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Let $A\subset B$ be two integrally closed Noetherian domains with $B$ finitely generated as an $A$-module. Then $B$ is reflexive.

Could you explain me why, please?

Reflexive means that the natural map from $B$ to $B^{**}$, where $B^{**}=\mathrm{Hom}_A(\mathrm{Hom}_A(B,A),A)$, is an isomorphism.

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