Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial x_j})\in B^{\times}$. Consider the multiplication map \begin{align*} m:A\otimes_A B\rightarrow B\\\ a\otimes b\mapsto ab^p \end{align*}

Q: How does one prove directly that $m$ is an isomorphism of rings?

Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial x_j})\in B^{\times}$. Consider the multiplication map \begin{align*} m:A\otimes_A B\rightarrow B\\\ a\otimes b\mapsto ab^p \end{align*} Note here that for $a\in A$ we have the tensor relation $1\otimes ab=a^p\otimes b$.

Q: How does one prove directly that $m$ is an isomorphism of rings?