"Numerical Criterion" for Flatness - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:37:37Z http://mathoverflow.net/feeds/question/82827 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82827/numerical-criterion-for-flatness "Numerical Criterion" for Flatness Xander Faber 2011-12-06T21:49:32Z 2011-12-07T07:41:04Z <p>Let $R$ be a Dedekind ring, let $S = \mathrm{Spec} R$, and let us suppose that $f: X \to S$ is a finite morphism. Note that $X$ is not required to be connected. Does there exist a "numerical criterion" that will produce a closed subscheme $S_0 \subset S$ such that $f$ is flat when restricted to $f^{-1}(S - S_0)$? </p> <p>Consider the following non-example: Let $F \in \mathbb{Z}[T]$ be a non-constant monic polynomial, and set <code>$X = \mathrm{Spec} \mathbb{Z}[T] / (F(T)), \qquad f: X \to \mathrm{Spec} \mathbb{Z}.$</code> The structure morphism $f \$ is finite and flat. A prime $p$ occurs as a factor of the discriminant $\Delta(F)$ if and only if the fiber of $f$ over $p$ contains a non-smooth point. So the discriminant can be viewed as a numerical criterion for detecting (non-)smoothness. All of the data needed to determine smoothness is contained in the discriminant. Does there exist a similar one-step gadget for detecting (non-)flatness? (I say "one-step" to mean that the vanishing of infinitely many Tor groups is not an adequate solution. Is there a single Tor group that captures what I'm after?) </p> http://mathoverflow.net/questions/82827/numerical-criterion-for-flatness/82853#82853 Answer by Will Sawin for "Numerical Criterion" for Flatness Will Sawin 2011-12-07T07:41:04Z 2011-12-07T07:41:04Z <p>Take a free resolution of $X$ as an $S$-module, or, more importantly, the first two steps. Look at the map between them, which is given by a matrix over $S$. You need to compute how the rank of this matrix varies across various prime ideals, since this is a constant minus the dimension of the cokernel.</p> <p>To compute the rank of a matrix, one of course uses the vanishing or nonvanishing of various determinants of $n$ by $n$ minors. These seem to be the kind of numerical criteria you are looking for.</p>