# When $K[X_1,X_2,…,X_n] \to K[Y_1,Y_2,…,Y_m]$ is a flat morphism

Let $$K$$ be a field and $$\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$$ a polynomial ring morphism. Assume $$n, m \ge 2$$. By definition $$\varphi$$ endows $$K[Y_1,Y_2,...,Y_m]$$ with a $$K[X_1,X_2,...,X_n]$$-module structure.

Are there any available criteria to decide when the $$K[X_1,X_2,...,X_n]$$-module $$K[Y_1,Y_2,...,Y_m]$$ induced in this way by $$\varphi$$ a flat $$K[X_1,X_2,...,X_n]$$-module?

I want also to note that this generalizes this MathSE question.

As far as we consider the case with more than one indeterminantes. The case $$n=m=1$$ always has a positive answer since $$K[X]$$ is a PID and in this setting flat = torsion-free. For $$n \ge 2$$ $$K[X_1,X_2,...,X_n]$$ is not a PID, so the criterion is not applicable.

An approach is to use a lemma that states that if $$R \to R', S \to S'$$ are flat modules, then $$R \otimes S \to R' \otimes S'$$ is a flat $$R \otimes S$$-module. The point is that obviously not every polynomial map $$\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$$ arises from such "atomic" pieces $$\varphi_i: K[X_i] \to K[Y_i]$$ as tensor product $$\bigotimes_i \varphi_i$$.

So I'm asking if there exist approaches dealing with this problem or is it too broad?

• The geometric way to look at it is that $\phi \colon \mathbf A^m \to \mathbf A^n$ is flat if and only if all fibres have dimension $m-n$. This follows from "miracle flatness"; see e.g. Tag 00R4. – R. van Dobben de Bruyn May 30 at 0:24
• I suspect you mean "$K$-algebra homomorphism" rather than just "ring homomorphism". – YCor May 30 at 2:50