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Here's a candidate definition for "total ramification" which jives jibes quite well with pure inseparability:

A finite map f:X -> Y is totally ramified at y if the scheme theoretic fibre X_y -> y is a universal homeomorphism.

If k is a field, and A is a finite k-algebra, then A is totally ramified over k in the above sense if and only if a) A is local, and b) the last condition holds after all base changes on k. If k has characteristic 0, this is equivalent to requiring that A be local with residue field k. This means that in the case X and Y are curves over a field of characteristic 0, this gives the usual notion. On the other hand, a finite extension L/K of fields is totally ramified in the above sense iff L/K is geometrically connected iff L/K is purely inseparable.

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Here's a candidate definition for "total ramification" which jives quite well with pure inseparability:

A finite map f:X -> Y is totally ramified at y if the scheme theoretic fibre X_y -> y is a universal homeomorphism.

If k is a field, and A is a finite k-algebra, then A is totally ramified over k in the above sense if and only if a) A is local, and b) the last condition holds after all base changes on k. If k has characteristic 0, this is equivalent to requiring that A be local with residue field k. This means that in the case X and Y are curves over a field of characteristic 0, this gives the usual notion. On the other hand, a finite extension L/K of fields is totally ramified in the above sense iff L/K is geometrically connected iff L/K is purely inseparable.