Irreducible morphisms and the radical of a category. (English) Zbl 0594.16020

Let \({\mathcal C}=mod(R)\), where R is a finite dimensional K-algebra. We recall that a map \(f: X\to Y\) in \({\mathcal C}\) is called irreducible if f can not be performed in the form \(f=gh\), where h is not a splittable monomorphism and g is not a splittable epimorphism. The aim of this paper is to study irreducible maps in \({\mathcal C}\) in terms of the radical rad \({\mathcal C}\) of \({\mathcal C}\). We recall that given X and Y in \({\mathcal C}\) we have \[ rad(X,Y)=\{f: X\to Y| \quad 1_ X-gf\quad is\quad invertible\quad for\quad all\quad g: Y\to X\}. \] It is proved that if \(X=C_ 1\oplus..\oplus C_ n\) and \(C_ 1=...=C_ n=C\), Y are indecomposable then \(f=(f_ 1,...,f_ n): X\to Y\), with \(f_ i: C_ i\to Y\), is irreducible iff \(\bar f_ 1,...,\bar f_ n\in rad(C,Y)/rad^ 2(C,Y)\) are linearly independent over \(K_ C=End(C)/rad End(C).\) Similarly almost split sequences in \({\mathcal C}\) are characterized.
Let I(X,Y) be the set of all \(\bar g\in rad(X,Y)/rad^ 2(X,Y)\), where \(g: X\to Y\) runs through all irreducible maps, and let \(K^*_ C\) be the group of units in \(K_ C\). The author studies the algebraic variety I(X,Y) with an obvious action of the algebraic group \(G_{XY}=K^*_ X\times K^*_ Y\). It is proved that I(X,Y) is an open subvariety of \(rad(X,Y)/rad^ 2(X,Y)\) and if R is of finite representation type then I(X,Y) has finitely many \(G_{XY}\)-orbits. It is concluded that \(\dim_ KI(X,Y)\leq 1\) for X,Y indecomposable, provided K is algebraically closed. Moreover, if \(0\to X\to Z\to Y\to 0\) is an almost split sequence in C and \(Z=Z_ 1^{n_ 1}\oplus..\oplus Z_ t^{n_ t}\), where \(Z_ 1,...,Z_ t\) are pairwise nonisomorphic and indecomposable, then \(n_ 1,...,n_ t\leq 3\) and if \(n_ i\geq 2\) then \(n_ j=1\) for \(j\neq i.\)
The author also studies \({\mathcal C}\) modulo \(rad^{\infty}(C)=\cap^{\infty}_{n=1}rad^ n(C)\) and the graph \(\Gamma_ R\) of \({\mathcal C}\) whose points are isoclasses of indecomposables in \({\mathcal C}\) and there is an arrow [X]\(\to [Y]\) in \(\Gamma_ R\) iff I(X,Y)\(\neq 0\). The shapes of \(\Gamma_ R\) are presented for hereditary algebras R of finite representation type corresponding to Dynkin diagrams with fixed orientations.
Part of results in this paper was announced by the author [in Bull. Am. Math. Soc., New. Ser. 2, 177-180 (1980; Zbl 0428.16029)]. Part of them was also presented by other authors [see C. M. Ringel in Lect. Notes Math. 831, 104-136, 137-287 (1980; Zbl 0444.16019 and Zbl 0448.16019)]. The diagram \(\Gamma_ R\) was introduced earlier by M. Auslander [in Lect. Notes Pure Appl. Math. 37, 245-327 (1978; Zbl 0404.16007)].
Reviewer: D.Simson


16Gxx Representation theory of associative rings and algebras
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
14L30 Group actions on varieties or schemes (quotients)
16P10 Finite rings and finite-dimensional associative algebras