m

R k ( ) S R S i

the local ring . Indeed is isomorphic to the localization for

i= i 1

the set of non zero-divisors of R , those elements contained in no minimal prime ([B-

AC] II Sect. 4 no. 3 Cor. 3 aÁ Prop. 14, IV Sect. 2 no. 5 Prop. 10). By hypothesis, for

1 0 1 1 0

f k ( )= f S R = H (F S R = S F ) f 2 F

i R

each i 0, so 0in .If

0 1 1

2 H (F ) y 2 S F

is a representative of the class f , this means there exists a

0 1 0 1

= f S F s 2 S y 2 F such that @y in . Then for some there exists such

0 1 0

= @y F y R ! F f R ! F

that sf in . The elements : and : determine a map

s s

(R ! R ) ! F (R ! R ) R of complexes : ,where is the complex with in

degrees 0 and 1, 0 in the other degrees, and where the non-trivial differential is

s 2 S R

given by s.As is a non zero-divisor, this complex of -modules is quasi- s

isomorphic to the complex which is R =sR concentrated in degree 0. Again as

(R=s) ! (R )

is a non zero-divisor the closed immersion i:Spec Spec is a regular

i D (R=s) ! D (R)

qc qc

closed immersion and the direct image functor : preserves

i D (R=s) !

perfect complexes and so restricts to a triangulated functor : parf

D (R ) R ! i R=s = R=s

parf ([SGA6] VII 1.4, 1.9, III 4.8.1, 2.5). Let : be the

(R ! R ) ! F

canonical quotient map. Replacing : by its composite with

D (R ) i (R=s) ' (R ! R)

the isomorphism in parf whose composite with is

(R ! R ) D (R )

the inclusion of R into degree 0 of , I get a factorization in parf

f = R ! i (R=s) ! F

of . This factorization and the naturality of various

(R )

standard quasi-isomorphisms gives the following commutative diagram in D parf >

for all n 0

1 n

f i ( i f ) R=s

This diagram gives a factorization of through .But is a

R=s

(R=s) 6 (R ) 6

noetherian ring, and as s is a non zero-divisor one has dim dim 1

d 1 ([EGA] 0IV 16.1.2.2). Then by the induction hypothesis that 3.6.5 is known

6 d i f k (x) = f k (x) = D (k (x))

for dimensions 1 and since R=s 0in

2 (R=s) (R ) n>

for all x Spec Spec , one gets that there exists an 0 such that

n 1

i f = D (R=s) f = D (R ) 0in . Then the factorization gives 0in .

parf R parf R=s This completes the proof of the induction step for 3.6.5, and hence the proof of the

theorem.

D (k )

REMARK 3.7. For k a ®eld the derived category is equivalent (by the functor k

sending a complex to its cohomology groups) to the category of Z-graded -vector

D (X ) f E ! F

spaces. Thus the Theorem 3.6 says that given a map in qc ,: , with

x 2 XH (f k(x))

E perfect and such that for all is the zero map, then there

f = D (X ) exists an integer n>0 such that 0in . To the more sophisticated this

comp3750.tex; 8/05/1997; 8:14; v.5; p.16 THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 17

is a surprisingly strong conclusion, and in any case the naive attempts to strengthen

it fail. +

n 1

R F

= =

? ?

( f ) f

N N

n n

( F ) F ( R ) R

@ 6

1 1

F ) R

(3.6.7.1) (

N N

n n

( F ) i (R=s) ( R ) i (R=s)

f )

( 1

= =

? ?

N N

n n

i ( i F ) i ( R=s)

R=s R=s

i ( i f )

(f )= H (f

For example, it is not suf®cient to suppose that H 0 rather than

k (x)) = x X = (Z ) E F

p) 0forall .Forlet Spec ( , and let the complexes , be

given by the rows in the diagram below, with f given by the columns and where

the center column is in degree 0

! ! ! Z ! ! Z !

(p) p)

00( 0

? ? ? ? ?

( : : )

y y y y

y 1 371

! ! ! Z ! ! !

00( 00

H (f )= n> H (( f ) Z=p) Z=p

Then 0, but for all 0is an isomorphism = =p

Z .

