Math 2.0

Associative rings and algebras Definition

Auslander–Reiten theory

Builds on Associative algebraModule over a ringIndecomposable moduleProjective moduleAlmost split sequenceAuslander–Reiten quiver

Definition

Let kk be a field and let AA be a finite-dimensional associative algebra over kk. Let mod⁡A\operatorname{mod} A be the category of finitely generated left AA-modules.

  1. A morphism f ⁣:M→Nf \colon M \to N between indecomposable modules is irreducible if it is neither a split monomorphism nor a split epimorphism, and the following holds. In every factorization f=ghf = g h, either hh is a split monomorphism or gg is a split epimorphism.
  2. For MM in mod⁡A\operatorname{mod} A with minimal projective presentation P1→pP0→M→0P_1 \xrightarrow{p} P_0 \to M \to 0, the transpose Tr⁡M\operatorname{Tr} M is the cokernel of Hom⁡A(p,A) ⁣:Hom⁡A(P0,A)→Hom⁡A(P1,A)\Hom_A(p, A) \colon \Hom_A(P_0, A) \to \Hom_A(P_1, A), a right AA-module.
  3. The Auslander–Reiten translation is τM=DTr⁡M\tau M = D \operatorname{Tr} M, where D=Hom⁡k(⋅,k)D = \Hom_k(\cdot, k).

Auslander–Reiten theory is the study of mod⁡A\operatorname{mod} A through irreducible morphisms, the translation τ\tau, almost split sequences and the Auslander–Reiten quiver.

Remarks

The basic existence theorem: if MM is indecomposable and not projective, there is an almost split sequence 0→τM→E→M→00 \to \tau M \to E \to M \to 0. Dually, for NN indecomposable and not injective there is an almost split sequence starting at NN, and it ends at τ−1N=Tr⁡DN\tau^{-1} N = \operatorname{Tr} D N. Example: for the path algebra of 1→21 \to 2, with simple modules S1,S2S_1, S_2 and the projective cover P1P_1 of S1S_1, one has τS1=S2\tau S_1 = S_2. The almost split sequence is 0→S2→P1→S1→00 \to S_2 \to P_1 \to S_1 \to 0. The translation τ\tau sends projective modules to 00. The theory works more generally for Artin algebras over a commutative Artinian ring RR, with D=Hom⁡R(⋅,E)D = \Hom_R(\cdot, E) for the injective envelope EE of R/rad⁡RR/\operatorname{rad} R.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Almost split sequence, concept, no evidence levelAlmost split sequenceconceptAssociative algebra, concept, no evidence levelAssociative algebraconceptAuslander–Reiten quiver, concept, no evidence levelAuslander–Reiten quiv…conceptIndecomposable module, concept, no evidence levelIndecomposable moduleconceptModule over a ring, concept, no evidence levelModule over a ringconceptProjective module, concept, no evidence levelProjective moduleconceptAuslander–Reiten theory, definition, no evidence levelAuslander–Reiten theo…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost split sequence concept
  2. Associative algebra concept
  3. Auslander–Reiten quiver concept
  4. Indecomposable module concept
  5. Module over a ring concept
  6. Projective module concept
  7. Auslander–Reiten theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Auslander%E2%80%93Reiten_theory
About this page
Tag40F8KNF8Q1
URIm2://40F8KNF8Q1
Kinddefinition
WritingW0 stub
MSC16G70, 16G10
DefinesAuslander–Reiten theory
UsesAssociative algebra, Module over a ring, Indecomposable module, Projective module, Almost split sequence, Auslander–Reiten quiver
Changed2 November 2026

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