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By V. B Dlab

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T / -+ (sk sk x )k I I o with o: = we have ker o= ker t(= imu), kej. ,ta, there exisfs pk Xkhrith p1t = o. , with t(b) define then p = ( or) Now, if that =3= q. 5o = . gi tU", r ti ti X + o is not surjective, I=LOP, r,rherey= for k +i € f , P€ L ( ": " :\ = a. t(b) k + i€r so t xi = (si X is a monomorphism. 3) r 1 ,{ A ) w i t h y i = X i j€ 1), and surn of copies of Fn, in 34 - particular projective. is , then -K indecornposable, K then of composition =o. o=oandslsln. o has any two maps be surjective is bijective, is I Finalfy, ls rhus trre + ena(si s* si lt Iil snd (S-x) e n c l( x ) ti -K in K' E"d1s+x1 -5- nnd(si si ll-5 Iil is a Iocal hence and s[s; sf, I.

E" + E^lq tz nonisomorphic with is x) is The Coxeter (Xr, respect (X,X), Extl lengths 2 a subcategory' forn of not finite tYPe. an End(X)-binodule. (X, This x) -having is E1, over ({) E2 are of End € End (x) (x) } two is indecomposable all on the objects of over be uniserial. to set of This Ext'(X,X) is also L (X) of X by is with X linearly then left, a infinite objects dimensions the (X, X),then extensions say If simple one dimensional some m > 1. for subcategory 11 one only L (x) implies End(X)-binodule the semisimple.

Es of F*. r analogous statements hold also. a) It is definition enough to consider the k-th component. t / -+ (sk sk x )k I I o with o: = we have ker o= ker t(= imu), kej. ,ta, there exisfs pk Xkhrith p1t = o. , with t(b) define then p = ( or) Now, if that =3= q. 5o = . gi tU", r ti ti X + o is not surjective, I=LOP, r,rherey= for k +i € f , P€ L ( ": " :\ = a. t(b) k + i€r so t xi = (si X is a monomorphism. 3) r 1 ,{ A ) w i t h y i = X i j€ 1), and surn of copies of Fn, in 34 - particular projective.

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