Let \(k\) be an algebraically closed field and let \(A=k\vec\Delta\) be the path algebra of a finite connected quiver \(\vec\Delta\) without oriented cycles and with \(n+1\) vertices. An \(A\)-module \(T\) is called tilting provided: (a) \(\text{Ext}^1_A(T,T)=0\) and (b) the number \(\delta(T)\) of the nonisomorphic indecomposable direct summands of \(T\) equals \(n+1\). The set of tilting modules over \(A\) forms a simplicial complex \(\Sigma\), the zero simplices of \(\Sigma\) being the isomorphism classes of indecomposable direct summands of tilting modules, and \(\{T_0,\dots,T_n\}\) span an \(r\)-simplex whenever \(\text{Ext}^1_A(\oplus T_i,\oplus T_i)=0\). One can also assign a quiver \(\vec K_0\) as follows. The vertices of \(\vec K_0\) are the isomorphism classes of the multiplicity-free tilting \(A\)-modules and there exists an arrow \(T\to T'\) if \(T\) and \(T'\) have a common direct summand \(M\) with \(\delta(M)=\delta(T)-1\) and \(\text{Ext}^1_A(T,T')=0\). In this paper the authoress studies the complex \(\Sigma\) concerning the representation type of the given algebra \(A\) and the number of one-dimensional links in the corresponding \(\Sigma\). She also studies the shape of full subquivers of \(\vec K_0\) with vertices the \(n\)-simplices with a given face \(\{M_0,\dots,M_{n-2}\}\).

Reviewer: F.U.Coelho (São Paulo)