Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ring containing a field of characterstic 0, then $B_{red}$ is also flat module over this ring. Then he goes and writes in Exercise 5.18 that if the conjecture were true then the Macaulay curve over an algebraically closed field of characteristic 0 (this is the parametric projective curve $(s^4:s^3t:st^3:t^4)$ in $\mathbb P^3$) is not a set-theoertic complete intersection. I cannot immediately see how this statement on Macaulay curve follows from this result on flatness. Can anybody help me understand this or give a hint?
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1$\begingroup$ Hint: $B $ flat $\,\Longleftrightarrow \, B$ Cohen-Macaulay, and same for $B_{red}$. $\endgroup$– abxCommented Mar 6, 2022 at 20:31
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$\begingroup$ And how would you use this on the curve? Are you going to do something with the coordinate ring of the curve? This curve is known to be non-arithmetically Cohen-Macaulay? $\endgroup$– Jose CapcoCommented Mar 6, 2022 at 20:43
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1$\begingroup$ Yes, almost trivially — it is not projectively normal. $\endgroup$– abxCommented Mar 7, 2022 at 5:33
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