Signatur/F und differentiell/Positive Charakteristik/Textabschnitt

Theorem

Let ${}R$ denote a local noetherian ring in positive characteristic.

If the ${}F$-signature of ${}R$ is positive, then also the differential signature of ${}R$ is positive, and we have the estimate

${}{\frac {\mu ^{d}}{d!}}\operatorname {DS} (R)\geq \operatorname {FS} (R)\,,$

where

${}\mu =\operatorname {dim} _{K^{p}}(R/{\mathfrak {m}}^{[p]})\,.$

If ${}R$ is ${}F$-pure and the differential signature is positive, then also the ${}F$-signature is positive. Moreover, if in this setting the differential signature exists as a limit, then the estimate

${}\operatorname {FS} (R)\geq {\frac {1}{d!}}\operatorname {DS} (R)\,$

holds.

The following example shows that without the ${}F$-pure hypothesis the second implication is not true.

Example

We consider in characteristic ${}p$ the ring given by the equation

${}X^{p}+Y^{p+1}+Z^{p+1}=0\,.$

This is a normal hypersurface ring. This ring is not ${}F$-pure, since ${}X\notin (Y,Z)$, but ${}X^{p}\in (Y^{p},Z^{p})$. Hence it is not strongly ${}F$-regular and so its ${}F$-signature is ${}0$. However, the differential signature is positive. This example is also an easy counterexample to the Zariski-Lipman conjecture.