\(I\)-sequences in the category of modules (Q1604354)

Let \(k\) be a field and \(\Lambda\) a finite dimensional \(k\)-algebra. If for all \(X,Y\in\text{mod }\Lambda\), \(I(X,Y)\) is an ideal of \(\Hom_\Lambda(X,Y)\), then \(I\) is called an ideal of \(\text{mod }\Lambda\). A short exact sequence \[ 0\to K@>f>>M@>g>>V\to 0 \] in \(\text{mod }\Lambda\) is called a right \(I\)-sequence, if for all \(X\in\text{mod }\Lambda\), \(\text{Im}\Hom(X,g)=I(X,V)\). A particular case of this concept is that of an Auslander-Reiten sequence. Left and bilateral \(I\)-sequences are also defined. The author investigates in detail the so called minimal \(I\)-sequences. The general results are then applied to the case when \(\Lambda\) is the group algebra \(kG\) of the finite group \(G\), and \(I(X,Y)\) is the ideal of maps which are projective relative to a set \(K\) of subgroups of \(G\). The author obtains an alternative proof for a result of R.~Knörr on the existence and properties of relatively \(K\)-projective covers of \(kG\)-modules.