Ramification and cleanliness

TL;DR

This paper studies the ramification behavior of Galois torsors and $ ext{l}$-adic sheaves in characteristic $p>0$, introducing a cleanliness condition that generalizes Kato's and proposing a conjectural Riemann-Roch formula.

Contribution

It extends the notion of cleanliness for sheaves, studies its properties, and proposes a conjectural Riemann-Roch formula in the context of ramification in characteristic $p>0$.

Findings

Introduces a boundedness condition on ramification along divisors.

Establishes properties of the cleanliness condition and its implications.

Proposes a conjectural Riemann-Roch formula for sheaves with controlled ramification.

Abstract

This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ be a finite local ${\mathbb Z}_\ell$-algebra, $F$ be a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a boundedness condition on the ramification of $F$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of $F$. The cleanliness condition extends the one introduced by Kato for rank one sheaves. Roughly speaking, it means that the ramification of $F$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for $F$. Some cases of this formula have been previously proved by Kato and by the second author (T.S.).