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Here is an algebraic construction. The way I think about it is based on these two facts: 1) when $A$ is regular domain, a module-finite A-algebra is flat iff it is Cohen-Macaulay (CM) of same dimensi …

The answer is yes. 1) $A$ is $I$-adic complete implies that $I \subset rad(A)$, the intersection of all maximal primes. Indeed, pick any $a \in I$. Look at $1-a + a^2 -a^3 ... \in A $ (this is where …

The point is the so-called miracle flatness, (in the graded form for this example to ease notations). Let $S=k[l_1,...,l_d]$ where the $l$s form a system of parameters for $R/I$ (and also $R/J$). …

It is hard for me not to mention the following splendid result by Auslander-Lichtenbaum: If $R$ is regular local with Krull dimension $d$, then for any finitely generated module $M$, $M^{\otimes n}$ …

Flatness can be checked locally on maximal ideals, so we may as well let $R=A_m, k = R/mR$ and considering: If $Tor_i^R(N,k) =0$ for $i>0$, when is $N$ flat? … You can also check out the paper "A local flatness criterion ..." by Hans Schoutens (available on his website). …