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Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and cocontinuous (in particular, exact). Since $R\text{-Mod}$ is both complete and cocomplete, $\operatorname{Forget}$ has both a left adjoint $\operatorname{Free}: \text{AbGp} \to R\text{-Mod}$ and a right adjoint $\operatorname{Cofree}: \text{AbGp} \to R\text{-Mod}$.

You can see what these functors are explicitly. Let me write $_R R_{\mathbb Z}$ for "$R$ as a left module" and $_{\mathbb Z} R _R$ for "$R$ as a right module". The $\operatorname{Forget}$ functor is (isomorphic to) the functor $\operatorname{Hom}_R({_R R_{\mathbb Z}},-)$ — this description makes it clearly continuous, and its left adjoint is $\operatorname{Free} \cong {_R R_{\mathbb Z}} \otimes_\mathbb Z (-)$. But we also have $\operatorname{Forget} \cong {_{\mathbb Z} R _R}\otimes_R (-)$, whence its right adjoint is $\operatorname{Cofree} \cong \operatorname{Hom}_{\mathbb Z}({_{\mathbb Z} R _R},-)$.

I feel like I have some positive amount of experience with free modules. (I would say, given the above, that the correct definition of "free module" is "object in the essential image of $\operatorname{Free}$", although what's actually used is "object of the form $\operatorname{Free}(\mathbb Z^{\oplus \kappa})$ for some cardinal $\kappa$.) But I hardly ever come across the essential image of $\operatorname{Cofree}$, or indeed the cofree functor at all. (Again, maybe the "standard" definition of "cofree module" is "module isomorphic to $\operatorname{Cofree}((\mathbb Q/\mathbb Z)^{\times \kappa})$," or something.) The functors are not the same: when $ R = \mathbb Z/2$, then $\operatorname{Free}(\mathbb Z) = \mathbb Z/2$, whereas $\operatorname{Cofree}(\mathbb Z) = 0$. If you would rather replace $\mathbb Z$ by a field throughout, then they are still not the same when $R$ is infinite-dimensional (for example).

So: Do people use cofree modules? If so, how? If not, why not? Are free modules just a lot nicer than cofree ones, and if so, how?

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Hilton and Stammbach has a section titled Cofree Modules, iirc, though I don't have my copy here. On the other hand, I would take both "free" and "cofree" (without further qualifications) to mean the (essential) images of the appropriate adjoints to the forgetful functor into Sets, not AbGp (although of course, at least in the free case, it comes down to the same thing). –  Peter LeFanu Lumsdaine Sep 8 '10 at 19:15
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up vote 5 down vote accepted

This construction is used frequently (at least, I use it frequently in my work). For example, it appears in the usual proof that module categories have enough injectives. (In this case one studies $Cofree(\mathbb Q/\mathbb Z)$, as you anticipated.)

If we generalize slightly, and replace $\mathbb Z$ by the group ring $k[H]$ and $R$ by the group ring $k[G]$ (with $H$ being a subgroup of $G$), then $Hom_{k[H]}(k[G],\text{--})$ is precisely the functor of induction from $H$-representations to $G$-representations, and the adjointness you note is a form of Frobenius reciprocity.

If $R$ is a Hecke algebra (over $\mathbb Z$) on a space of weight $k$-cuspforms of some level, then $Cofree(\mathbb Z)$ is the space of modular forms of weight $k$ with coefficients in $\mathbb Z$. (This technical relationship between Hecke operators and the space of modular forms on which they operate is used frequently by number theorists working on the arithmetic of modular forms.)

There are lots of other contexts in which this functor (and its variants, replacing $\mathbb Z$ by other rings) appear, but maybe I've said enough for now.

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Matt, you're not a number theorem (see end of 3rd paragraph) –  KConrad Sep 8 '10 at 23:51
    
Thanks Keith! (Now fixed.) –  Emerton Sep 9 '10 at 0:25
    
For the beginners, it might be worth pointing out that many people use the term "co-induction" for the cofree functor, instead of "induction", and use "induction" for the free functor. And I think some people even do both ways, though in different books. –  JBorger Sep 9 '10 at 8:43
    
Dear Jim, Good point. My (somewhat vague) impression is that people focussing on finite groups use induction for the free functor and coinduction for the cofree functor, while those doing topological groups, unitary reps. of Lie groups, and so on use induction for the cofree functor (and typically it is not precisely the cofree functor that is used, but some variant that takes into account topologies on the various objects involved). –  Emerton Sep 9 '10 at 15:11
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