# Rigidity of finitely generated projective modules

I am currently reading a lecture notes on module theory and more specifically on Morita Theory. Here is a porition of the lecture note that I do not understand.

Finitely generated projective modules are `rigid' in the following sense. Let $$A$$ be a ring and $$P$$ be a finitely generated projective left $$A$$-module. Then its $$A$$-dual $$P^{\vee} := \textrm{Hom}_A(P,A)$$ is a finitely generated projective right $$A$$-module (by additivity) and for every left $$A$$-module $$M$$ the natural homomorphism $$P^{\vee} \otimes_A M \to \textrm{Hom}_A(P,M)$$ given by $$f \otimes m \mapsto (p \mapsto f(p)m)$$ is an isomorphism. Also, we have a natural transformation $$\overline{\omega}_P : P \to (P^{\vee})^{\vee}$$ given by the usual formula $$x \mapsto \textrm{ev}_x$$ where $$\textrm{ev}_x : P^{\vee} \to A$$ with $$f \mapsto f(x)$$ is the evaluation map. As this is an isomorphism for $$P = A$$ and as both sides are additive, it follows that $$P$$ is an isomorphism for any finitely generated projective $$A$$-module $$P$$.

I do not get what the author means by "$$P^\vee$$ is a finitely generated projective right $$A$$-module by additivity. I think this relates to the fact that a module $$P$$ is projective iff it's a direct summand of a free module. Also, why is it that a module $$P$$ is projective and finitely generated iff it's a direct summand of a finitely generated free module? I also am not sure how the map $$f \otimes m \mapsto (p \mapsto f(p)m)$$ is an isomorphism. I have proven the case when $$P$$ is a free $$A$$-module of rank $$n$$, but I am not sure how to write the explicit inverse for the map when $$P$$ is a finitely generated projecive module. Likewise, I get that the map $$x \mapsto \textrm{ev}_x$$ is an isomorphism for $$P = A$$, but I am not sure how to generalize.

Lastly, what is the intuition behind the author's use of the term "rigid"? Can you provide intuitive description of why he uses this term?

Let us start with the assertion that $$P$$ finitely generated and projective if and only if $$P$$ is a direct summand of a finitely generated free module. Of course, if $$P$$ is a summand of any free module, it is projective. Conversely, assume that $$P$$ is finitely generated and projective. Then we find a surjection $$F\to P$$ for some finite free module $$F$$. Completing this surjection to a short exact sequence $$0 \to K\to F\to P\to 0$$ gives the assertion by the splitting lemma (equivalenly, a module may be defined to be projective if and only if every sequence of the above form splits).
Now note that any additive functor preserves split-exact sequences. Indeed, a short exact sequence is split-exact if and only if either the projection on the right admits a section or the embedding on the left admits a retraction. Both conditions are preserved by functoriality, hence so is the split. If now $$0 \to K\to F\to P\to 0$$ is split-exact with $$F$$ free of finite rank, then $$0 \to P^\vee\to F^\vee\to K^\vee\to 0$$ remains short exact. If $$F\cong A^n$$, then $$F^\vee\cong A^n$$. Since the dual sequence is still split-exact and $$F^\vee$$ is free, $$P^\vee$$ is projective. Note that this requires the full strength of the splitting lemma.
For checking that the natural maps $$P^\vee\otimes_A M\to\operatorname{Hom}_A(P,M)$$ and $$P\to(P^\vee)^\vee$$ are isomorphisms, try writing $$F\cong K\oplus P$$ by explictly fixing either a retract of $$K\to F$$ or a section of $$F\to P$$. Then combine the semi-explicit isomorphisms with the cases you already deduced. Even though the isomorphism $$F\cong K\oplus P$$ depends on a choice, fixing this isomorphism and checking commutativity of the necessary diagrams by hand should suffice. However, I have not checked this in detail.
Both, the formula $$P^\vee\otimes_A M\to\operatorname{Hom}_A(P,M)$$ as well as the canonical identification with the double dual space are important properties of finite dimensional vector spaces, which fail for general modules. Assuming finite generation and projectivity allows a reasonably well-behaved theory, close to that of ordinary linear algebra. This would be my interpreation of "rigidity". A helpful keyword here would be that of dualizable module (a notion agreeing with finite generation and projectivity, but a distinct concept nonetheless).