The de Rham-Witt complex and p-adic vanishing cycles

@article{Geisser2003TheDR,
  title={The de Rham-Witt complex and p-adic vanishing cycles},
  author={Thomas H. Geisser and Lars Hesselholt},
  journal={Journal of the American Mathematical Society},
  year={2003},
  volume={19},
  pages={1-36}
}
We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is isomorphic to the sheaf of p-adic vanishing cycles. We use this together with results of the second author and Madsen to evaluate the $K$-theory with finite coefficients of the quotient field K of the henselian local ring of X at a generic point of Y. The result… 
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References

SHOWING 1-10 OF 37 REFERENCES
On the K-theory of local fields
In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. The…
Algebraic Jf-theory and Trace Invariants
The cyclotomic trace of Bokstedt-Hsiang-Madsen, the subject of Bokstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K-fiA) ^.TR;(A;p) from Quillen's algebraic A"-theory to a…
p-adic Etale Cohomology
What follows is a report on joint work with O. Gabber and K. Kato.* A manuscript with complete proofs exists and is currently being revised. For compelling physical reasons (viz. time, space, and…
The cyclotomic trace and algebraic K-theory of spaces
The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and…
ON A THEOREM OF HUREWICZ AND $ K$-THEORY OF COMPLETE DISCRETE VALUATION RINGS
It is proved that for a complete discrete valuation ring of zero characteristic with residue field of positive characteristic and maximal ideal , the natural homomorphism of -groups with coefficients…
On the de Rham–Witt complex in mixed characteristic
Topological Cyclic Homology of Schemes
In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic K-theory. It is known that for finite algebras over the ring of Witt vectors…
On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring
We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic (0,p), the p-adic etale K-theory and p-adic topological cyclic homology agree.
DE RHAM–WITT COHOMOLOGY FOR A PROPER AND SMOOTH MORPHISM
  • A. Langer, T. Zink
  • Mathematics
    Journal of the Institute of Mathematics of Jussieu
  • 2004
We construct a relative de Rham–Witt complex $W\varOmega^{\cdot}_{X/S}$ for a scheme $X$ over a base scheme $S$. It coincides with the complex defined by Illusie (Annls Sci. Ec. Norm. Super.12…
the absolute and relative de rham–witt complexes
we compare the absolute and relative de rham–witt complexes considered by hesselholt and madsen (ann. sci. école norm. sup. 37 (2004), 1–43; ann. of math. (2) 158 (2003), 1–113) and by langer and…
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