= D (X ) 9n

Nor can one strengthen the conclusion to f 0in rather than

f = (f ) = f k (x) = 0, even if one supposes H 0 in addition to 0for

comp3750.tex; 8/05/1997; 8:14; v.5; p.17

18 R. W. THOMASON

x X = (Z )

all . For again with Spec ( consider the map of complexes give by

! ! ! Z ! Z ! !

p) (p)

00( 0

? ? ? ? ?

( : : )

y y y y y 372

! ! Z ! Z ! ! !

p) (p)

0( 00

0 2 2 2

f 6= D (Z ) H (f Z=p )=p Z=p ! Z=p f Z=p

Here 0in ( parf as :, although

and f are 0.

X = (Z )

In both counterexamples Spec ( is about as nice a scheme as one could Z

imagine except for ®elds and ;, and the counterexamples generalize to Spec( )and

[T ] to Spec(k ). Having entered this far into the question, one should abandon all hope.

THEOREM 3.8 (Tensor nilpotence with parameters). Let X be a quasi-compact

G X F

and quasi-separated scheme, E and perfect complexes on , and a complex

O f E ! F

of sheaves of X -modules with quasi-coherent cohomology. Let : be

D (X ) x 2 (G ) f k (x) =

a morphism in qc . Suppose for all Supph that 0 in

(k (x)) n> G ( f )=

D . Then there is an integer 0such that 0as a morphism

n n

( E ) ! G ( F ) D (X )

G in .

2 X (G f ) k (x)= D (k (x))

Proof. (c.f. [Ho] Thm. 10ii) For each x 0in

k (x) ' x 2 (G ) f k (x)=

as either G 0orelse Supph and 0. So by Tensor

(G f )= D (X )

Nilpotence 3.6 there is an n> 0 such that 0in . A fortiori

n n n

n 1

( R H (G ; O )) ( G ) ( f )= G ( f )

om X 0. I will show that is

a retract of this morphism and so is also 0.

For this, it suf®ces to show by induction on n that is a direct summand in

n 1

D (X ) ( R H (G ; O )) ( G ) n =

parf of om X .For 1 this is trivial. To do the

n 2

n > ( R H (G ; O ))

induction step to prove it for 2 it suf®ces to prove that om X

n n

n 1 1

( G ) ( R H (G ; O )) ( G ) n

is a direct summand of om X .For 3

n 3 2

( R H (G ; O )) ( G )

this follows upon tensoring with om X from the

n = G R H (G ; O ) ( G )

case 2, namely that is a direct summand of om X .

Hence it suf®ces to prove the latter.

D (X ) As G is perfect, there is in a natural isomorphism (easy, or [SGA6] I

7.7)

L L

R H (G ; O ) G ! R H (G ; O ) X

: om X om

R H (O ;G ) ! RH (G ;G ) ( : : )

om X om 381

R H (G ;G ) G

Thus the problem is to show G is a direct summand of om .

G ! R H (G ;G ) G ( ) G O !

But if : om is of the morphism X

R H (G ;G ) G ! G R H (G ;G )

om corresponding to 1 G : ,andifeval: om

comp3750.tex; 8/05/1997; 8:14; v.5; p.18

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 19

G ! G = G

is the evaluation map, then eval 1 G .So splits off of

H (G ;G ) G 2

R om .

TT ! DEFINITION 3.9 Let T be a triangulated category with a ®xed functor :

T which is a covariant triangulated functor in each variable.

A T

A full triangulated left- -subcategory of is a full triangulated subcategory

T 2T A 2A T A

of T such that for all objects and ,is also an object in the

A T 2A

subcategory A. Similarly a full triangulated right- -subcategory has

2A T 2T whenever A and .

A full triangulated left- -subcategory is said to be respectively strictly full, thick,ordense if it is such as a triangulated subcategory (1.2±1.4).

The condition of being a strictly full left- -subcategory is invariant under

replacing by any naturally isomorphic functor .

When is commutative up to isomorphism, the concepts of strictly full tri-

angulated left- -subcategory and of strictly full triangulated right- -subcategory

are equivalent, and one says simply strictly full triangulated -subcategory. I

emphasize that despite what this terminology suggests, the condition of A being a

-subcategory is stronger than merely requiring that on restricts to a functor

A T

AA ! A;itsays is a sort of `ideal' in the `ring' with as `multiplication'. Y (However, use of the term `ideal' for such an A as in 3.9.1 with a closed subscheme would lead immediately to a bad terminological singularity at `the ideal

associated to Y'.)

