Auslander–Reiten theory
Encyclopedia
In algebra, Auslander–Reiten theory studies the representation theory of Artinian ring
s using techniques such as Auslander–Reiten sequences (also called almost split sequences) and Auslander–Reiten quivers. Auslander–Reiten theory was introduced by and developed by them in several subsequent papers.
For survey articles on Auslander–Reiten theory see , , , and the book . Many of the original papers on Auslander–Reiten theory are reprinted in .
of finitely generated left modules over R is called an almost-split sequence (or Auslander–Reiten sequence) if it has the following properties:
For any finitely generated left module C that is indecomposable but not projective there is an almost-split sequence as above, which is unique up to isomorphism. Similarly for any finitely generated left module A that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism.
The module A in the almost split sequence is isomorphic to D Tr C, the dual
of the transpose
of C.
for 1 ≤ m < n. The first morphism takes a to (xa, a) and the second takes (b,c) to b − xc.
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s using techniques such as Auslander–Reiten sequences (also called almost split sequences) and Auslander–Reiten quivers. Auslander–Reiten theory was introduced by and developed by them in several subsequent papers.
For survey articles on Auslander–Reiten theory see , , , and the book . Many of the original papers on Auslander–Reiten theory are reprinted in .
Almost-split sequences
Suppose that R is an Artinian algebra. A sequence- 0→ A → B → C → 0
of finitely generated left modules over R is called an almost-split sequence (or Auslander–Reiten sequence) if it has the following properties:
- The sequence is not split
- C is indecomposable and any homomorphism from an indecomposable module to C that is not an isomorphism factors through B.
- A is indecomposable and any homomorphism from A to an indecomposable module that is not an isomorphism factors through B.
For any finitely generated left module C that is indecomposable but not projective there is an almost-split sequence as above, which is unique up to isomorphism. Similarly for any finitely generated left module A that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism.
The module A in the almost split sequence is isomorphic to D Tr C, the dual
Artin algebra
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin.Every Artin algebra is an Artin ring.-Dual and transpose:...
of the transpose
Artin algebra
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin.Every Artin algebra is an Artin ring.-Dual and transpose:...
of C.
Example
Suppose that R is the ring k[x]/(xn) for a field k and an integer n≥1. The indecomposable modules are isomorphic to one of k[x]/(xm) for 1≤ m ≤ n, and the only projective one has m=n. The almost split sequences are isomorphic tofor 1 ≤ m < n. The first morphism takes a to (xa, a) and the second takes (b,c) to b − xc.