22 3 F I L T E R E D D E R I V E D C A T E G O R I E S

3 Filtered Derived Categories

Now we are going to study our fundamental example of an exact category, namely filtered categories associated to abelian categories.

3.1 C o n s t r u c t i o n s

Let r be an abelian category. We are going to consider filtered objects and complexes of filtered objects. A filtration will always be a finite filtration.

Let ~4~lt be the category of the ascendingly filtered objects of Jr. This fil- tration will usually be called weight filtration W,. Let fl~bifilt be the category of bifiltered objects of ,4. Morphisms of fl[bifilt a r e morphisms of r which respect both filtrations. Our convention is that one of them (weight filtra- tion W,) is ascending, the other descending (Hodge filtration F*). They are both finite.

To simplify notation, let in the sequel B be either A~it or Abi~lt.

In B any morphism has a kernel and a cokernel. We get them by taking the kernel or cokernel in ,4 with the induced filtration. The category is not abelian, since in general the image and the coimage of a morphism are not isomorphic.

Def in i t ion 3.1.1 A morphism in B is called strict if its image and coimage are isomorphic in B. A sequence

0 ~ A - 2 + B - ~ C ~0

in B is called short exact sequence if a is a strict monomorphism and b a strict epimorphism with kernel a.

A strict morphism factors by definition as a strict epimorphism followed by a strict monomorphism. Kernels and cokernels are always strict.

P r o p o s i t i o n 3.1.2 B is an exact category.

Proof. ' We check the conditions in [44] 1.0.2. It is only an exercise in computations with kernels and cokernels. []

There are other useful characterizations of strictness.

3.1 Constructions 23

L e m m a 3.1.3 A sequence

S = [ 0 ~ A - ~ B - ~ C ~0]

in B is exact if and only if one of the following equivalent conditions holds:

1. For all indices the sequence W~(S) or Wn N F'~(S) resp. is exact in ,4.

2. For all indices the sequence GrW(S) or Gr TM (3 GrT(S ) = Gr~ GrW(S) resp. is exact in ,4.

P r o o f i 1.) is an assertion of the kind tha t becomes trivial once one has made clear what one is talking about. A sequence is exact if and only a is the kernel of b and b the cokernel of a. The filtrations on the kernels and cokernels are induced by those on B.

To see the equivalence 1.) and 2.), we only have to work in ,4. There is (in the filtered case) a commutative diagram

o

whose rows are exact. With any two columns the third is also exact (a nice application of the long exact sequence of homology). The finiteness condition gives the start ing point for the induction, for small n we have W,~(S) = GrW(S) = 0. By induction on n we get the equivalence of of 1.) and 2.). In the bifiltered case we have to apply this criterion twice, the auxiliary step being the diagram Fro(S). []

This criterion shows tha t strictness in the bifiltered case is stronger than strictness with respect to both F* and W.. It is also clear tha t our definition of strictness agrees with the usual notion. Danger: the strict morphisms in Hom~fi,, (A, B) are not a subgroup in general. This is really a nuisance.

As in paragraph 2.1 we can consider the categories C?(B), C~tr)(B), C2tr(,4~lt ). There is a good criterion for strictness of differentials:

L e m m a 3.1.4 (De l igne , B e i l i n s o n ) A complex K in C?(`4~lt) has strict differentials if and only if the E1-spectral sequence for the filtered complex degenerates at El.

For a bifiltered complex K*, the following are equivalent:

a) The ]unctors Gr W Gr F commute with cohomology on K*.

24 3 FILTERED DERIVED CATEGORIES

b) The differentials of K" are strict.

c) The differentials are strict with respect to W~, and the differentials of Gr W K* with the filtration induced by F are strict with respect to F,

P r o o f : The assertion for filtered complexes is [11] Hodge II, Prop. 1.3.2.

The equivalence of a) and b) for strictness follows from 3.1.3. The equiv- alence to c) follows by applying the definition of strictness of a filtered mor- phism first to (g*, W) and then to (Gr W g*, F). []

The next steps are again the introduction of the homotopy category and of the derived category.

Def in i t ion 3.1.5 A morphism f in K ? (~4~lt) is a filtered quasi-isomorphism / f H i G r , ( f ) is an isomorphism for all i ,n E 7].

A morphism f in K?(Jlbifilt) is a bifiltered quasi-isomorphism if g ' Gr TM Gr~( f ) is an isomorphism for all i, n, m e 7/.

L e m m a 3.1.6 (Bi)-filtered quasi-isomorphisms in K?(B) are simple quasi- isomorphisms if we consider them in K? (~4).

A morphism in K ? (B) is a (bi)-filtered quasi-isomorphism if and only if it is a quasi-isomorphism in the sense of exact categories.

P r o o f : The first assertion is checked elementary by induction on the short exact sequences W. ~ W~+I ~ Gr W in ,4.

We show that a complex K is acyclic if and only if the morphism 0 �9 ~ K is a (bi)-filtered quasi-isomorphism. In fact the complex is acyclic if and only if all morphisms K n-1 ) Ker d n are strict epimorphisms. This means that all differentials are strict and all naive cohomology objects vanish. In addition a (bi)-filtered quasi-isomorphism is always a quasi-isomorphism in ,4, hence the equivalence of the notions is clear. []

This proves also that the filtered derived category defined by Laumon [44] agrees with the one introduced in Hodge II [11].

P r o p o s i t i o n 3.1.7 Strictness of differentials is invariant under (bi)-filtered quasi-isomorphisms.

3.1 Constructions 25

Proof i A filtered quasi-isomorphism induces an isomorphism of the El- spectral sequences of the filtered complexes. By 3.1.4 the assertion follows in the filtered case. The bifiltered assertion follows using the characterization c). []

This is the first half of being derivably strict as introduced in 2.1.2. The second condition also holds.

