On functors with given derivatives (Q1261268)

For a functor \(F: {_ \Lambda\text{Mod}}\to{_ \mathbb{Z}\text{Mod}}\) from the category of \(\Lambda\)-modules to the category of Abelian groups let \(L_ i F\) (resp. \(R_ i F\)) denote the \(i\)th left (resp., right) derived functor of \(F\). The author examines the following (A. V. Jakovlev's) property of \(\Lambda\): For given \(\Lambda\)-modules \(A\) and \(B\) there is a covariant functor \(_ B F_ A: { _ \Lambda\text{Mod}}\to {_ \mathbb{Z}\text{Mod}}\) such that \(L_{iB} F_ A=\text{Tor}_ \Lambda^ i(A,-)\) and \(R_{iB} F_ A=\text{Ext}^ i_ \Lambda(B,-)\) for \(i=1,2,\dots\;\). Then some sufficient (but not necessary) nice conditions for rings \(\Lambda\) satisfying that property are presented.