COMPARISON OF KUMMER LOGARITHMIC TOPOLOGIES WITH CLASSICAL TOPOLOGIES

We compare the Kummer flat (resp., Kummer étale) cohomology with the flat (resp., étale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes, and Kato’s logarithmic multiplicative group. We are particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.

KUMMER COVERINGS AND SPECIALISATION

We prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

Logarithmically regular morphisms

We consider the stack $${\mathcal {L}}og_{X}$$ L o g X parametrizing log schemes over a log scheme X, and weak and strong properties of log morphisms via $${\mathcal {L}}og_{X}$$ L o g X , as defined by Olsson. We give a concrete combinatorial presentation of $${\mathcal {L}}og_{X}$$ L o g X , and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato’s chart criterion of logarithmic smoothness.

Log-isocristaux surconvergents et holonomie

Let 𝒱 be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let $\X $ be a separated smooth formal 𝒱-scheme, 𝒵 be a normal crossing divisor of $\X $, $\X ^\#:= (\X , \ZZ )$ be the induced formal log-scheme over 𝒱 and $u: \X ^\# \rightarrow \X $ be the canonical morphism. Let X and Z be the special fibers of $\X $ and 𝒵, T be a divisor of X and ℰ be a log-isocrystal on $\X ^\#$ overconvergent along T, that is, a coherent left $\D ^\dag _{\X ^\#} (\hdag T) _{\Q }$-module, locally projective of finite type over $ \O _{\X } (\hdag T) _{\Q }$. We check the relative duality isomorphism: $u_{T,+} (\E ) \riso u_{T,!} (\E (\ZZ ))$. We prove the isomorphism $u_{T,+} (\E ) \riso \D ^\dag _{\X } (\hdag T) _{\Q } \otimes _{\D ^\dag _{\X ^\#} (\hdag T) _{\Q }} \E (\ZZ )$, which implies their holonomicity as $\D ^\dag _{\X } (\hdag T) _{\Q }$-modules. We obtain the canonical morphism ρℰ : uT,+(ℰ)→ℰ(†Z). When ℰ is moreover an isocrystal on $\X $ overconvergent along T, we prove that ρℰ is an isomorphism.