For a constructible etale sheaf on a smooth variety of positive characteristic ramified along an effective divisor, the largest slope in Abbes and Saitos ramification theory of the sheaf gives a divisor with rational coefficients called the conductor divisor. In this article, we prove decreasing properties of the conductor divisor after pull-backs. The main ingredient behind is the construction of etale sheaves with pure ramifications. As applications, we first prove a lower semi-continuity property for conductors of etale sheaves on relative curves in the equal characteristic case, which supplement Deligne and Laumons lower semi-continuity property of Swan conductors and is also an $ell$-adic analogue of Andres semi-continuity result of Poincare-Katz ranks for meromorphic connections on complex relative curves. Secondly, we give a ramification bound for the nearby cycle complex of an etale sheaf ramified along the special fiber of a regular scheme semi-stable over an equal characteristic henselian trait, which extends a main result in a joint work with Teyssier and answers a conjecture of Leal in a geometric situation.
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