Desingularization of quasi-excellent schemes in characteristic zero

2007/03/31 by Michael Temkin
Computer Science · Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Algebraic structures and combinatorial models #Conjecture #Converse #Geometry #Gravitational singularity #Mathematical analysis #Mathematics #Noetherian #Polynomial and algebraic computation #Pure mathematics #Resolution (logic) #Resolution of singularities #Scheme (mathematics) #Type (biology) #Zero (linguistics) #math.AG

paper · pdf · doi:10.1016/j.aim.2008.05.006

published as Advances in Mathematics 219 (2008), pp. 488-522 · 35 pages, revised version

openalex publication_date 2008/06/19 · arxiv created 2008/09/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent Q-schemes.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie
• Smoothness, semi-stability and alterations
• Crit�res de platitude et de projectivit�
• Éléments de géométrie algébrique : I. Le langage des schémas

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