P r o p o s i t i o n 3.1.8 Let A ~'m be a double complex in 13 with A n'm = 0 for n << 0 or m << 0. The complexes A N'* are assumed to have strict differentials and the complex morphisms d : A n'* ) A ~+1'* to be in C+r(13). Then Tot(A) is in C~tr)(13), i.e. the complex has strict differentials.

Proof : Because of 3.1.4 c) it suffices to prove the proposition for filtered complexes. First we consider the special case of cones. By [3] 3.1 the assertion holds. (One checks strictness elementwise.) The same method as in the proof of 2.2.3 reduces the general case to this special one. []

P r o p o s i t i o n 3.1.9 I f A ~ B is a morphism of double complexes con- centrated in the first quadrant such that all A T M ~ B n'* are (bi)-filtered

quasi-isomorphisms, then Tot(A) ~ Tot(B) is also a (bi)-filtered quasi- isomorphism.

Proof : As taking the total complexes commutes with the functor Gr TM or W m Gr, Gr f resp., this immediately reduces to an assertion about the abelian

category .4. As in the proof of 3.1.8 it suffices to consider the case of cones. But in this case it is an axiom of derived categories. []

The following operation will also be needed.

Def in i t ion 3.1.10 The functor Dec : C + (13) ~ C+(13) (ddcalage) is given on objects as follows: to an ascendingly filtered complex (K, W . ) , we assign the pair (K, Dec(W).) where

Dec(W)p(K n) = Ker (d: Wp_n(K n) ) K~+l /Wp_n_l (Kn+l) )

The Hodge filtration remains unchanged if there is one.

L e m m a 3.1.11 We have

Gr W H n ( g ) Dec(W) =c,pn H (K)

26 3 FILTERED DERIVED CATEGORIES

P r o o f i By [11] Hodge II 1.3.4 there is a morphism

E0 p'~-p(Dec K) ) E~+n'-P(K) ,

which induces a quasi-isomorphism of all the E~(Dec K) ) E~+~(K). As the spectral sequences converge, this gives an isomorphism of the limit terms. Hence we have

G DeeW H n ( K ) _-- E~n-P(Dec K) -~ E~+~'-P(K) = Gr_Wp_n Hn(K) r_p

[]

3 .2 D e r i v i n g F u n c t o r s

Now we want to have a look at derivatives of functors on filtered categories.

L e m m a 3.2.1 I f . 4 has enough injectives, then so do A~lt and -Abifilt-

P r o o f i An object of the form I = ~ , In (or I = ~n,~ In,~ resp.) where In is pure of weight n (or Into pure of bidegree n, m) and injective in A will be injective in J[~lt (in ~bif i l t ) - For this, one simply checks the universal property. Note tha t Hom~i, t (M , In) = Hom~(GrW(M),In) .

Consider A E Ob(A~lt). Then there is a (bi)-filtered resolution I* of A such tha t all I " and all Gr W I n (Gr TM Gr F I n) are injective (standard trick for compatible resolutions of exact sequences, in fact it is the direct sum of injective resolutions of the graded pieces). This is the desired injective resolution. []

Functors between abelian categories induce functors between the cor- responding filtered categories. Their derived functors are closely related, too.

L e m m a 3.2.2 Let .4 be an abelian category with enough injectives. : .A ~ ~4' be a left exact additive functor. The derived functors

exist, and we have

R r : D+(A) ~ D + ( A ')

R ~ l t : D+(A~,t) ) D+(A~lt)

Let

forget o Rr = Rff o forget

Here "forget" is the canonical exact functor which assigns to a filtered object the underlying object.

3.3 An R-Category 27

P r o o f i The lemma follows immediately if we compute the derived functor with the injective resolution tha t was constructed in 3.2.1. []

R e m a r k : This lemma can be applied to the computat ion of Ext-Groups. We get

RHom.~fi,~ = W0 RHomA

If Ext i vanishes in ,4 for i > n then the same is true in the filtered category.

3 .3 A n R - C a t e g o r y

Now let A be a rigid Q-category ([19] Def. 1.7). The unit object is denoted by 11. Further assume we are given a Tate-object ]1(1). Let ]1(n) = ]1(1) | (As usual we mean the dual of ]1(-n) for negative n.) We assume tha t the functors (n) = �9 | ]l(n) are exact. We have defined twist functors on ,4.

This s tructure passes to the derived filtered category.

P r o p o s i t i o n 3.3.1 Let either B = Atilt or B = ` 4 b i f i l t . The object ]1 in B is the unit of A, concentrated in degree 0 of the weight (and Hodge) filtration. The Tare-object 11(1) is concentrated in degree -2 of the weight filtration (resp. in degree -1 of the Hodge filtration). Let

( n ) : D + ( B ) ~ 9+(13)

K ~-+ K | ]l(n) .

These functors respect the subcategories D~tr)(13 ) and D+~(13).

With these definitions and the t-structure of 2.1, D?(13) becomes an R- category with weights.

Proof." The t-s tructure and its heart were given in section 2.1. 13 is a subcategory of the heart , and in this way we consider ]1 as an object of the heart .

By assumption the functor | is exact, commutes with (bi)-filtered quasi-isomorphisms and defines directly a functor on the filtered derived category.

The Wn are well-defined tr iangulated functors which commute with trun- cation. As ]1(1) is pure of weight -2, we get the necessary shift of the weight filtration. []

In the easiest case, i.e. if .4 is a category of vector spaces, then ]1 and ]I(1) are both equal to the ground field, and the twist functors (n) simply perform a shift of the weight filtration by - 2 n (and a shift of the Hodge filtration by - n ) .