# On faithfully flat and faithfully projective modules

Let $$R$$ be a commutative Noetherian ring. Let $$P,Q$$ be some $$R$$-modules such that $$-\otimes_R P$$ and $$Hom_R(Q,-)$$ are faithfully exact functors i.e., for any sequence of modules

$$A \xrightarrow{f} B \xrightarrow{g} C$$ , we have

$$A \xrightarrow{f} B \xrightarrow{g} C$$ is exact if and only if $$A\otimes_RP \xrightarrow{id\otimes_f} B\otimes_R P \xrightarrow{id\otimes g} C\otimes_RP$$ if and only if $$Hom(Q,A) \xrightarrow{hom(f)} Hom(Q,B) \xrightarrow{hom(g)} Hom(Q,C)$$ is exact. Also note that all this if and only if conditions are equivalent to saying : $$P$$ is faithfully flat and $$Q$$ is projective and $$Hom(Q,X)\ne 0$$ for every $$R$$-module $$X$$, and such $$Q$$ is also called faithfully projective.

My question is: Under the above conditions, when can we say that the functors $$-\otimes_R P$$ and $$Hom_R(Q,-)$$ are naturally isomorphic ? What are some examples when they are not isomorphic ? Is it true that if they are isomorphic, then $$P\cong Q$$ ?

Another, much more concrete question : Is it true that for every faithfully flat module $$P$$, there exists a faithfully projective module $$Q$$ such that the functors $$-\otimes_R P$$ and $$Hom_R(Q,-)$$ are naturally isomorphic ?

(NOTE: Of course if they are isomorphic, then $$P\cong Hom_R(Q,R)$$ )

Obviously , the functors $$-\otimes_R P$$ and $$Hom_R(Q,-)$$ are naturally isomorphic when $$P\cong Q$$ is free of finite rank. Apart from that I don't know any examples. Also, I would like to see some canonical counter-examples when they are not isomorphic .

Thanks

• If I have not misunderstood, take $R$ to be a local Noetherian ring, $P$ its completion and $Q=R$. If $R$ is not complete, clearly $Q$ and $P$ are not isomorphic. – Mohan May 18 at 2:33
• A simple example where the functors are not isomorphic is where $P$ and $Q$ are free of different nonzero finite ranks. – Jeremy Rickard May 18 at 10:34
• It’s not exactly a duplicate, but I think that most of your questions are answered here: math.stackexchange.com/questions/2522672/… – Jeremy Rickard May 18 at 10:40