Base Change and Grothendieck Duality for Cohen-macaulay Maps
نویسنده
چکیده
Let f : X → Y be a Cohen-Macaulay map of finite type between Noetherian schemes, and g : Y ′ → Y a base change map, with Y ′ Noetherian. Let f ′ : X → Y ′ be the base change of f under g and g : X → X the base change of g under f . We show that there is a canonical isomorphism θ g : g ∗ ωf ≃ ωf ′ , where ωf and ωf ′ are the relative dualizing sheaves. The map θ g f is easily described when f is proper, and has a subtler description when f is not. If f is smooth we show that θ g corresponds to the canonical identification gΩ f = Ω f ′ of differential forms, where r is the relative dimension of f . Our results generalize the results of B. Conrad in two directions we don’t need the properness assumption, and we do not need to assume that Y and Y ′ carry dualizing complexes. Residual Complexes do not appear in this paper.
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