= Invertible module
{title2=$L\otimes_A L^*\cong A$}
= Algebraic line bundle
{synonym}
An invertible module over a <commutative ring> $A$ is a <finitely generated module> that is a <projective module> and whose <localization at a prime ideal> is free of rank one at every prime. Its <dual module> $L^*=\operatorname{Hom}_A(L,A)$ is also finite projective. Evaluation $L\otimes_A L^*\to A$ is an <isomorphism> after every prime localization, hence is an isomorphism globally by <localization detects zero elements>. The <Picard group of a ring> consists of the isomorphism classes of these modules under the <tensor product of modules>.
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