The Stacks project

Example 56.3.2. Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. Let $K$ be a $A \otimes _ R B$-module. Then we can consider the functor
\begin{equation} \label{functors-equation-FM-modules} F : \text{Mod}_ A \longrightarrow \text{Mod}_ B,\quad M \longmapsto M \otimes _ A K \end{equation}

This functor is $R$-linear, right exact, commutes with arbitrary direct sums, commutes with all colimits, has a right adjoint (Lemma 56.3.1).

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