Abstract: Let X be a smooth proper scheme over a perfect field k of positive characteristic p. For p-primary torsion sheaves on X, there are two duality theories which are fundamentally different: one is the Serre-Grothendieck duality for coherent sheaves on X, which is later generalized by Ekedahl to a duality theory for coherent sheaves on W_nX with the top dRW sheaf being the dualizing sheaf; the other one is the Milnor-Kato-duality, which works for a much bigger class of sheaves, and has the top log dRW sheaf as the dualizing sheaf.
The basis of Ekedahl's work is that regular dRW differentials admit a perfect pairing via the wedge product, just as regular Kähler differentials under the Serre-Grothendieck duality.
Then it is natural to ask, what if we allow the differentials to have certain poles along a divisor D, do we still get a good duality theory? What should be the natural definition for a "dRW sheaf with ramification along D"? Moreover, what is the dual of a given ramified dRW sheaf, can it also be defined naturally?
In this talk, we will discuss our solutions to these questions, and how our ramified duality theory extends to a ramified version of the Milne-Kato duality.
