Points in the fppf topology

Abstract

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, étale, syntomic, and fppf. Contents 7 3. Applications to algebraic geometry 11 4. fppf-local rings 13 5. Construction of fppf-local rings 16 6. Points in the fppf topos 20 References 24

Geometry on totally separably closed schemes

“…Absolute integral closure in characteristic p > 0 provides examples of flat ring extensions that are not a filtered colimit of finitely presented flat ring extensions, as Bhatt [6] noted. I have used absolute closure to study points in the fppf topos [57].…”

The p-radical closure of local noetherian rings

“…It is trivial if R is noetherian, or more generally if R, viewed as a topological ring with respect to the m R -adic topology, is Hausdorff. On the other hand, the kernel coincides with the maximal ideal if R is an fppf-local ring, as studied by Gabber and Kelly in [10], Section 3 and myself in [38], Section 4. Using [4], Chapter III, §2, No.…”

Azumaya toposes

“…In [3,Lemma 3.3], some properties of points for the flat (fppf) topology are given, but this case is more difficult and still lacks a concrete description. Schröer in [13] also studies points for the fppf topology, but he considers the category of arbitrary commutative rings instead of only finitely generated ones.…”