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 N X/S [-1], thanks to Arinkin-Cldraru, Arinkin-Cldraru-Hablicsek, and Grivaux.In this paper, 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 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.