Conversely, assume each localization is reduced. Consider a proper strong subobject W of X . This implies that W is not unipotent as the pullback of any unipotent map is unipotent. The pullback of any epi along a localization is epic as any localization is coflat. Conversely assume f is not epic.
P be the pullback of t along p .
The following commutative diagram implies that t and f are not disjoint, thus f is unipotent. Y is a localization of Y at a prime P . V s are the induced maps. We have proved that for any prime P we can find a small open neighborhood U P of P such that the restriction of t on U P can be factored through the restriction of f on U P . Since the analytic category is strict, these factorization can be glue together to obtain a global factorization for t by f . n Proposition 5.5.7.