Duality of sheaves with actions

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As an analogue of the duality of quasi-coherent sheaves with actions over schemes, we studied the duality with actions over formal schemes. As a result, we have proved the following : Let S be a universally catenary Nagata Noetherian scheme, f : X→Y a surjective universally open S-morphism of S-schemes. If X is of finite type over S and Y is reduced, then Y is of finite type over S. As an appliatlon, we obtained a new proof of the finiteness of geometric quotients due to Fogarty. As another application, we proved that if S is a Noetherian scheme, f : X→Y a faithfully flat S-morphism of S-schemes, and X is of finite type over S, then Y is of finite type over S. Later, as a generalization, we proved that we may replace the faithful flatness by purity. Moreover, we obtained a new proof of the global F-regularity of Schubert varieties due to Lauritzen, Raben-Pedersen, and Thomsen. Moreover, we obtained a new geometric proof of the computation of invariant subrings first provedin 70's by De Concini and Procesi. Moreover, we extended the class of rings of which we take the invariant subrings, from the class of polynomial rings to that of determinantal rings. Lastly, we proved the following : Let R be a Dedekind domain, G an affine flat R-group scheme, B a flat R-algebra on which G acts. If a Noetherian R-algebra and an R-algebra map A→B^G is given, and if for any R-algebra which is an algebraically closed field, the induced map Kotimes A→(Kotimes B)^{Kotimes G} is an isomorphism, then for any R-algebra S, the induced map S otimes A→(Sotimes B)^{Sotimes G} is an isomorphism.