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Local cohomology on diagrams of schemes

We define local cohomology on a diagram of schemes, and discuss basic properties of it. In particular, we prove the independence theorem and the flat base change for local cohomology on diagrams of schemes. We also introduce the notion of G-localness of a G-scheme. It is an equivariant analogue of localness of a scheme. As an application, we prove that Cohen–Macaulay property is inherited by the affine geometric quotient under the action of a linearly reductive group scheme over a field. This generalizes the special case (that the ring in question contains a field) of the theorem of Hochster and Eagon on the Cohen–Macaulay property of an invariant subring under the action of a finite group.