Kato's Ramification filtration via de Rham-Witt complex and applications
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909993439068160 |
|---|---|
| author | Krishna, Amalendu Majumder, Subhadip |
| author_facet | Krishna, Amalendu Majumder, Subhadip |
| contents | Given an $F$-finite regular scheme $X$ of positive characteristic and a simple normal crossing divisor $E$ on $X$, we introduce a filtration on the de Rham-Witt complex $W_mΩ^\bullet_{X\setminus E}$. When $X$ is the spectrum of a henselian discrete valuation ring $A$ with quotient field $K$, this extends the classical filtration on $W_m(K)$ due to Brylinski. We show that Kato's ramification filtration on $H^q_\et(X \setminus E, {\Q}/{\Z}(q-1))$ for $q \ge 1$ admits an explicit description in terms of the above filtration of the de Rham-Witt complex of $X \setminus E$. When $q =1$, this specializes to the results of Kato and Kerz-Saito.
As applications, we prove refinements of the duality theorem of Jannsen-Saito-Zhao for smooth projective schemes over finite fields and the duality theorem of Zhao for semi-stable schemes over henselian discrete valuation rings of positive characteristic with finiteresidue fields. We also prove a modulus version of the duality theorem of Ekedahl. As another application, we prove Lefschetz theorems for Kato's ramification filtrations for smooth projective varieties over $F$-finite fields. This extends a result of Kerz-Saito for $H^1$ to higher cohomology. Similar results are proven for the Brauer group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12177 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kato's Ramification filtration via de Rham-Witt complex and applications Krishna, Amalendu Majumder, Subhadip Algebraic Geometry Primary 14G17, Secondary 19F15 Given an $F$-finite regular scheme $X$ of positive characteristic and a simple normal crossing divisor $E$ on $X$, we introduce a filtration on the de Rham-Witt complex $W_mΩ^\bullet_{X\setminus E}$. When $X$ is the spectrum of a henselian discrete valuation ring $A$ with quotient field $K$, this extends the classical filtration on $W_m(K)$ due to Brylinski. We show that Kato's ramification filtration on $H^q_\et(X \setminus E, {\Q}/{\Z}(q-1))$ for $q \ge 1$ admits an explicit description in terms of the above filtration of the de Rham-Witt complex of $X \setminus E$. When $q =1$, this specializes to the results of Kato and Kerz-Saito. As applications, we prove refinements of the duality theorem of Jannsen-Saito-Zhao for smooth projective schemes over finite fields and the duality theorem of Zhao for semi-stable schemes over henselian discrete valuation rings of positive characteristic with finiteresidue fields. We also prove a modulus version of the duality theorem of Ekedahl. As another application, we prove Lefschetz theorems for Kato's ramification filtrations for smooth projective varieties over $F$-finite fields. This extends a result of Kerz-Saito for $H^1$ to higher cohomology. Similar results are proven for the Brauer group. |
| title | Kato's Ramification filtration via de Rham-Witt complex and applications |
| topic | Algebraic Geometry Primary 14G17, Secondary 19F15 |
| url | https://arxiv.org/abs/2601.12177 |