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Work locally, assume that $f: R\to S$ is a local homomorphism. Let $cmd(R) \operatorname{cmd}(R) =dim R-depth \operatorname{dim} R-\operatorname{depth} R$ (this is the so-called Cohen-Macaulay defect of $R$). Claim: cmd $\operatorname{cmd}$ is preserved by l.c.i maps (easy, essentially because both depth and dimension drop by one when you kill a regular element).

Now since the map $\phi: Spec(S) \operatorname{Spec}(S) \to Spec \operatorname{Spec} (R)$ is finite and surjective, $dim \operatorname{dim} R= dim \operatorname{dim} S$, which combines with the last claim to show that $depth \operatorname{depth} R = depth \operatorname{depth} S$. But since l.c.i also implies finite flat dimension, we have $depth \operatorname{depth} R -depth \operatorname{depth} S = pd_RS$, so $S$ is flat over $R$.

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Work locally, assume that $f: R\to S$ is a local homomorphism. Let $cmd(R) =dim R-depth R$ (this is the so-called Cohen-Macaulay defect of $R$). Show that Claim: cmd is preserved by l.c.i mapmaps (easy, essentially because both depth and dimension drop by one when you kill a regular element).

Now since the map $\phi: Spec(S) \to Spec (R)$ is finite and surjective, $dim R= dim S$, which combines with the last sentence claim to show that $depth R = depth S$. But since l.c.i also implies finite flat dimension, we have $depth R -depth S = pd_RS$, so $S$ is flat over $R$.

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Work locally, assume that $f: R\to S$ is a local homomorphism. Let $cmd(R) =dim R-depth R$ (this is the so-called Cohen-Macaulay defect of $R$). Show that cmd is preserved by l.c.i map. Now since the map $\phi: Spec(S) \to Spec (R)$ is finite and surjective, $dim R= dim S$, which combines with the last sentence show that $depth R = depth S$. But since l.c.i also implies finite flat dimension, we have $depth R -depth S = pd_RS$, so $S$ is flat over $R$.
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