Generalized Serre conditions and perverse coherent sheaves

Abstract

In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism Y→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→X is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition Sr. In this paper, the authors introduce generalized Serre conditions—these are local cohomology conditions which include Sr and the Cohen–Macaulay condition as special cases. To any generalized Serre condition Sρ, 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 Sρ-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 Sρ-extension.