  We realise Buchweitz and Flenner's semiregularity map (and hence a fortiori
Bloch's semiregularity map) for a smooth variety $X$ as the tangent of a
generalised Abel--Jacobi map on the derived moduli stack of perfect complexes
on $X$. The target of this map is an analogue of Deligne cohomology defined in
terms of cyclic homology, and Goodwillie's theorem on nilpotent ideals ensures
that it has the desired tangent space (a truncated de Rham complex).
  Immediate consequences are the semiregularity conjectures: that the
semiregularity maps annihilate all obstructions, and that if $X$ is deformed,
semiregularity measures the failure of the Chern character to remain a Hodge
class. This gives rise to reduced obstruction theories of the type featuring in
the study of reduced Gromov--Witten and Pandharipande--Thomas invariants. We
also give generalisations allowing $X$ to be singular, and even a derived Artin
stack.
