TITLE: Weil Conjectures for nonprojective algebraic varieties
QUESTION [12 upvotes]: If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?

REPLY [21 votes]: Correctly restated, the conjectures hold for any variety $V$ (not necessarily complete or nonsingular) over a finite field $k$.
Dwork proved that the zeta function $Z(V,t)$ of $V$ is a rational function of $t$.
Grothendieck (et al.) expressed $Z(V,t)$ as the alternating product of the characteristic
polynomials of the Frobenius map $F$ acting on the etale cohomology groups of $V$ with compact support.
Deligne showed (Weil II) that for each positive integer $i$ and each eigenvalue $a$ of $F$ acting on the $i$th etale cohomology group of $V$ with compact support, there exists an integer $j\leq i$ such that all the complex conjugates of $a$ have absolute value $q^{j/2}$ where $q=|k|$.
When $V$ is nonsingular and complete, these statements together with Poincare duality, give Weil's original conjectures.