Quasitilted extensions of algebras. I (Q2701620)

By definition, a quasi-tilted algebra \(A\) is the endomorphism ring of a tilting object in a hereditary category. If \(A\) is not simple, then it can be written as a one-point extension \(A=B[M]\) for some (smaller) quasi-tilted algebra \(B\). Conversely, \textit{D. Happel, I. Reiten} and \textit{S. O. Smalø} [Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011)] have observed that whenever \(A\) can be written as \(A=B[M]\), then \(B\) must be quasi-tilted.NEWLINENEWLINENEWLINEThe paper under review, and its second part [J. Algebra 227, No. 2, 582-594 (2000; Zbl 1064.16010)], study in more detail the following situation: The connected quasi-tilted algebra \(A=B[M]\) is a one-point extension with the module \(M\) not being indecomposable. It is shown that every indecomposable direct summand of \(M\) must be directing, and moreover \(A\) itself must be tilted, not just quasi-tilted. To prove the statement on the summands of \(M\), Galois coverings are used which are multiple one-point extensions of a product of copies of \(A\).