CHAPTER 1
The ^-adic Formalism
1. Quotients of t-Categories
For the definition of t-categories we refer to [4, 1.3]. Roughly speaking,
a t-category is a triangulated category X, having, for every i G Z, truncation
functors ri and ri and cohomology functors
hl
: V C, where C is the
heart of D, which is an abelian subcategory of V. We will discuss methods
of constructing sub- and quotient-t-categories of a given t-category V.
Let V be a t-category with heart C. Let
hl
: T C denote the associated
cohomology functors.
First, we will construct t-subcategories of V.
L E M M A
1.1. Let C be a full non-empty subcategory of C. Then the
following are equivalent.
(1) The category C is abelian and closed under extensions in C. The
inclusion functor Cf —* C is exact.
(2) The category C is closed under kernels cokernels and extensions in
C.
(3) / /
Ax
M
A2

A3

A4 -U A5
is an exact sequence in C and A\, A2, A4 and A$ are in C', then so
is A3.
PROOF. Considering the exact sequence
0 cok / As ker g 0
this is immediate.
DEFINITIO N
1.2. We call a subcategory C of C closed if it is a full
non-empty subcategory satisfying any of the conditions of Lemma 1.1.
Now let C be a closed subcategory of C. We may define the full subcat-
egory V of V by requiring an object M of V to be in V, if for every i G Z
the object
hlM
of C is contained in C'. Then using the above lemma it is
an easy exercise to prove that V is a triangulated subcategory of V. One
defines a triangle in V to be distinguished if it is distinguished as a triangle
in V. It is just as immediate (see also [4, 1.3.19]) that we get an induced
t-structure on V as follows. If ( P - ° , D - ° ) is the t-structure on D, then
{V n £^°, V n P - ° ) is the induced t-structure on V. The truncation and
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