Lemma 11.4.1. Let $A$, $A'$ be $k$-algebras. Let $B \subset A$, $B' \subset A'$ be subalgebras with centralizers $C$, $C'$. Then the centralizer of $B \otimes _ k B'$ in $A \otimes _ k A'$ is $C \otimes _ k C'$.
Proof. Denote $C'' \subset A \otimes _ k A'$ the centralizer of $B \otimes _ k B'$. It is clear that $C \otimes _ k C' \subset C''$. Conversely, every element of $C''$ commutes with $B \otimes 1$ hence is contained in $C \otimes _ k A'$. Similarly $C'' \subset A \otimes _ k C'$. Thus $C'' \subset C \otimes _ k A' \cap A \otimes _ k C' = C \otimes _ k C'$. $\square$
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