Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes (Q1092136)

The author considers a flat algebraic group scheme G over a noetherian base scheme S acting on a noetherian scheme X. He proves that in certain cases, any sheaf of finitely generated modules with G-action on X admits an equivariant resolution by finitely generated locally free modules with G-action. The restrictions on X, S and G are of the following sort: X and S are regular or affine, or are quasiprojective over an affine scheme; G is semisimple, or is reductive over a normal base S, or G is affine and smooth with connected fibers over a regular noetherian base S of dimension \(\leq 2\). The statement about reductive groups is a conjecture of Seshadri, who showed that it implies finite generation of rings of invariants of reductive group actions [\textit{C. S. Seshadri}, Adv. Math. 26, 225-274 (1977; Zbl 0371.14009)]. The resolution results also yield proofs that semisimple groups over any base S (and certain other types of groups as well) are linear in that they can be embedded as closed subgroups of Aut(V) for V a vector bundle over S. Similarly, if X is non- equivariantly affine (resp. normal and quasiprojective over S) then X can be equivariantly embedded into a linear action on a vector space bundle (resp. projective space bundle), strengthening the results of \textit{H. Sumihiro} in J. Math. Kyoto Univ. 15, 573-605 (1975; Zbl 0331.14008).