#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 891-918

** Nobuo Tsuzuki **
On Base Change Theorem
and Coherence in Rigid Cohomology

We prove that the base change theorem in rigid cohomology holds when the
rigid cohomology sheaves both for the given morphism and for its base extension
morphism are coherent. Applying this result, we give a condition under
which the rigid cohomology of families becomes an overconvergent isocrystal.
Finally, we establish generic coherence of rigid cohomology of proper smooth
families under the assumption of existence of a smooth lift of the generic
fiber. Then the rigid cohomology becomes an overconvergent isocrystal generically.
The assumption is satisfied in the case of families of curves. This example
relates to P. Berthelot's conjecture of the overconvergence of rigid cohomology
for proper smooth families.

2000 Mathematics Subject Classification: 14F30, 14F20, 14D15

Keywords and Phrases: rigid cohomology, overconvergent isocrystal, base change theorem, Gauss-Manin
connection, coherence

Full text: dvi.gz 50 k,
dvi 142 k,
ps.gz 773 k,
pdf 283 k.

Home Page of
DOCUMENTA MATHEMATICA