# When does the forgetful functor from modules to vector spaces have a right adjoint?

Given any algebra $$R,$$ when does the forgetful functor $$R\text{-}Mod \rightarrow Vec$$ have a right adjoint? Does this imply any finiteness conditions on R? Is there a book/paper discussing this?

I've assumed $$R$$ is $$k$$ algebra where $$k$$ is a field. but if $$k$$ is not a field, and just a commutative ring then Marco's answer should hold up still with replacing $$Vec$$ by $$k-Mod$$.

• I don't understand why this forgetful functor exists, e.g. if $R=\mathbb{Z}$. Are you assuming $R$ is an algebra over some field $k$? And Vec is the category of vector spaces over $k$? – Alex Kruckman May 23 '19 at 0:35
• I have written algebra in the question, just not the $k$s. – AMaths May 24 '19 at 7:36
• Well, "algebra" doesn't always mean algebra over a field. – Alex Kruckman May 24 '19 at 18:00
• Technically Very true, in my mind it always is, Thank you for pointing that out! – AMaths May 24 '19 at 19:23
• I beleive marco's answer still holds thou – AMaths May 24 '19 at 19:24

Hom is right adjoint to tensor product. Have you tried to write the forgetfull functor as $$F(M)=R\otimes_R M$$?
$$Hom_k(Forget(M),W)=Hom_k(M,W)\cong Hom_k(R\otimes_RM,W)\cong Hom_R(M,Hom_k(R,W))$$ where $$M$$ is an $$R$$-module, $$W$$ is a vector space, and the $$R$$-structure on $$Hom(R,W)$$ is from the right structure of $$R$$, that is $$(r\cdot f)(x)=f(xr)$$.
• Are you aware of a similar construction for the forgetful functor from $Forg : R-Comod \rightarrow Vec$ having a LEFT adjoint? – AMaths May 21 '19 at 20:37
• mmm.. that seems more dificult. Now $R$ is a coalgebra, if $M$ is a (right) comodule then $M$ is a (left) $R^*$-module, and – Marco Farinati May 21 '19 at 21:30