Log-isocristaux surconvergents et holonomie

Daniel Caro
2009 Compositio Mathematica  
AbstractLetbe a complete discrete valuation ring of unequal characteristic with perfect residue field. Let$\X $be a separated smooth formal-scheme,be a normal crossing divisor of$\X $,$\X ^\#:= (\X , \ZZ )$be the induced formal log-scheme overand$u: \X ^\# \rightarrow \X $be the canonical morphism. LetXandZbe the special fibers of$\X $and,Tbe a divisor ofXandbe a log-isocrystal on$\X ^\#$overconvergent alongT, that is, a coherent left$\D ^\dag _{\X ^\#} (\hdag T) _{\Q }$-module, locally
more » ... le, locally projective of finite type over$ \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 }$-modules. We obtain the canonical morphismρ:uT,+()→(†Z). Whenis moreover an isocrystal on$\X $overconvergent alongT, we prove thatρis an isomorphism.
doi:10.1112/s0010437x09004199 fatcat:deu5mqjzz5a2lcmjxd2keqjweu