# Rigid cohomology

• ## General Introduction

Rigid cohomology usually refers to the p-adic cohomology theory described for example in the book of Le Stum, but there is a paper by Voevodsky (Open problems...) in which he uses the terms rigid homotopy and rigid homology for something completely different.

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• ## Online References

Something might be here

Berthelot, P.: Cohomologie rigide et cohomologie rigide `a supports propres. Pr´epublication (1996) (is this online??)

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• ## Paper References

Book by Le Stum?

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• ## Some Research Articles

Petrequin on Chern classes and cycle classes

Kedlaya on finiteness (long review)

arXiv:1205.4702 Rigid Cohomology and de Rham-Witt complexes from arXiv Front: math.AG by Pierre Berthelot Let \$k\$ be a perfect field of characteristic \$p > 0\$, \$Wn = Wn(k)\$. For separated \$k\$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to \$Wn\$ with coefficients in a crystal that is only supposed to be flat over \$Wn\$.

arXiv:1008.0305 Overconvergent Witt Vectors from arXiv Front: math.AG by Christopher Davis, Andreas Langer, Thomas Zink Let A be a finitely generated algebra over a field K of characteristic p >0. We introduce a subring of the ring of Witt vectors W(A). We call it the ring of overconvergent Witt vectors. We prove that on a scheme X of finite type over K the overconvergent Witt vectors are an étale sheaf. In a forthcoming paper (Annales ENS) we define an overconvergent de Rham-Witt complex on a smooth scheme X over a perfect field K whose hypercohomology is the rigid cohomology of X in the sense of Berthelot.

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• ## Other Information

Andreas Langer (Exeter) "An integral structure on rigid cohomology" (MR13, 2:30pm as usual)

Abstract: For a quasiprojective smooth variety over a perfect field k of char p we introduce an overconvergent de Rham-Witt complex by imposing a growth condition on the de Rham-Witt complex of Deligne-Illusie using Gauus norms and prove that its hypercohomology defines an integral structure on rigid cohomology, i.e. its image in rigid cohomology is a canonical lattice. As a corollary we obtain that the integral Monsky-Washnitzer cohomology (considered before inverting p) of a smooth k-algebra is of finite type modulo torsion. This is joint work with Thomas Zink.

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