#### Vol. 15, No. 6, 2021

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Wild ramification, nearby cycle complexes, and characteristic cycles of $\ell$-adic sheaves

### Hiroki Kato

Vol. 15 (2021), No. 6, 1523–1564
##### Abstract

We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at that point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale sheaf at a point is determined by wild ramification at that point.

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