In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p-derivation is related to that of a derivation in differential algebra.
Definition
Let p be a prime number. A p-derivation or Buium derivative on a ring $R$ is a map of sets $\backslash delta:R\backslash to\; R$ that satisfies the following "product rule":
- $\backslash delta(ab)\; =\; \backslash delta\; (a)b^p\; +\; a^p\backslash delta\; (b)\; +\; p\backslash delta\; (a)\backslash delta\; (b)$
and "sum rule":
- $\backslash delta(a+b)\; =\; \backslash delta\; (a)\; +\; \backslash delta(b)\; +\; \backslash frac\{a^p\; +b^p\; -\; (a+b)^p\; \}\{p\}$.
Note that in the "sum rule" we are not really dividing by p, since all the relevant binomial coefficients in the numerator are divisible by p, so this definition applies in the case when $R$ has p-torsion.
Relation to Frobenius Endomorphisms
A map $\backslash sigma:\; R\backslash to\; R$ is a lift of the Frobenius endomorphism provided $\backslash sigma(x)\; =\; x^p\; \backslash mod\; pR$. An example such lift could come from the Artin map.
If $(R,\backslash delta)$ is a ring with a p-derivation, then the map
$\backslash sigma(x):=\; x^p\; +\; p\backslash delta(x)$ defines a ring endomorphism which is a lift of the frobenius endomorphism. When the ring R is p-torsion free the correspondence is a bijection.
Examples
- For $R\; =\; \backslash mathbb\; Z$ the unique p-derivation is the map
- $\backslash delta(x)\; =\; \backslash frac\{x-x^p\}\{p\}.$
The quotient is well-defined because of Fermat's Little Theorem.
- If R is any p-torsion free ring and $\backslash sigma:R\; \backslash to\; R$ is a lift of the Frobenius endomorphism then
- $\backslash delta(x)\; =\; \backslash frac\{\backslash sigma(x)-x^p\}\{p\}$
defines a p-derivation.
See also
References
External links
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