Cyclic forms on DG-Lie algebroids and semiregularity

Given a transitive DG-Lie algebroid (A; p) over a smooth separated scheme X of finite type over a field K of characteristic zero, we define a notion of connection ∇: RΓ(X; Ker p) → RΓ(X; ΩX1[ 1]⊗Kerp)and construct anL∞-morphism between DG-Lie algebras f: RΓ(X; Ker p) RΓ(X; Ω≤1 X [2]) associated to a connection and to a cyclic form on the DG-Lie algebroid. In this way, we obtain a lifting of the first component of the modified Buchweitz–Flenner semiregularity map in the algebraic context, which has an application to the deformation theory of coherent sheaves on X admitting a finite locally free resolution. Another application is to the deformations of (Zariski) principal bundles on X.