View source for Frobenius endomorphism

An [[Endomorphism|endomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417701.png" /> of a [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417702.png" /> over the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417703.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417704.png" /> elements such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417705.png" /> is the identity mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417706.png" />, and the mapping of the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417707.png" /> is that of raising to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417708.png" />-th power. The Frobenius endomorphism is a purely-inseparable morphism and has zero differential. For an affine variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f0417709.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177010.png" />, the Frobenius endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177011.png" /> takes the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177013.png" />.

The number of geometric points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177014.png" /> that are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177015.png" /> is the same as the number of fixed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177016.png" />, which enables one to use the [[Lefschetz formula|Lefschetz formula]] to determine the number of such points.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR></table>



====Comments====
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177017.png" /> is the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177018.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177019.png" />, i.e. the set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177020.png" /> that are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041770/f04177021.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR></table>