  In the noncommutative geometry of Artin, Van den Bergh, and others, the
twisted homogeneous coordinate ring is one of the basic constructions. Such a
ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism
of a projective scheme X. Many open questions regarding $\sigma$-ample divisors
have remained.
  We derive a relatively simple necessary and sufficient condition for a
divisor on X to be $\sigma$-ample. As a consequence, we show right and left
$\sigma$-ampleness are equivalent and any associated noncommutative homogeneous
coordinate ring must be noetherian and have finite, integral GK-dimension. We
also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.
