Tensor Product of Projective Modules is Projective
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Theorem
Let $A$ be a commutative ring with unity.
Let $P$ and $Q$ be projective $A$-modules.
Then the tensor product $P \otimes_A Q$ is a projective $A$-module.
Proof
By Projective iff Direct Summand of Free Module, there exist $A$-modules $P'$ and $Q'$, such that $P \oplus P'$ and $Q \oplus Q'$ are free.
By Tensor Product Distributes over Direct Sum, there is an isomorphism
- $\paren {P \oplus P'} \otimes_A \paren {Q \oplus Q'} \cong \paren {P \otimes_A Q} \oplus \paren {P' \otimes_A Q} \oplus \paren {P \otimes_A Q'} \oplus \paren {P' \otimes_A Q'}$
By Tensor Product of Free Modules is Free the left hand side is free.
Hence $P \otimes_A Q$ is a direct summand of a free module.
By Projective iff Direct Summand of Free Module $P \otimes_A Q$ is projective.
$\blacksquare$