Definition 15.55.5. Let $R$ be a ring.

1. For any $R$-module $M$ over $R$ we denote $M^\vee = \mathop{\mathrm{Hom}}\nolimits (M, \mathbf{Q}/\mathbf{Z})$ with its natural $R$-module structure. We think of $M \mapsto M^\vee$ as a contravariant functor from the category of $R$-modules to itself.

2. For any $R$-module $M$ we denote

$F(M) = \bigoplus \nolimits _{m \in M} R[m]$

the free module with basis given by the elements $[m]$ with $m \in M$. We let $F(M)\to M$, $\sum f_ i [m_ i] \mapsto \sum f_ i m_ i$ be the natural surjection of $R$-modules. We think of $M \mapsto (F(M) \to M)$ as a functor from the category of $R$-modules to the category of arrows in $R$-modules.

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