  We give a short new computation of the quantum cohomology of an arbitrary
smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of
Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The
proof is based on the purely algebraic fact that a class of generalised
Jacobian rings associated to $X$ are free as modules over the Novikov ring. In
contrast to previous results of this kind, $X$ need not be compact. When $X$ is
monotone the presentation we obtain is completely explicit, using only
well-known computations with the standard complex structure.
