Example 56.3.2 (0GNT)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 56: Functors and Morphisms
Section 56.3: Functors between categories of modules
Example 56.3.2 (cite)

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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

56.3.2.1

$\begin{equation} \label{functors-equation-FM-modules} F : \text{Mod}_ A \longrightarrow \text{Mod}_ B,\quad M \longmapsto M \otimes _ A K \end{equation}$

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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