## 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

### MSC:

 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

### Citations:

Zbl 0428.16029; Zbl 0444.16019; Zbl 0448.16019; Zbl 0404.16007