On tense modules for extended algebras over rational fields. (Q2572614)

Let \(\Lambda\) be a finite dimensional algebra over a field \(k\) and \(J\) the radical of \(\Lambda\). Denote by \(P^1(\Lambda)\) the category of \(\Lambda\)-module homomorphisms \(\alpha\colon P_1\to P_2\) such that \(P_1,P_2\) are projective modules and \(\alpha(P_1)\subseteq JP_2\). A minimal projective presentation of a \(\Lambda\)-module \(M\) detemines an object \(\alpha_M\colon P_M\to Q_M\) in \(P^1(\Lambda)\). A \(\Lambda\)-module \(M\) is tense if \(\text{Ext}_{P^1(\Lambda)}(\alpha_M,\alpha_M)=0\). Let \(k(X)\) be the field of rational functions in one variable \(X\). Then any tense module \(M\) over \(\Lambda\otimes k(X)\) is extended from \(\Lambda\) in the sense that \(M\simeq N\otimes_kk(X)\) for some \(\Lambda\)-module \(N\). In particular there is a bijection between isoclasses of tilting modules over \(\Lambda\) and over \(\Lambda\otimes k(X)\).