T = D (X ) For X a quasi-compact and quasi-separated scheme and parf ,

henceforth I consider strictly full triangulated -subcategories where by default

is the usual derived tensor product.

Y X

EXAMPLE 3.9.1 If X is a quasi-compact and quasi-separated scheme and

(X )

is any subspace, the full subcategory of D parf consisting of those perfect com-

(E ) Y

plexes E such that Supph is a thick triangulated -subcategory of

(X ) D parf .

REMARKS 3.10 (a) The intersection of any set of full (resp. strictly full, resp. T

thick) triangulated left- -subcategories of is again a full (resp. strictly full, resp.

T T thick) triangulated left- -subcategory of . Thus each subcategory of has a

smallest full (resp. strictly full, resp. thick) left- -subcategory containing it.

(b) Let A be a full triangulated left- -subcategory of . By 1.5, the smallest

T A

thick triangulated subcategory A of containing has as objects all direct sum-

A A

mands in T of objects isomorphic to objects in . So an easy check shows is T in fact a thick triangulated left- -subcategory of , and is also the smallest such

containing A. X PROPOSITION 3.11 ( -subcategory test). Let be a quasi-compact and quasi-

comp3750.tex; 8/05/1997; 8:14; v.5; p.19

20 R. W. THOMASON

fL g

separated scheme with an ample family of line bundles ([SGA6] II 2.2.4).

AD(X) L

Suppose parf is a thick triangulated subcategory, and that for each ,

A L A

a line bundle in the given ample family, and for each A in one has that

A D (X )

is in A.Then is a thick triangulated -subcategory of parf .

= (R )

COROLLARY 3.11.1 (a) If X Spec is an af®ne scheme, all thick triangu-

(X ) lated subcategories of D parf are thick triangulated -subcategories.

(b) If X is a quasi-compact and quasi-separated scheme with an ample line

O ( )

bundle X 1 ([EGA] II 4.5), e.g. any scheme quasi-projective over some ring, and

AD(X) O ( ) AA

if parf is a thick triangulated subcategory such that X 1 ,

then A is a -subcategory.

(X )

D parf be a thick triangulated subcategory which is locally determined. (That

E 2 D (X ) fU g

is, one supposes that if parf and if there exists an open cover of

X U F 2A E jU F jU

such that for each there is an and an isomorphism

D (U ) E 2A A

in parf ,then .) Then is a -subcategory.

2A L

Proof of Cor. (c) Suppose A . As any line bundle is locally isomorphic

O L A A A

to X , is locally isomorphic to .As is locally determined, this gives

A 2A L

L for any line bundle and then Proposition 3.11 yields the result.

O ( ) fO ( )g O ( )

X X (b) If X 1 is an ample line bundle the set 1 consisting of 1

alone is an ample family, and (b) gives the result.

X O O A A

= X (a) If is af®ne, X is an ample line bundle. As 3.11 applies

to give the result.

D (X ) B

Proof of Proposition. Let B parf be the full subcategory of those

2A B A 2A A

such that for each A one has .As is a strictly full triangulated

(X ) B A subcategory of D parf , is also a strictly full triangulated subcategory. As is

thick, B is also closed under the taking of direct summands, i.e. a thick subcategory

D (X ) B D (X ) L

of parf . The proposition asserts that parf . By hypothesis, for all in

L k > A

the ample family one has A . Then by induction on 1 one gets

k k

L L 2B

AA,i.e. . The proof of the proposition is thus completed by

the following Lemma 3.12.

LEMMA 3.12 Let X be a quasi-compact and quasi-separated scheme with an

fL g B D (X )

ample family of line bundles .Let parf be a thick triangulated

k> L B D (X ) subcategory such that for all integers 0and all , 2B.Then .

parf

O B

Proof. For any complex of sheaves of X -modules quasi-isomorphic to a

bounded complex which in each degree is a ®nite direct sum of L for various

B 2B

and k>0, one has . One sees this by an easy induction on the total number

of factors L in the direct sums.

