Grothendieck Duality via Diagonally Supported Sheaves

2023/03/28 by Andy Jiang, Jiang, Andy
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2303.16086

openalex publication_date 2023/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Following a formula found in the paper of Avramov, Iyengar, Lipman, and Nayak (2010) and ideas of Neeman and Khusyairi, we indicate that Grothendieck duality for finite tor-amplitude maps can be developed from scratch via the formula f^! := δ^*π₁^×f^*. Our strategy centers on the subcategory Γ_Δ(QCoh(X × X)) of quasicoherent sheaves on X × X supported on the diagonal. By exclusively using this subcategory instead of the full category QCoh(X × X) we give systematic categorical proofs of results in Grothendieck duality and reprove many formulas found in Neeman (2018). We also relate some results in Grothendieck duality with properties of the sheaf of (derived) Grothendieck differential operators.

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