For a finite dimensional algebra \(A\) over an algebraically closed field \(K\), let \(S_ 1,\dots, S_ n\) denote the isomorphism classes of simple \(A\)-modules. Assume that the quiver of \(A\) has no oriented cycles and that the numbering of the \(S_ i\) is such that \(\text{Ext}(S_ i,S_ j) = 0\) for \(i > j\). Consider the bilinear form \(( , )\) on \(\mathbb{Z}^ n = \mathbb{Z} S_ 1 \oplus \dots \oplus \mathbb{Z} S_ n\) with \((S_ i, S_ j) = \sum^ 2_{k = 0}(-1)^ k \dim_ K \text{Ext}^ k_ A (S_ i,S_ j)\). On the double cone \(\pm \mathbb{N}^ n \subset \mathbb{Z}^ n\) let \(\sigma_ i\) be the truncated reflection \(\sigma_ i(X) = X - ((X,S_ i) + (S_ i,X))S_ i\) if this expression is in \(\pm \mathbb{N}^ n\), and \(\sigma_ n(X) = X\) otherwise. The author proves the following theorem. Suppose that the Auslander-Reiten quiver of \(A\) has a preprojective component \(C\), and let \(M \in C\) be non-projective with dimension vector \(\underline{\text{dim} }M\), i.e. (\(\underline{\text{dim} }M)_ i\) is the multiplicity of \(S_ i\) in a Jordan-Hölder series of \(M\). If \(\tau M\) denotes the Auslander-Reiten translate of \(M\), then \(\underline{\text{dim} }M = \sigma_ n \dots \sigma_ 1(\underline{\text{dim} }\tau M)\). Similarly, the dimension vectors of the projectives and injectives in \(C\) can be calculated by means of the \(\sigma_ i\).

Reviewer: W.Rump (Eichstätt)