2011/06/14 by Christopher L. Bremer, Daniel S. Sage · 1 citation
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Algebraic structure #Algebraic structures and combinatorial models #Base change #Coherent sheaf #Cohomology #Commutative Algebra and Its Applications #Computer science #Extension (predicate logic) #Functor #Mathematical analysis #Mathematics #Morphism #Property (philosophy) #Pure mathematics #Scheme (mathematics) #math.AC #math.AG #msc:14B14 #msc:14F43
paper · pdf · doi:10.1016/j.jalgebra.2013.06.018
published as J. Algebra 392 (2013) 85-96 · 13 pages
arxiv created 2011/06/14 · openalex publication_date 2013/07/18 · arxiv updated 2013/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 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 Srho, 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 Srho-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 Srho-extension.