Auslander-Reiten Quiver: Definition
An additive category is something like a ring.
is an addition,
Call a map f
irreducible, provided it is not split-mono
and the composition behaves like the
multiplication in a ring,
however, it is only partially defined...)
and given any factorization f = f'f",
then f' is split-epi, or f" is split mono.
Note that: factorizations f = f'f" with
f' is split-epi, or f" is split mono cannot be avoided
An exact sequence 0 → X → Y → Z → 0
is called an Auslander-Reiten sequence
(they are just trivial factorizations):
provided the maps
X → Y and Y → Z are irreducible
(as a consequence, X, Z are indecomposable).
The Auslander-Reiten quiver of an exact category is defined as follows:
- Its vertices are the isomorphism classes [X]
of indecomposable objects X
- There is an arrow [X] → [Y], provided
there exists an irreducible map X → Y.
- If 0 → X → Y → Z → 0 is an Auslander-Reiten sequence,
and Yi is an
indecomposable direct summand of Y
is a 2-simplex.