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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?

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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).

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