# Arithmetic and geometric Frobenius

- Arithmetic and geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring "R" that has characteristic "p", where "p" is a prime number. Namely, the mapping φ that takes "r" in "R" to "r"^{"p"} is a ring endomorphism of "R".

The image of φ is then "R"^{"p"}, the subring of "R" consisting of "p"-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring "automorphism".

The terminology of **geometric Frobenius** arises by applying the spectrum of a ring construction to φ. This gives a mapping

:φ*: Spec("R"^{"p"}) → Spec("R")

of affine schemes. Even in cases where "R"^{"p"} = "R" this is not the identity, unless "R" is the prime field.

Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called "geometric Frobenius". The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Glossary of arithmetic and Diophantine geometry** — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… … Wikipedia

**Heidelberg University Faculty of Mathematics and Computer Science** — Infobox University Faculty name = Faculty of Mathematics and Computer Science native name = Fakultät für Mathematik und Informatik established = 2002 dean = Prof. Dr. R. Rannacher staff = 27 students = 1100 website = http://www.math.uni… … Wikipedia

**List of mathematics articles (A)** — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A … Wikipedia

**Hasse-Witt matrix** — In mathematics, the Hasse Witt matrix H of a non singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping ( p th power mapping where F has q elements, q a power of the prime number p ) with respect to a basis for… … Wikipedia

**Hasse–Witt matrix** — In mathematics, the Hasse–Witt matrix H of a non singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the… … Wikipedia

**algebra** — /al jeuh breuh/, n. 1. the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. 2. any of… … Universalium

**Coin problem** — With only 2 pence and 5 pence coins, one cannot make 3 pence, but one can make any higher amount. The coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a… … Wikipedia

**History of Grandi's series** — Geometry and infinite zerosGrandiGuido Grandi (1671 – 1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into nowrap|1=1 − 1 + 1 − 1 + · · · produced varying results: either:(1 1) + (1 1) + … Wikipedia

**Timeline of mathematics** — A timeline of pure and applied mathematics history. Contents 1 Before 1000 BC 2 1st millennium BC 3 1st millennium AD 4 1000–1500 … Wikipedia

**Group (mathematics)** — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia