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 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 ρ , 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 ρ -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 ρ -extension.