By the global resolution of perfect complexes on schemes with an ample family

of line bundles ([SGA6] II 2.2.8, or as a porism to [TT] 2.3.1), each perfect complex

X B E on is quasi-isomorphic to a bounded above complex which in each degree

comp3750.tex; 8/05/1997; 8:14; v.5; p.20

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 21

k >n

n B

is a ®nite direct sum of various L . For each integer denote by the brutal

truncation of B , the subcomplex given by

B i > n

n i

> if

B ) : ( : : ) ( 312 1

0ifi

>n >n

B B 2B

Then is bounded and by the preceding paragraph . I claim

E ' B B

that for all n suf®ciently less than 0 that is a direct summand of

(X ) B E 2B B D (X ) in D parf .As is thick this will imply . Thus parf , which will

prove the lemma. Thus it suf®ces to prove the claim.

B B

For this, I will show for n 0 that the inclusion of complexes

(X ) B ! B

is a split epimorphism in D parf , that the identity morphism 1: factors

n >n

6 1

B B = B

through it. Denote by the opposite brutal truncation, .There

n 6n

> 1

(X ) B ! B ! B is an exact triangle in D parf: . By the long exact

Puppe sequence ([Ve] I Sect. 1 1±2, [Ha] I 1.1)

! (B ; B ) ! (B ;B )

(X ) D (X )

MorD Mor

! B ) ! ( : : ) (B ;

(X )

MorD 312 2

B ! B D (X )

it suf®ces to show for n 0that is 0 in parf .

F 2 D (X )

For any qc there is a natural strongly converging spectral sequence

(e.g. [TT] 2.4.1.5±6)

p;q p+q

E = H (X cH (R H (B ;F ))) =) B ;F )

; om Ext (

(X )

2 D

( B ;F ) ( : : ) =

(X )

MorD 312 3

R H (B ; ) As B is perfect the functor om has ®nite cohomological dimen-

sion (e.g. [TT] 2.4.1.b), and in particular there exists an integer a such that for

+ q

F 2 D (X ) m H (F ) = q > m

all qc with an integer such that 0if ,then

q q

(R H (B ;F )) = q > m a q H (R H (B ;F ))

H om 0for Moreover, for all om X is a quasi-coherent sheaf on X (e.g. [TT] 2.4.1.c). As is quasi-compact and

quasi-separated there exists an integer b such that for all quasi-coherent sheaves

X p > b H (X F ) =

F on and all one has ; 0 ([EGA] III 1.4.12 applied to

X ! (Z) F 2 D (X )

Spec and boosted by IV 1.7.21 or [TT] B.11). Then if qc

q q

(F ) = q > a b H (R H (B ;F )) = q > b

with H 0for ,then om 0for ,and

p q

(X H (R H (B ;F ))) = p + q > p 6 b q<b

H ; om 0for 0, indeed unless and .

(B ;F ) = F

(X )

Thus the spectral sequence (3.12.3) gives MorD 0forsuch .In

(B ; B )= n 6 a b 2

(X )

particular Mor D 0if .

EXAMPLE 3.13 Proposition 3.11 and its corollary show that the `usual kinds' of

(X ) thick subcategories of D parf are triangulated -subcategories. Here I give an

example of the existence of an `exotic' thick subcategory which is not a -subcategory.

comp3750.tex; 8/05/1997; 8:14; v.5; p.21 22 R. W. THOMASON

1 1

= P k P ! (k )

Let X be the projective line over a ®eld , : Spec its struc-

k k

( )

ture map, and O 1 its fundamental line bundle. As is smooth and projective R

the derived direct image functor preserves perfect complexes and induces a

(P ) ! D (k ) triangulated functor D ([SGA6] III 4.8.1, 2.5 or [TT] 2.5.4).

k parf parf

D (P ) Let A be the thick triangulated subcategory whose objects are

k parf

E " L R E ! E

those for which the natural adjunction morphism : is an

D (P ) ! R L

isomorphism in . As the other adjunction morphism :Id

k parf

(k )

is always a natural isomorphism in D parf (e.g. [ThFP] Lemme 3), using the

R " R = "L L =

adjunction identities 1and 1 one checks easily

D (k ) A A D (k )

that L parf , and that in fact is equivalent to parf via the functors

R L

and .

D (P ) F 6' But A is not a -subcategory of .Forif 0 is any nonzero object

k parf

D (k ) L F 2A O ( ) L F 2A= L R (O ( )

of parf , while 1 . Indeed 1

F ) '

L 0 since by the projection formula ([SGA6] III 3.7, or [Ha] II 5.6)

R (O ( ) L F ) ' (R O ( )) F R O ( ) '

1 1 and 1 0 by Serre ([EGA] III 2.12±16).

