Local cohomology and base change

Abstract

Let X⟶fS be a morphism of Noetherian schemes, with S reduced. For any closed subscheme Z of X finite over S, let j denote the open immersion X∖Z↪X. Then for any coherent sheaf F on X∖Z and any index r≥1, the sheaf f⁎(Rrj⁎F) is generically free on S and commutes with base change. We prove this by proving a related statement about local cohomology: Let R be Noetherian algebra over a Noetherian domain A, and let I⊂R be an ideal such that R/I is finitely generated as an A-module. Let M be a finitely generated R-module. Then there exists a non-zero g∈A such that the local cohomology modules HIr(M)⊗AAg are free over Ag and for any ring map A→L factoring through Ag, we have HIr(M)⊗AL≅HI⊗ALr(M⊗AL) for all r.