Proof of the centraliser theorem in Lorenz's "Algebra II"

Question

I'm working through Lorenz's "Algebra II" and don't understand a step in his proof of the double centraliser theorem.

The definitions and conventions used:

• Two simple artinian $K$-algebras $A,B$ are called similar (denoted $A\sim B)$ if there is a division $K$-algebra $D$ and $n,m\in\mathbb N$ such that $A\cong M_n(D)$ and $B\cong M_m(D)$.
• A $K$-algebra is central if its centre coincides with $K$ (no assumptions on dimensionality), simple if it has no proper two-sided ideals, artinian if it's left artinian as a ring.
• Tensor products over the base field $K$ are denoted simply $\otimes$.

The statement of the theorem:

Double centraliser theorem: Let $A$ be a simple, central, artinian $K$-algebra, $B\subset A$ a simple finite-dimensional subalgebra, and $C=Z_A(B)$ its centraliser in $A$. The following then holds

• $C$ is simple artinian.
• $C$ is similar to $B^\circ\otimes A$, and in particular $Z(C)=Z(B)$.
• If $L$ is the centre of $B$, then $B\otimes_LC$ is similar to $A\otimes L$.

My question is about the third statement. As proof, Lorenz simply presents the following chain of similarities without comment (other than to say that this follows from the first two statements) $$B\otimes_LC\sim B\otimes_L(B^\circ\otimes A)\cong(B\otimes_LB^\circ)\otimes A\sim L\otimes A\cong A\otimes L.$$

Although this is not stated explicitly, I assume it is meant that the isomorphisms and similarities are that of $L$-algebras.

I can't work out the first similarity, i.e. why $B\otimes_LC\sim B\otimes_L(B^\circ\otimes A)$.

Answer 1

I think I finally figured it out. Since $C$ and $B^\circ\otimes A$ are similar $K$-algebras there are $m,n$ with $C\otimes M_m(K)\cong(B^\circ\otimes A)\otimes M_n(K)$, so using properties of base extension (namely that $M_n(K)\otimes L\cong M_n(L)$ and $B\otimes A\cong B\otimes_L(A\otimes L)$ as $L$-algebras) and the fact that $B$ is an $L$-CSA $$(B\otimes_LC)\otimes_LM_m(L)\cong(B\otimes_LC)\otimes M_m(K)\cong B\otimes_L(C\otimes M_m(K))\cong B\otimes_L((B^\circ\otimes A)\otimes M_n(K))\\ \cong (B\otimes_L(B^\circ\otimes A))\otimes M_n(K)\cong((B\otimes_LB^\circ)\otimes A)\otimes M_n(K)\cong((B\otimes_LB^\circ)\otimes_L(A\otimes L))\otimes M_n(K)\\ \cong(M_d(L)\otimes_L(A\otimes L))\otimes M_n(K)\cong M_d(A\otimes L)\otimes M_n(K)\cong M_d(A\otimes L)\otimes_LM_n(L)\cong (A\otimes L)\otimes_LM_{nd}(L),$$ so $B\otimes_LC\sim A_L$ as $L$-algebras.