Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth $\mathbb{C}$-linear representations of $K$ preserve projective objects?

One is tempted to make an argument using the right adjoint $\mathrm{Ind}_K^G$, right exactness of which would suffice. But does the openness of $K$ imply the right exactness of the induction functor $\mathrm{Ind}_K^G$?


The induction functor is actually exact, not just right-exact, so yes, you can do this. Seeing as you've essentially asked two questions about the induction functors on these groups, you might want to take a look at Bushnell--Henniart's Local Langlands for GL2; section 2 of this gives a pretty simple introduction to the two functors and their basic properties.

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  • $\begingroup$ Thanks, I didn't realize that this was in Bushnell-Henniart. Inspecting the proof there, I see that my assumptions were too restrictive: the same conclusion holds even if $K$ is merely closed (instead of being open) and with $\mathbb{C}$ replaced by any commutative ring $R$ such that $G$ has a compact open subgroup with pro-order in $R^*$. $\endgroup$ – Question Mark Oct 24 '14 at 16:26

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