8

1. THE ^-ADIC FORMALISM

cohomology functors are obtained by restricting from P to D'. The heart of

V is V fl C, which is equal to C. So we have proved

P R O P O S I T I O N

1.3. The category V is a t-category with heart C'. The

inclusion functor V —± V is exact. IfV is non-degenerate, then so is V.

We now consider the case of quotients of V. For this construction we

need a thick subcategory C of C. This means that C is a non-empty full

subcategory satisfying the following condition. Whenever

0 — A ! — A — • A" — • 0

is a short exact sequence in C, then A is in C if and only if A' and A"

are. Note that this condition implies that C is closed in C. Thus the above

construction can be carried out and we get a t-subcategory V of V with

heart C. Note that V satisfies the following condition. If M — N is a

morphism in V whose cone is in V and which factors through an object of

V', then M and N are in V. In other words, V is a thick subcategory of

V.

P R O P O S I T I O N

1.4. The quotient category V/V is in a natural way a

t-category with heart C/C. The quotient functor V — + V/V is exact. If V

is non-degenerate, then so is V/V.

P R O O F . We define a morphism A — * B in C to be an isomorphism

modulo C if its kernel and cokernel are in C. The category C/C is endowed

with a functor C —* C/C that turns every isomorphism modulo C into an

isomorphism. Moreover, C/C is universal with this property.

Call a morphism M —» T V in V a quasi-isomorphism modulo C if for

every i E Z the homomorphism hlM — hlN is an isomorphism modulo C.

Note that M —• N is a quasi-isomorphism modulo C if and only if the cone

of M — N is in V. This proves that the quasi-isomorphisms modulo C

form a multiplicative system and the category V/V is the universal cat-

egory turning the quasi-isomorphisms modulo C into isomorphisms. It is

standard (see for example [14]) that V/V is a triangulated category if we

call a triangle in V/V distinguished if it is isomorphic to the image of a

distinguished triangle in V.

If (X-°,P-°) defines the t-structure on V, then the essential images

of V-° and V-° in V/V define a t-structure on V/V whose associated

truncation and cohomology functors are obtained from the corresponding

functors for V through factorization. It is not difficult to prove that the

inclusion C — » V factors to give a functor C/C — V/V that identifies C/C

as the heart of the t-structure on V/V. •

R E M A R K

1.5. Let V\ and V2 be t-categories with hearts C\ and C2,

respectively. Let C[ and C2 be thick subcategories of C\ and C2, giving rise

to thick subcategories V[ and V2 as above. Let F : V\ —• V2 be an exact

functor. If F maps V[ to V2, then F induces a functor F : V\/Vx — V2/V2.