A question about derived functors

[Jim Kata]

Is it true that if two functors are adjoint, then their derived category functors are adjoint? I'm thinking in particular of ext^n and tor_n. The answer seems like it would be obviously yes to me, but I don't think I've seen it spelled out, and I am too lazy to try and prove it. Is there a theorem saying something like if F and G are adjoint functors in two abelien categories then there nth derived functors are also adjoint to one another.

 

[mathwonk]

i don't think so, after a little reading on wikipedia. namely it says there that every left adjoint functor is right exact. but i seem to recall the only right exact functor that commutes with direct sums is tensor product, since tor also commutes with direct sums, it must not be right exact, hence not a left adjoint. does this seem ok?

 

[Terandol]

Edit: Better answer above. My answer wasn't really applicable to the question because the theorem I quoted requires too many assumptions on the category being localized.

By the way, the theorem mathwonk mentioned which says that all right exact functors (at least functors between categories of modules) which preserve coproducts are naturally isomorphic to a tensor product functor is called the Eilenberg-Watts theorem if you are interested.

 

[Jim Kata]

Thank you for your answer mathwonk. It seems to work.

 

[mathwonk]

uh, yes, eilenberg watts, it goes something like this: write a module M as a quotient of free modules, i.e. direct sums of copies of the ring:

SUM(Ri)-->SUM(Rj)-->M-->0. then do two things: apply the functor F to this sequence, and then separately apply the functor F(R)tensor.

The two results are this: SUM(F(Ri))-->SUM(F(Rj))-->F(M)-->0, and SUM(F(Ri))-->SUM(F(Rj))-->F(R)tensorM-->0. (using the facts that both functors are right exact and commute with direct sums, and that F(R)tensorRi ≈ F(R) ≈ F(Ri), since we are tensoring over R≈Ri.)

Note the two sequences are the same at the left, so they are also the same at the right. I.e. F(M) and FG(R)tensorM are both quotients of the same two

modules, so at least if you believe the maps are the same, they are isomorphic.