  Using methods from commutative algebra and topos-theory, we construct
topos-theoretical points for the fppf topology of a scheme. These points are
indexed by both a geometric point and a limit ordinal. The resulting stalks of
the structure sheaf are what we call fppf-local rings. We show that for such
rings all localizations at primes are henselian with algebraically closed
residue field, and relate them to AIC and TIC rings. Furthermore, we give an
abstract criterion ensuring that two sites have point spaces with identical
sobrification. This applies in particular to some standard Grothendieck
topologies considered in algebraic geometry: Zariski, etale, syntomic, and
fppf.
