# Finiteness of rigid cohomology with coefficients

@article{Kedlaya2002FinitenessOR, title={Finiteness of rigid cohomology with coefficients}, author={Kiran S. Kedlaya}, journal={Duke Mathematical Journal}, year={2002}, volume={134}, pages={15-97} }

We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology with compact supports H^i_c(X,E) are finite dimensional vector spaces. We also establish Poincare duality and the Kunneth formula with coefficients. The arguments use a pushforward construction in relative dimension 1, based on a relative version of Crew's conjecture on the quasi-unipotence of certain…

## 75 Citations

Finiteness of cohomology of local systems on rigid analytic spaces

- Mathematics
- 2016

We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite…

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

- Mathematics
- 2013

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method…

A p-adic local monodromy theorem

- Mathematics
- 2001

We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential…

Rigid Cohomology for Algebraic Stacks

- Mathematics
- 2010

We extend le Stum's construction of the overconvergent site to algebraic stacks. We prove that etale morphisms are morphisms of cohomological descent for finitely presnted crystals on the…

Cohomology of arithmetic families of (,Γ)-modules

- Mathematics
- 2012

We prove the niteness of the ( ’; )-cohomology and the Iwasawa cohomology of arithmetic families of (’; )-modules. Using this niteness theorem, we show that a family of Galois representations that is…

On Dwork cohomology for singular hypersurfaces

- Mathematics
- 2003

Let Z be a projective hypersurface over a finite field. With no smoothness assumption, we relate the p-adic cohomology spaces constructed by Dwork in his study of the zeta function of Z (cf. [29],…

A Variational Tate Conjecture in crystalline cohomology

- MathematicsJournal of the European Mathematical Society
- 2019

Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate a Variational Tate Conjecture characterising, in terms of the crystalline…

Semistable Reduction of overconvergent F-isocrystals I : Isocrystals and Rigid Cohomology

- Mathematics
- 2007

Notation 1.1.2. By a k-variety, I meant a reduced (not necessarily irreducible) separated scheme of finite type over k. (It could be shown that the theory only depends on the reduced scheme…

## References

SHOWING 1-10 OF 76 REFERENCES

A p-adic local monodromy theorem

- Mathematics
- 2001

We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential…

The cohomology of Monsky and Washnitzer

- Mathematics
- 1986

The Zeta-function of an algebraic variety over a finite field can be expressed in terms of a Frobenius operator acting on p-adic cohomology groups of this variety. Those cohomology groups, based on…

On base change theorem and coherence in rigid cohomology.

- Mathematics
- 2003

We prove that the base change theorem in rigid coho- mology holds when the rigid cohomology sheaves both for the given morphism and for its base extension morphism are coherent. Apply- ing this…

The Convergent Topos in Characteristic p

- Mathematics
- 1990

The purpose of this note is to investigate some of the foundational questions concerning convergent cohomology as introduced in [?] and [?], using the language and techniques of Grothendieck…

A note on weakly complete algebras

- Mathematics
- 1969

Fix a commutative noetherian ring R with unit and an ideal / in R. P. Monsky and G. Washnitzer have developed the notion of a weakly complete finitely generated algebra over (i?, I) [l], [2]; we…

More étale covers of affine spaces in positive characteristic

- Mathematics
- 2002

We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at…

Semistable reduction for overconvergent $F$-isocrystals on a curve

- Mathematics
- 2002

Given a smooth affine curve X over a field k of positive characteristic, and an overconvergent F-isocrystal on X, we prove after replacing k by a finite purely inseparable extension, there exists a…

Finite local monodromy of overconvergent unit-root F-isocrystals on a curve

- Mathematics
- 1998

<abstract abstract-type="TeX"><p>We prove that the category of <i>p</i>-adic continuous representations of a fundamental group of a curve over a perfect field of a positive characteristic <i>p</i>…