Functoriality of HKR isomorphisms

2020/01/31 by Shengyuan Huang, Huang, Shengyuan
Mathematics · #14F05(primary) 17B(secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2002.00017

openalex publication_date 2020/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a closed embedding of smooth schemes X\hookrightarrow S with a fixed first order splitting, one can construct HKR isomorphisms between the derived scheme X×^R_S X and the total space of the shifted normal bundle ℕ_X/S[-1], due to Arinkin-Căldăraru, Arinkin-Căldăraru-Hablicsek, and Grivaux. In this paper, we study functoriality property of the HKR isomorphisms for a sequence of closed embeddings X\hookrightarrow Y\hookrightarrow S. The HKR isomorphism is functorial when a certain cohomology class, which we call the Bass-Quillen class, vanishes. We obtain Lie theoretic interpretations for the HKR isomorphisms and for the Bass-Quillen class as well.

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