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For a smooth surface X over an algebraically closed field of positivecharacteristic, we consider the ramification of an Artin-Schreier extension ofX. A ramification at a point of codimension 1 of X is understood by the Swanconductor. A ramification at a closed point of X is understood by the invariantr_x defined by Kato [2]. The main theme of this paper is to give a simpleformula to compute r_x' defined in [4], which is equal to r_x for goodArtin-Schreier extension. We also prove Kato's conjecture for upper bound ofr_x.

On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic III