Projectivity and flatness over the endomorphism ring of a finitely generated module. (Q1774777)

The paper is motivated by the observation of the authors that the problem of characterizing projective (respectively flat) modules over subrings of fixed elements, or rings of invariants in specific situations [e.g. \textit{J. J. García} and \textit{A. del Río}, Math. Scand. 76, No. 2, 179-193 (1995; Zbl 0842.16023) and \textit{T. Guédénon}, Commun. Algebra 29, No. 10, 4357-4376 (2001; Zbl 0995.16030)], can be obtained from a more general, but elementary, problem of characterizing projective (respectively flat) modules over the endomorphism ring of some finitely generated (respectively finitely presented) module. Given a ring \(A\), a finitely generated right \(A\)-module \(\Lambda\) and setting \(B:=\text{End}(\Lambda_A)\), the first main result, Theorem 2.2., provides some sufficient and some necessary conditions for a right \(B\)-module \(P\) to be projective. In case \(\Lambda_A\) is semi \(\Sigma\)-projective (in particular quasi-projective), it is shown that \(P_B\) is projective if and only if \(P\simeq\Hom_A(\Lambda,Q)\) as right \(B\)-modules for some direct summand \(Q\subseteq_\oplus\Lambda^{(I)}\). Given a ring \(A\), a finitely presented right \(A\)-module \(\Lambda\) and setting \(B:=\text{End}(\Lambda_A)\), the second main result, Theorem 2.3., provides characterizations of a right \(B\)-module \(P\) to be flat. In particular it is shown that \(P_B\) is flat if and only if \(P\simeq\Hom_A(\Lambda,Q)\) as right \(B\)-modules, where \(Q\simeq\varinjlim\Lambda^{n_i}\) for some positive integers \(\{n_i\}_I\). An interesting application in the paper is Example 3.3.: let \(\Lambda\) be a ring, \(({\mathcal C},\Delta,\varepsilon)\) be a \(\Lambda\)-coring with a group-like element \(x\) and consider the left dual ring \(^*{\mathcal C}:={_\Lambda\Hom}({\mathcal C},\Lambda)\) with the convolution product. Then \(\Lambda\) becomes a cyclic \(^*{\mathcal C}\)-module through \(\lambda\leftharpoonup f:=f(x\lambda)\) for all \(\lambda\in\Lambda\) and \(f\in{^*{\mathcal C}}\). Moreover, the endomorphism ring \(B:=\text{End}(\Lambda_{^*{\mathcal C}})\) can be described as \( B=\Lambda^{^*{\mathcal C}}:=\{b\in\Lambda\mid bf(x)=f(xb)\) \(\forall f\in{^*{\mathcal C}}\}\). Sufficient and necessary conditions can be obtained then for the projectivity or flatness of any right \(B\)-module.