  Given a transitive DG-Lie algebroid $(\mathcal{A}, \rho)$ over a smooth
separated scheme $X$ of finite type over a field $\mathbb{K}$ of characteristic
$0$ we define a notion of connection $\nabla \colon
\mathbf{R}\Gamma(X,\mathrm{Ker} \rho) \to \mathbf{R}\Gamma
(X,\Omega_X^1[-1]\otimes \mathrm{Ker} \rho)$ and construct an $L_\infty$
morphism between DG-Lie algebras $f \colon \mathbf{R}\Gamma(X, \mathrm{Ker}
\rho) \rightsquigarrow\mathbf{R}\Gamma(X, \Omega_X^{\leq 1} [2])$ 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 $X$.
