Functoriality of HKR isomorphisms

For a closed embedding of smooth schemes X ,→ S with a fixed first-order splitting, one can construct Hochschild–Kostant–Rosenberg (HKR) isomorphisms between the derived scheme X ×R S X and the total space of the shifted normal bundle NX/S[−1], thanks to Arinkin–C˘ald˘araru, Arinkin–C˘ald˘araru–Hablicsek, and Grivaux. In this pa per, we study the functoriality property of the HKR isomorphisms for a sequence of closed embeddings X ,→ Y ,→ S. The HKR isomorphism is functorial when a certain co homology 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.