**Yoshifumi Tsuchimoto**

such that . The -th

Elements of (respectively, ) are often referred to as

A bit of category theory:

- A collection
of
**objects**of . - For each pair of objects
, a set
**morphisms**. - For each triple of objects
,
a map(``composition (rule)'')

- unless .
- (Existence of an identity) For any
,
there exists an element
such that
- (Associativity)
For any objects
, and for any morphisms
,
we have

Morphisms are the basic actor/actoress in category theory.

An additive category is a category in which one may ``add'' some morphisms.

- A4-1.
- Every morphism in has a kernel .
- A4-2.
- Every morphism in has a cokernel .
- A4-3.
- For any given morphism
, we have
a suitably defined isomorphism

- An object
in
is said to be
**injective**if it satisfies the following condition: For any morphism and for any monic morphism , ``extends'' to a morphism . - An object
in
is said to be
**projective**if it satisfies the following condition: For any morphism and for any epic morphism , ``lifts'' to a morphism .

be an exact sequence of -modules. Assume furthermore that is projective. Then show that the sequence

is exact.