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In algebra, Auslander–Reiten theory studies the representation theory of Artinian rings 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 Artin 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 of the transpose of ''C''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Auslander–Reiten theory」の詳細全文を読む スポンサード リンク
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