Yoga for Fourier--Mukai partnership

2025/12/18 by Lank, Pat, Manali-Rahul, Kabeer, Pavic, Nebojsa
#14A30 (primary) #14B05 #14F08 #18G80 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.16503

Abstract

We study the behavior of integral transforms under various forms of base change. In particular, we establish a yoga of local algebra and fibers to test for derived equivalences via integral transforms. This partially generalizes a result of Orlov to singular varieties and strengthens many results in the literature to arbitrary fields. Additionally, it provides new insight into fibrations and their singularities arising in more arithmetically flavored contexts (e.g. projective and flat schemes over a DVR).

Citations

• Integral transforms on singularity categories for Noetherian schemes
• Weakly approximable triangulated categories and enhancements: a survey
• The passage among the subcategories of weakly approximable triangulated categories
• Finite approximations as a tool for studying triangulated categories
• Reconstruction of projective curves from the derived category
• Simultaneous normalization and algebra husks
• Equivalences of derived categories of sheaves on quasi-projective schemes
• Relative Integral Functors for Singular Fibrations and Singular Partners
• Fourier Mukai Transforms for Gorenstein Schemes
• Generators and representability of functors in commutative and noncommutative geometry
• Flops and derived categories
• Derived categories of coherent sheaves on abelian varieties and equivalences between them

Related

• Fourier-Mukai Transforms
• Fourier Mukai Transforms for Gorenstein Schemes
• The Fourier-Mukai transform made easy
• Fourier--Mukai transforms commuting with Frobenius
• On a relative Fourier-Mukai transform on genus one fibrations