Foreigners Riding Quiver

The foreigner riding quiver - named after Maurice foreigner (1926–1994) and Idun riding (* 1942) - is a translation quiver that is used for the combinatorial description of Abelian categories . He was originally introduced to the category of representations of a quiver or - more generally - of modules over Artin - algebras to describe.

Definition for the category of representations of a quiver

Be a body and an acyclic quiver. Let the category of representations of the quiver over the body . Then the points of the foreigner riding quiver are the isomorphism classes of the indecomposable representations in . The arrows between two points and are defined as follows: Select a representative and select a representative . The arrows from to form a basis of the space of the irreducible mappings from to . The translation is a mapping of a subset of the points in into the set of points . For every point , the elements of which are not projective, there is a foreigner-riding sequence (almost disintegrating, short exact sequence ) of the form with . Then is . ${\displaystyle k}$ ${\displaystyle Q}$ ${\displaystyle C}$ ${\displaystyle k}$ ${\displaystyle \Gamma _{C}}$ ${\displaystyle C}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \tau }$ ${\displaystyle \Gamma _{C}}$ ${\displaystyle \Gamma _{C}}$ ${\displaystyle z}$ ${\displaystyle 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0}$ ${\displaystyle Z\in z}$ ${\displaystyle \tau (z)=x}$

Correspondingly, there is also a foreigner-riding sequence of the form with for every point whose elements are not injective . ${\displaystyle x}$ ${\displaystyle 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0}$ ${\displaystyle X\in x}$

Explanations

The foreigner riding quiver provides a description of the objects in the category on the basis of the theorem of Krull-Remak-Schmidt (every non-trivial representation of a quiver is the direct sum of indivisible representations) . ${\displaystyle C}$

If is representational, every non-trivial mapping can be decomposed as a composition of finitely many irreducible maps. Therefore, in this case, the foreigner riding quiver also provides a description of the morphisms . ${\displaystyle C}$

literature

• M. Auslander, I. Riding: Representation theory of artin algebras III. Almost split sequences , Comm. Algebra 3: 239-294 (1975)
• M. Auslander, I. Riding, SO Smalø: Representation theory of artin algebras , Cambridge Studies in Advanced Mathematics 36, Cambridge University Press (1994)
• Karsten Schmidt: foreigner riding theory for simply connected differential graded algebras . Dissertation, University of Paderborn 2007