The mechanism of this example might suggest that the classi®cation of thick

D (X ) -subcategories of parf given by 3.15 below might be extended to classify all

thick subcategories on an X with an ample family of line bundles by incorporating

X ) some additional structure related to Pic ( . However, it seems dif®cult to ®nd a

candidate structure which reduces to the usual classi®cation when X is af®ne and

so all thick subcategories are -subcategories.

LEMMA 3.14 Let X be a quasi-compact and quasi-separated scheme, and

;F 2 D(X) X (E )

E parf two perfect complexes on . Suppose that Supph

F ) E D (X )

Supph( .Then is in the smallest thick triangulated -subcategory of parf

containing F .

Proof. (c.f. [Ho] proof of Thm. 7, [Ne] 1.2) Denote by A this smallest thick

triangulated -subcategory containing .

a G !O D (X ) C (a) a

For a morphism : X in parf , denote by the cone of .Then

O ( )( )

by de®nition and the minor abuse of considering X , one has exact >

triangles for n 1

- -

G O G O

X X

J] J]

J J

C ( a) (a) (3.14.1) C

comp3750.tex; 8/05/1997; 8:14; v.5; p.22

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 23

( a)

G ( G ) G

G C ( a)

+ n

n 1

a a (G ( a)) As is identi®ed to the composition , the octahedral

axiom ([Ve] I Sect. no.1±1 TR4, or [Ha] I Sect. TR4) gives from these a further >

exact triangle for all n 1

n n+

G C ( a) C ( a)

(3.14.2)

C (a)

Using the exact triangle resulting from tensoring this one with E , by induction

E C (a) A

on n one sees that if is in the thick triangulated -subcategory ,then

C ( a) n >

so is E for all 1.

F R H (F ; O ) F =

As is perfect, there is an isomorphism (3.8.1) om X

(F ;F ) R H (F ;F ) A H

R om ,so om is an object of the -subcategory .Let

f O ! R H (F ;F ) F ! F

: X om be the morphism corresponding to 1: ,and

a G ! O f

let : X be the `homotopy ®bre' of , the edge opposite the vertex

H (F ;F ) f C (a) ' R H

R om in the exact triangle having as an edge. Then om

F ;F ) A E C (a) 2 A

( is in the -subcategory ,so and as above, for all

> E C ( a) 2A

n 1 .

> E ( a)= D (X )

I claim there exists an n 1suchthat 0in parf. Then consid-

( ;E )

(X )

ering the long exact Puppe sequence which results from applying MorD

to the exact triangle which is the tensor with E of the second exact triangle of

E E O ! E C( a) =

(3.14.1), one will obtain that X is a split monomor-

E C ( a) A

phism, i.e. that E is a direct summand of .As is thick, this will

A imply that E is in , proving the lemma.

It remains to prove the claim. By tensor nilpotence with parameters (Thm. 3.8),

2 (E ) a k (x) = D (k (x))

it suf®ces to show for all x Supph that 0in .But

2 (E ) (F ) F k (x) 6' k (x) !

if x Supph Supph ,then 0. Thus the map

R H F k (x) ! F k (x) (F k (x) ;F k(x))

(x)

omk corresponding to 1:

(k (x))

is not zero, and so is a split monomorphism in D parf as any non null-

(x) k (x)

homotopic map from the ®eld k to a chain complex of -vector spaces

splits. But for F perfect one has an easy isomorphism (e.g. [SGA6] I 7.1.2)

R H (F ;F ) k(x) RH (F k (x) ;F k(x))

(x)

om omk , under which this

k (x) fa = a split monomorphism is identi®ed to f .As 0since is the homotopy

comp3750.tex; 8/05/1997; 8:14; v.5; p.23

24 R. W. THOMASON

f k (x) a k (x)= 2

®bre of f ,andas is a split monomorphism, one gets 0. X THEOREM 3.15 (Classi®cation of thick -subcategories). Let be a quasi-

compact and quasi-separated scheme. Denote by C the set of thick triangulated

D (X )

-subcategories (3.9) of the derived category parf of perfect complexes (3.1)

X S Y X Y = Y

on . Denote by the set of those subspaces such that is a

Y X X Y

union of closed subspaces of such that is quasi-compact. S

Then there is a bijective correspondence between C and .

