We define the characteristic cycle of a locally constant \'etale sheaf on a
smooth variety in positive characteristic ramified along boundary as a cycle in
the cotangent bundle of the variety, at least on a neighborhood of the generic
point of the divisor on the boundary. The crucial ingredient in the definition
is an additive structure on the boundary induced by the groupoid structure of
multiple self products.
We prove a compatibility with pull-back and local acyclicity in
non-characteristic situations. We also give a relation with the characteristic
cohomology class under a certain condition and a concrete example where the
intersection with the 0-section computes the Euler-Poincar\'e characteristic.Comment: 56 pages. In v2, the local acyclicity is proved in Proposition 3.14.
In v3, errors in Examples 2.18.2 and 3.18 are corrected. In v4, the
assumption in Proposition 3.14 on local acyclicity is weakened. In v5,
Conjectures on the integrality of the characteristic cycle and on the total
dimension of nearby cycles are formulated. In v6, some corrections are made
and explanations are adde