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We prove a purely local form of a result of Saito and Yatagawa. They provedthat the characteristic cycle of a constructible \'etale sheaf is determined bywild ramification of the sheaf along the boundary of a compactification. Butthey had to consider ramification at all the points of the compactification. Wegive a pointwise result, that is, we prove that the characteristic cycle of aconstructible \'etale sheaf around a point is determined by wild ramificationat the point. The key ingredient is to prove that wild ramification of thestalk of the nearby cycle complex of a constructible \'etale sheaf at a pointis determined by wild ramification at the point.

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