# The Stacks Project

## Tag 08WQ

Definition 34.4.9. Let $R$ be a ring. Define the contravariant functor $C$ $: \text{Mod}_R \to \text{Mod}_R$ by setting $$C(M) = \mathop{\rm Hom}\nolimits_{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),$$ with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 942–950 (see updates for more information).

\begin{definition}
\label{definition-C}
Let $R$ be a ring. Define the contravariant functor
{\it $C$} $: \text{Mod}_R \to \text{Mod}_R$ by setting
$$C(M) = \Hom_{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),$$
with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.
\end{definition}

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