Let $A$ be a $\mathbb{C}$-algebra. Let $\phi:\mathbb{C}[X_1,...,X_n] \otimes_{\mathbb{C}} A \to \mathbb{C}[t] \otimes_{\mathbb{C}} A$ be a ring homomorphism, sending $X_i$ to say $f_i \in \mathbb{C}[t] \otimes_{\mathbb{C}} A$. I am looking for conditions on $f_i$ such that $A[X_1,...,X_n]/(\mbox{ker}(\phi))$ is $A$-flat. Any reference will be most welcome.


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