Positive Charakteristik/F-Signatur/Differentielle Signatur/Vergleich/Fakt/en

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.