Generalized Serre conditions and perverse coherent sheaves

TL;DR

This paper introduces generalized Serre conditions in algebraic geometry, linking them to perverse coherent sheaves and providing criteria for their existence and properties in scheme modifications.

Contribution

It defines generalized Serre conditions, connects them to perverse t-structures, and characterizes schemes admitting canonical S_rho-ifications and extensions.

Findings

Generalized Serre conditions include S_r and Cohen-Macaulay as special cases.

Existence of canonical S_rho-ifications characterized by intermediate extension functors.

Universal properties established for S_rho-extensions in scheme modifications.

Abstract

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.