Variation of the Swan conductor of an $\mathbb{F}_{\ell}$-sheaf on a rigid annulus
| Autor: | Bah, Amadou |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $C=A(r, r')$ be a closed annulus of radii $r$ and $r'$ ($r < r' \in \mathbb{Q}_{\geq 0}$) over a complete discrete valuation field with algebraically closed residue field of characteristic $p>0$. To an \'etale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on $C$, ramified at most at a finite set of rigid points of $C$, we associate an Abbes-Saito Swan conductor function $\mathrm{sw}_{\mathrm{AS}}(\mathcal{F}, \cdot): [r, r']\cap \mathbb{Q}_{\geq 0} \to \mathbb{Q}$ which, for the variable $t$, measures the ramification of $\mathcal{F}\lvert C^{[t]}$ - the restriction of $\mathcal{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-thickness - along the special fiber of the normalized integral model of $C^{[t]}$. We show that this function is continuous, convex and piecewise linear outside the radii of the ramification points of $\mathcal{F}$, with finitely many slopes which are all integers. For two distinct radii $t$ and $t'$ lying between consecutive radii of ramification points of $\mathcal{F}$, we compute the difference of the slopes of $\mathrm{sw}_{\mathrm{AS}}(\mathcal{F}, \cdot)$ at $t$ and $t'$ as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$. Comment: 34 pages. Comments are welcome |
| Databáze: | arXiv |
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