show/hide this revision's text 3 added particular case of what was last isomorphism ; made small stylistic changes in first sentence.

People

For those interested in general statementsmight like to notice the following results stating that , here is a summary of assumptions under which the canonical morphisms of $A$-modules below are isomorphismsunder the stated assumptions.:

If $P$ is finitely generated projective: $$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad$$

A module $P$ is finitely generated projective iff the following canonical map below is an isomorphism

$$ \quad P\otimes P^\vee \otimes P \stackrel{\sim }{\to} End(P) $$

If $P$ or $P'$ is finitely generated projective

$$ P^\vee \otimes P' \stackrel{\sim }{\to} Hom(P,P') $$

If both $P$ and $P'$ or both $P$ and $M$ or both $P'$ and $M'$are finitely generated projective $$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M') $$

In particular for $P$ or $P'$ finitely generated projective

$$ P^\vee \otimes P'^\vee \stackrel{\sim }{\to} (P \otimes P')^\vee $$

(this is mephisto's answer follows from to the case $M=M'=A$)OP's question)
show/hide this revision's text 2 Changed hypotheses in order to have more general statements

People interested in general statements might like to notice the following results stating that the canonical morphisms of $A$-modules below are isomorphisms under the stated assumptions.

If $P$ is finitely generated projective: $$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad and quad$$

A module $P$ is finitely generated projective iff the canonical map below is an isomorphism

$$ \quad P\otimes P^\vee \stackrel{\sim }{\to} End(P)$$End(P) $$

If moreover $M$ is an arbitrary P$ or $A$- moduleP'$ is finitely generated projective

$$ P\otimes M P^\vee \otimes P' \stackrel{\sim }{\to} Hom(P^\vee,MHom(P,P') $$

If moreover both $P$ and $P'$ is a finitely generated projective or both $P$ and $M'$ an arbitrary M$ or both $A$-moduleP'$ and $M'$are finitely generated projective $$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M') $$

(The OP's question is, of course, mephisto's answer follows from the case $M=M'=A$)
show/hide this revision's text 1

People interested in general statements might like to notice the following results stating that the canonical morphisms of $A$-modules below are isomorphisms under the stated assumptions.

If $P$ is finitely generated projective: $$P\stackrel{\sim }{\to} P^{\vee }{^\vee} \quad and \quad P\otimes P^\vee \stackrel{\sim }{\to} End(P)$$

If moreover $M$ is an arbitrary $A$- module

$$ P\otimes M \stackrel{\sim }{\to} Hom(P^\vee,M) $$

If moreover $P'$ is a finitely generated projective and $M'$ an arbitrary $A$-module $$Hom(P,M) \otimes Hom(P',M') \stackrel{\sim }{\to} Hom(P\otimes P',M\otimes M') $$

(The OP's question is, of course, the case $M=M'=A$)