# Frobenius endomorphism

From Encyclopedia of Mathematics

An endomorphism of a scheme over the finite field of elements such that is the identity mapping on , and the mapping of the structure sheaf is that of raising to the -th power. The Frobenius endomorphism is a purely-inseparable morphism and has zero differential. For an affine variety defined over , the Frobenius endomorphism takes the point to .

The number of geometric points of that are defined over is the same as the number of fixed points of , which enables one to use the Lefschetz formula to determine the number of such points.

#### References

[1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |

#### Comments

Here, is the set of -points of , i.e. the set of points of that are defined over .

#### References

[a1] | P. Gabriel, "Etude infinitésimal des schémas en groupes" M. Demazure (ed.) A. Grothendieck (ed.) , SGA 3. Exp. VII , Lect. notes in math. , 151 , Springer (1970) |

**How to Cite This Entry:**

Frobenius endomorphism.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Frobenius_endomorphism&oldid=23834

This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article