A remark on the Morita theorem for operads. (Q2897386)

The author extends a ``Morita theorem'' of \textit{M. M. Kapranov} and \textit{Y. Manin} [Am. J. Math. 123, No. 5, 811--838 (2001; Zbl 1001.18004)] for algebras over operads in \(k\)-modules for some commutative ring \(k\). This extension is concerned with \(A\)-modules which are symmetric sequences equipped with an action of an operad \(A\). These generalize the notion of \(A\)-algebras. Let \(P\) be a right \(A\)-module which is a direct summand of a finitely generated free \(A\)-module and which admits an epimorphism \(\bigoplus _\Lambda P\rightarrow A\). Then it is proved that the categories of (left or right) modules over \(A\) and over the \(A\)-endomorphism operad of \(P\) are equivalent. If \(P\) is finitely generated free the result of \textit{M. M. Kapranov} and \textit{Y. Manin} [Am. J. Math. 123, No. 5, 811--838 (2001; Zbl 1001.18004)] for algebras is recovered. In this special case, the author also proves a version for cyclic algebras over a cyclic operad.