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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