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Duality for Hodge-Witt cohomology with modulus

Fei Ren, Kay Rülling

Abstract

Given an effective Cartier divisor D with simple normal crossing support on a smooth and proper scheme X over a perfect field of positive characteristic p, there is a natural notion of de Rham-Witt sheaves on X with zeros along D. We show that these sheaves correspond under Grothendieck duality for coherent sheaves to de Rham-Witt sheaves on X with modulus (X,D), as defined in the theory of cube invariant modulus sheaves with transfers developed by Kahn-Miyazaki-Saito-Yamazaki. From this we deduce refined versions of Ekedahl - and Poincaré duality for crystalline cohomology generalizing results of Mokrane and Nakkajima for reduced D, and a modulus version of Milne-Kato duality for étale motivic cohomology with p-primary torsion coefficients, which refines a result of Jannsen-Saito-Zhao. We furthermore get new integral models for rigid cohomology with compact supports on the complement of D and a modulus version of Milne's perfect Brauer group pairing for smooth projective surfaces over finite fields.

Duality for Hodge-Witt cohomology with modulus

Abstract

Given an effective Cartier divisor D with simple normal crossing support on a smooth and proper scheme X over a perfect field of positive characteristic p, there is a natural notion of de Rham-Witt sheaves on X with zeros along D. We show that these sheaves correspond under Grothendieck duality for coherent sheaves to de Rham-Witt sheaves on X with modulus (X,D), as defined in the theory of cube invariant modulus sheaves with transfers developed by Kahn-Miyazaki-Saito-Yamazaki. From this we deduce refined versions of Ekedahl - and Poincaré duality for crystalline cohomology generalizing results of Mokrane and Nakkajima for reduced D, and a modulus version of Milne-Kato duality for étale motivic cohomology with p-primary torsion coefficients, which refines a result of Jannsen-Saito-Zhao. We furthermore get new integral models for rigid cohomology with compact supports on the complement of D and a modulus version of Milne's perfect Brauer group pairing for smooth projective surfaces over finite fields.
Paper Structure (15 sections, 80 theorems, 636 equations)

This paper contains 15 sections, 80 theorems, 636 equations.

Table of Contents

  1. Introduction
  2. The $p$-saturated filtration
  3. The $p$-saturated filtration defines a conductor
  4. The proof of \ref{['c3']}
  5. Some more properties of the $p$-saturated filtration
  6. Hodge-Witt sheaves with modulus
  7. The structure of Hodge-Witt sheaves with modulus
  8. Applications to Hodge-Witt cohomology with modulus
  9. The structure of Hodge-Witt sheaves with zeros
  10. Duality for Hodge-Witt cohomology with modulus: finite level
  11. Duality for Hodge-Witt cohomology with modulus: infinite level
  12. Milne-Kato duality with modulus
  13. Weights zero and one
  14. Weight zero
  15. Weight one

Key Result

Theorem 1

Multiplication induces isomorphisms, for all $q\ge 1$ and $n$, and

Theorems & Definitions (169)

  • Theorem 1: Theorem \ref{['thm:duality-mod']}
  • Corollary 1
  • Theorem 2: Theorem \ref{['thm:fil-cond']} and Theorem \ref{['thm:HW-modulus']}
  • Theorem 3: Theorem \ref{['thm:top-Lef']}
  • Theorem 4: Corollary \ref{['cor:limit-crys']}
  • Theorem 5: Theorem \ref{['thm:Zpnqfrp-duality']}
  • Corollary 2: Corollary \ref{['cor:Zpnq-duality-finite']}
  • Theorem 6: Theorem \ref{['thm:duality-Br-surf']}
  • Theorem 7: Theorem \ref{['thm:strHWM']}, Theorem \ref{['thm:HW-zeros']}
  • Definition 2.4
  • ...and 159 more