If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. Stephen Donkin's article Hopf complements and injective comodules for algebraic groups, section 3, he shows there is a right adjoint to the restriction. I am wondering if there is also a left-adjoint to the restriction?
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$\begingroup$ Doesn't taking the cotensor product with C over D suffice? $\endgroup$– user43326Commented Jun 25, 2014 at 16:50
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$\begingroup$ If I am not mistaken, that gives the right adjoint. $\endgroup$– Peter PatztCommented Jun 25, 2014 at 16:52
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$\begingroup$ Are we over a field? $\endgroup$– TilmanCommented Jul 27, 2014 at 18:15
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$\begingroup$ Yes everything is over a field. I am not interested in other cases right now. $\endgroup$– Peter PatztCommented Jul 27, 2014 at 18:19
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1 Answer
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If we only consider coalgebras over a fixed ground field, then Takeuchi's article Morita theorems for categories of comodules seem to prove the existence of left-adjoints of the restriction.
In 1.4 he defines cohomorphisms of $C$-comodules $\operatorname{Cohom}_C(V,V')$ for quasi-finite $V$, that is $\operatorname{Hom}_C(W,V)$ is finite dimensional for every finite dimensional $C$-comodule $W$. He then proves in 1.10 that if $V$ is a $(D,C)$-bicomodule, $\operatorname{Cohom}_C(V,-)$ is the left-adjoint to the cotensor product $-\square^D V$.
We figure that if $C$ is a quasi-finite $D$-comodule, we have with $\operatorname{Cohom}_D(C,-)$ the left-adjoint to $-\square^C C$, i.e. $$\operatorname{Hom}_C(\operatorname{Cohom}_D(C,V),W) \cong \operatorname{Hom}_D(V,W\square^CC)\cong \operatorname{Hom}_D(V,\operatorname{Res} W).$$
The only question that remains is, for $f\colon C\to D$ when is $C$ a quasi-finite $D$-comodule. But that can be left to the comments or to a new question if urgent.