S ! C Y X

The bijection : sends a subspace to the thick subcategory whose

(E ) Y Y

objects are those perfect E such that Supph , i.e. which are acyclic off .

C ! S A D(X)

The inverse bijection ': sends a triangulated subcategory parf

Y = (E )

to the subspace Supph .

E 2A

(A)

Proof. (c.f. [Ho] Thm. 11, [Ne] 1.5). Lemma 3.3 gives that the subspace '

A D (X ) Y X

is in S for any subcategory of parf , and clearly for any subspace

(Y ) C '

is a thick triangulated -subcategory, and so is in . So the functions and

' C

are de®ned. It is easy to see that both and preserve the partial orders on and

' (Y ) Y

on S given by inclusions of subcategories and of subspaces, and that

(A) A '

and ' . It remains to show and are mutually inverse, for which it

(Y ) Y Y 2 S '(A) A

suf®ces to prove the reverse inclusions ' for and C

for A2 .

' (Y ) Y Y 2 S Y X X Y

To prove ,since is a union of closed with Y

quasi-compact, it suf®ces to show for each such there exists a perfect complex

E (E ) Y E (Y ) Y Y

such that Supph ,suchan necessarily being in since .

But the existence of such an E is given by Lemma 3.4.

(A) A A

To show ' for a thick triangulated -subcategory, let an arbitrary

2 '(A) E 2 A '

E be given. I will show . By the de®nition of , ,thereare

F A (E ) (F (E )

in such that Supph Supph ). The closed subspaces Supph ,

Supph ) being the complements of quasi-compact opens (Lemma 3.3), they

are constructible subsets of X by de®nition ([EGA] 0III 9.1.2, IV 1.8.1). Then by

f (F )g f (F g

[EGA] IV 1.9.9 there is a ®nite subset Supph of Supph ) such that

n n

(F ) (E ) ( F ) F 2A

SupphE Supph . So Supph Supph ,andas ,

i i i

1 i1 1

2A 2 Lemma 3.14 gives E .

REMARKS 3.16.1 Recall from Corollary 3.11.1 that if X has an ample line bundle

O ( ) A D (X )

X 1 , then a thick triangulated subcategory of parf is a -subcategory

O ( ) AA A

if X 1 . This condition is satis®ed for the most `natural' ,e.g.if A

membership in A is locally determined (3.11.1.c), and is satis®ed for all thick

(R ) R

when X is Spec for a ring (3.11.1.a). X

3.16.2. If Y is an arbitrary subspace of a quasi-compact and quasi-separated , one

(Y ) D (X ) does have a thick triangulated -subcategory of parf , whose objects

are the perfect complexes acyclic off Y . But the subspace corresponding under

comp3750.tex; 8/05/1997; 8:14; v.5; p.24

THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 25

0 0

Y Y Y Y

3.15 to this thick -subcategory is not in general ,butrather where

Y Y Y X

is the union of all such that is closed in with quasi-compact com-

plement. For example, consider the polynomial ring in in®nitely many variables

[T ;T ;:::] X = (C [T ;T ;:::]) Y

C 1 2,set Spec 1 2and let be the closed point corre-

T ;T ;:::) X Y

sponding to the maximal ideal ( 1 2.Then is not quasi-compact, and

= ; Y

indeed Y , i.e. any perfect complex acyclic off is acyclic everywhere. Thus

' D (X ) this closed subspace Y is not of a thick triangulated -subcategory of parf . This example is due to Neeman ([Ne] 4.1).

But when X is noetherian, all subspaces are quasi-compact ([B-AC] II Sect. 4

X S Y

no. 2 Prop. 9, [EGA] I 2.7.1), and the subspaces Y in are just those Y which are unions of closed subspaces of X ,the `closed under specialization'.

HISTORY 3.17 Hopkins told me that he proved versions of the nilpotence theorems

= (R ) R 3.6 and 3.8, and of the classi®cation Theorem 3.15 in the case X Spec for a noetherian ring, but that while writing them up in his expository article on similar results in stable homotopy, he was inspired with a `more obvious proof' for 3.6,

which yielded statements for R any commutative ring ([Ho] Thm. 10, Thm. 11).

