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

8 events

when toggle format	what		by	license	comment
May 21, 2019 at 22:09	comment	added	AMaths		It is equivalent to R being finitely projective as a k module
May 21, 2019 at 22:09	comment	added	AMaths		I believe the answer to this question is in Prop 31 of projecteuclid.org/download/pdf_1/euclid.pjm/1102868049
May 21, 2019 at 21:37	comment	added	Marco Farinati		mmm.. that seems more dificult. Now $R$ is a coalgebra, if $M$ is a (right) comodule then $M$ is a (left) $R^$-module via $\phi\cdot m=\phi(m_1)m_0$, and $Hom^R(N,M)=Hom_{R^}(N,M)$ for every pair of comodules $M,N$. On the other side, we know that $$Hom_k(V,M)\cong Hom_{R^}(R^\otimes V,M)$$ but $R^\otimes V$ is not rational in general (i.e. is not an $R^$-module comming from an $R$-module, unless $R$ is finite dimensional), so, if it exists, I could try $(R^*\otimes V)_{rat}$, maybe $R^0\otimes V$, but it looks hard to work.
May 21, 2019 at 21:30	comment	added	Marco Farinati		mmm.. that seems more dificult. Now $R$ is a coalgebra, if $M$ is a (right) comodule then $M$ is a (left) $R^*$-module, and
May 21, 2019 at 20:37	comment	added	AMaths		Are you aware of a similar construction for the forgetful functor from $Forg : R-Comod \rightarrow Vec$ having a LEFT adjoint?
May 21, 2019 at 20:33	comment	added	AMaths		Thank you Marco, I thought Hom(R,-) would work but couldn’t think of a good reason why.
May 21, 2019 at 16:51	history	edited	Marco Farinati	CC BY-SA 4.0	added 312 characters in body
May 21, 2019 at 16:33	history	answered	Marco Farinati	CC BY-SA 4.0