Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am referring to Theorem 6.3 p. 143 from Lang's Algebra (but the question description will be as self-contained as possible).

Let $A$ be a commutative ring and $M$, $W$, $V$, $U$ be $A$-modules. Let the sequence $$0 \longrightarrow W \stackrel{\lambda}{\longrightarrow} V \stackrel{\phi}{\longrightarrow} U \longrightarrow 0 $$ be exact.

By Proposition 2.1 p. 122, the induced sequence $$0\longrightarrow Hom_A(U,M) \stackrel{\phi'}\longrightarrow Hom_A(V,M) \stackrel{\lambda'}\longrightarrow Hom_A(W,M) $$ is exact. If we let $M$ be $A$ as a module over itself, then we obtain $$0 \longrightarrow Hom_A(U,A) \stackrel{\phi'}{\longrightarrow} Hom_A(V,A) \stackrel{\lambda'}{\longrightarrow} Hom_A(W,A)$$ exact or in Lang's notation $$0 \longrightarrow U^{\vee} \stackrel{\phi'}{\longrightarrow} V^{\vee} \stackrel{\lambda'}{\longrightarrow} W^{\vee},$$ where $V^{\vee}$ is the dual module of $V$.

Now, this is where i have a problem: it is mentioned in the proof of theorem 6.3 that since $A$ is projective (because it is free), then we also have exactness from the right, i.e. $$0 \longrightarrow U^{\vee} \stackrel{\phi'}{\longrightarrow} V^{\vee} \stackrel{\lambda'}{\longrightarrow} W^{\vee} \longrightarrow 0$$ is exact. I don't see where this exactness from the right comes from since $A$ being projective means that the functor $Hom_A(A,\cdot)$ is exact, while to obtain dual spaces we use the functor $Hom_A(\cdot,A)$.

Any insights? Thank you:-)

share|cite|improve this question
up vote 1 down vote accepted

You are missing that Lang assumes that the $A$-modules $W,V,U$ are finite free. The exactness of the dual sequence is then immediate.

share|cite|improve this answer
I am studying the same problem after several months, i got stuck exactly at the same point and now i can not see why the exactness is immediate. Could you please give me a hint? Thanks. – Manos Dec 10 '11 at 18:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.