R. Tyrrell Rockafellar: Difference between revisions
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{{DISPLAYTITLE:''p''-derivation}} | |||
{{Merge to|Arithmetic_derivative|discuss=Talk:Arithmetic_derivative|date=September 2013}} | |||
In [[mathematics]], more specifically [[differential algebra]], a '''''p''-derivation''' (for ''p'' a prime number) on a [[Ring (mathematics)|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 [[Differential algebra|derivation]] in differential algebra. | |||
==Definition== | |||
Let ''p'' be a prime number. A '''''p''-derivation''' or Buium derivative on a ring <math> R </math> is a map of sets <math> \delta:R\to R </math> that satisfies the following "[[product rule]]": | |||
:<math> \delta_p(ab) = \delta_p (a)b^p + a^p\delta_p (b) + p\delta_p (a)\delta_p (b) </math> | |||
and "sum rule": | |||
:<math> \delta_p(a+b) = \delta_p (a) + \delta_p(b) + \frac{a^p +b^p - (a+b)^p }{p} </math>. | |||
as well as | |||
:<math> \delta_p(1) =0 </math>. | |||
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 <math> R </math> has ''p''-[[Torsion (algebra)|torsion]]. | |||
==Relation to Frobenius Endomorphisms== | |||
A map <math> \sigma: R\to R </math> is a lift of the [[Frobenius endomorphism]] provided <math> \sigma(x) = x^p \mod pR </math>. An example such lift could come from the [[Artin map]]. | |||
If <math> (R,\delta) </math> is a ring with a ''p''-derivation, then the map | |||
<math> \sigma(x) := x^p + p\delta(x) </math> 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 <math> R = \mathbb Z </math> the unique ''p''-derivation is the map | |||
:<math> \delta(x) = \frac{x-x^p}{p}. </math> | |||
The quotient is well-defined because of [[Fermat's Little Theorem]]. | |||
* If ''R'' is any ''p''-torsion free ring and <math>\sigma:R \to R</math> is a lift of the Frobenius endomorphism then | |||
:<math> \delta(x) = \frac{\sigma(x)-x^p}{p} </math> | |||
defines a ''p''-derivation. | |||
==See also== | |||
*[[Derivation (abstract algebra)|Derivation]] | |||
*[[Fermat Quotient]] | |||
==References== | |||
* {{Citation|first=Alex|last=Buium|title=Arithmetic Differential Equations|year=1989|publisher=Springer-Verlag|isbn=0-8218-3862-8|series=Mathematical Surveys and Monographs}}. | |||
==External links== | |||
*[http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.dmj/1077245037&page=record Project Euclid] | |||
[[Category:Differential algebra]] |
Revision as of 02:31, 29 January 2014
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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 is a map of sets that satisfies the following "product rule":
and "sum rule":
as well as
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 has p-torsion.
Relation to Frobenius Endomorphisms
A map is a lift of the Frobenius endomorphism provided . An example such lift could come from the Artin map.
If is a ring with a p-derivation, then the map 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
The quotient is well-defined because of Fermat's Little Theorem.
defines a p-derivation.
See also
References
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