As usual, `obvious' means wrong. Neeman pointed out the error ([Ne], after 1.4),

= (R ) R and gave a proof of 3.6 for X Spec of a noetherian ring ([Ne] 1.1). In fact 3.6.6±3.6.7 is essentially Neeman's proof, which Hopkins says is essentially the

same as his ®rst unpublished proof. Neeman asks ([Ne] 1.6) if tensor nilpotence

R ) R holds on Spec( when is not noetherian; Theorem 3.6 shows that it does (c.f. 3.6.2.2 vs. [Ne] 1.1).

Hopkins used his version of tensor nilpotence to deduce a classi®cation of

(R )

the thick triangulated subcategories of D parf ; Theorem 11 of [Ho] states that

(R )

each thick triangulated subcategory of D parf is the thick subcategory of perfect

(R ) complexes acyclic off some subspace Y Spec closed under specialization.

Hopkins says he had a valid proof of this for R noetherian, and in fact the literal

statement of ([Ho] Thm. 11) is true for all R by Theorem 3.15 and 3.16.2 above. But the remarks succeeding to Hopkins' statement suggest he believed he had established a bijective correspondence between the thick subcategories and the

subspaces Y closed under specialization. This is Neeman's interpretation of the

statement as given in ([Ne] Sect. 0), and proved by Neeman for R noetherian ([Ne] 1.5). Neeman also pointed out that this bijective correspondence fails for certain

non-noetherian R ([Ne] 4.1). My Theorem 3.15, boosted by 3.11.1.a, gives the

correct bijective correspondence for arbitrary commutative rings R .

( ) 5. Classi®cation of all strictly full -subcategories of D X parf

THEOREM 4.1 Let X be a quasi-compact and quasi-separated scheme. Recall the de®nitions of 1.2, 3.1, and 3.9. Then:

There is a bijective correspondence between the set of strictly full triangulated

A D (X ) (Y; H ) Y -subcategories of parf , and the set of data ,where is a subspace

comp3750.tex; 8/05/1997; 8:14; v.5; p.25

26 R. W. THOMASON

X Y X X Y

of which is a union of closed subspaces of with quasi-compact,

K (X Y ) K (X )

and H 0 on is a 0 -submodule of the Grothendieck group of the Y

triangulated category of perfect complexes on X acyclic off .

Y; H ) A D (X )

To ( corresponds the triangulated subcategory parf whose

Y [E ]

objects are those perfect complexes E acyclic off and such that the class

(X Y ) H in K0 on lies in . Proof. The theorem results easily on combining 2.1 and 3.15. For the strictly full

triangulated subcategory A is dense in the smallest thick triangulated subcategory

e e

A D (X ) A

A containing it by 1.5. As is a -subcategory of parf , is also a -sub-

e Y

category (3.10.b). Then by 3.15, A corresponds to some subspace of the type e

stated. By 2.1, the strictly full dense triangulated subcategories of A correspond

(A) K (X Y ) K (X )=

bijectively to subgroups of K0 0 on . As the ring structure on 0

(D(X) ) K (X Y )

K0parf and its action on 0 on are induced by the derived tensor

product , one checks easily that the strictly full dense triangulated -subcate-

A H K (X Y ) K (X ) gories A of correspond to those subgroups 0 on which are 0 -

submodules. 2

References [BBD] Beilinson, A. A., Bernstein, J. and Deligne, P.: Faisceaux pervers, le tout de: Analyse et Topologie sur les Espaces Singuliers I (colloque aÁ 8 Luminy 1981) AsterisqueÂ 100, (1982) Soc. Math. France. [BN] Bokstedt,È M. and Neeman, A.: Homotopy limits in triangulated categories, Compositio Math. 86 (1993) 209±234. [DHS] Devinatz, E. S., Hopkins, M. J. and Smith, J. H.: Nilpotence and stable homotopy theory I, Ann. Math. 128 (1988) 207±241. [Ha] Hartshorne, R.: Residues and Duality, Lecture Notes in Math. 20 (1966) Springer-Verlag.

[He] Heller, A.: Some exact sequences in algebraic K -theory, Topology 3 (1965) 389±408. [Ho] Hopkins, M. J.: Global methods in homotopy theory, in: Homotopy Theory (conference at Durham 1985), London Math. Soc. Lecture Notes 117 (1987) Cambridge Univ. Press, 73±96. [HS] Hopkins, M. J. and Smith, J. H.: Nilpotence and stable homotopy II, to appear in Ann. Math., currently available by anonymous ftp to hopf.math.purdue.edu.

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