  For a constructible \'etale sheaf on a smooth variety of positive
characteristic ramified along an effective divisor, the largest slope in Abbes
and Saito's 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 Laumon's lower semi-continuity property of Swan
conductors and is also an $\ell$-adic analogue of Andr\'e's semi-continuity
result of Poincar\'e-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.
