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

Abstract

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 affirms the Lichtenbaum-Quillen conjecture for the field K.

References in corpus (2)

• On the K-theory of local fields
• Algebraic K-theory and trace invariants

Cited by in corpus (10)

• Integral p-adic Hodge theory
• Topological Hochschild homology and integral $p$-adic Hodge theory
• Some recent advances in topological Hochschild homology
• Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields
• $p$-adic vanishing cycles as Frobenius-fixed points
• Syntomic complexes and $p$-adic étale Tate twists
• Towards Vorst's conjecture in positive characteristic
• Duality for relative logarithmic de Rham-Witt sheaves on semistable schemes over ${\mathbb{F}}_q[[t]]$
• On $K(1)$-local $\mathrm{TR}$
• Rigid cohomology over Laurent series field I: First definitions and basic properties