The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah–Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. (2007), doi: 10.1016/j.aim.2007.06.012], decomposes naturally in the derived category D(X) into ⊕p⩾0Sp(LX/Y[1]), the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case.Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah–Chern character, the exponential of the universal Atiyah class.We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah–Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.