Editing Lefschetz fixed-point theorem (section)

==Frobenius==
Let <math>X</math> be a variety defined over the [[finite field]] <math>k</math> with <math>q</math> elements and let <math>\bar X</math> be the base change of <math>X</math> to the [[algebraic closure]] of <math>k</math>. The '''[[Frobenius endomorphism]]''' of <math>\bar X</math> (often the ''geometric Frobenius'', or just ''the Frobenius''), denoted by  <math>F_q</math>, maps a point with coordinates <math>x_1,\ldots,x_n</math> to the point with coordinates <math>x_1^q,\ldots,x_n^q</math>. Thus the fixed points of <math>F_q</math> are exactly the points of <math>X</math> with coordinates in <math>k</math>; the set of such points is denoted by <math>X(k)</math>.  The Lefschetz trace formula holds in this context, and reads:

:<math>\#X(k)=\sum_i (-1)^i \mathrm{tr}(F_q^*| H^i_c(\bar{X},\Q_{\ell})).</math>

This formula involves the trace of the Frobenius on the [[étale cohomology]], with compact supports, of <math>\bar X</math> with values in the field of [[p-adic number|<math>\ell</math>-adic numbers]], where <math>\ell</math> is a prime coprime to <math>q</math>.

If <math>X</math> is smooth and [[equidimensionality|equidimensional]], this formula can be rewritten in terms of the ''arithmetic Frobenius'' <math>\Phi_q</math>, which acts as the inverse of <math>F_q</math> on cohomology:

:<math>\#X(k)=q^{\dim X}\sum_i (-1)^i \mathrm{tr}((\Phi_q^{-1})^*| H^i(\bar X,\Q_\ell)).</math>

This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to [[algebraic stack]]s over finite fields.