## Tohoku Mathematical Journal 2015 June SECOND SERIES VOL. 67, NO. 2

 Tohoku Math. J. 67 (2015), 153-194

Title RAMIFICATION AND NEARBY CYCLES FOR $\ELL$-ADIC SHEAVES ON RELATIVE CURVES

Author Haoyu Hu

(Received July 24, 2013, revised March 4, 2014)
Abstract. Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an $\ell$-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.

Mathematics Subject Classification. Primary 14F20; Secondary 11S15.

Key words and phrases. Nearby cycles, ramification, characteristic cycles, Deligne-Kato's formula.