In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism from Y to X where the geometry of Y is nicer than that of X. One version of this problem, first studied by Faltings, requires Y to be Cohen-Macaulay; in this case Y is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition S_r. In this paper, the authors introduce generalized Serre conditions--these are local cohomology conditions which include S_r and the Cohen-Macaulay condition as special cases. To any generalized Serre condition S_rho, there exists an associated perverse t-structure on the derived category of coherent sheaves on a suitable scheme X. Under appropriate hypotheses, the authors characterize those schemes for which a canonical finite S_rho-ification exists in terms of the intermediate extension functor for the associated perversity. Similar results, including a universal property, are obtained for a more general morphism extension problem called S_rho-extension.
Download