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
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 ) 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.