2019/03/05 by Suresh Nayak, Nayak, Suresh, Pramathanath Sastry +2
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1903.01783
openalex publication_date 2019/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a smooth map between noetherian schemes, Verdier relates the top relative\ndifferentials of the map with the twisted inverse image functor `upper shriek'.\nWe show that the associated traces for smooth proper maps can be rendered\nconcrete by showing that the resulting theory of residues satisfy the residue\nformulas (R1)--(R10) in Hartshorne's "Residues and Duality". We show that the\nresulting abstract transitivity map relating the twisted image functors for the\ncomposite of two smooth maps satisfies an explicit formula involving\ndifferential forms. We also give explicit formulas for traces of differential\nforms for finite flat maps (arising from Verdier's isomorphism) between schemes\nwhich are smooth over a common base, and use this to relate Verdier's\nisomorphism to Kunz and Waldi's regular differentials. These results also give\nconcrete realisations of traces and residues for Lipman's fundamental class map\nvia the results of Lipman and Neeman relating the fundamental class to\nVerdier's isomorphism.\n