  We analyse infinitesimal deformations of pairs $(X,\mathcal{F})$ with
$\mathcal{F}$ a coherent sheaf on a smooth projective manifold $X$ over an
algebraic closed field of characteristic $0$. We describe a differential graded
Lie algebra controlling the deformation problem, and we prove an analog of a
Mukai-Artamkin Theorem about the trace map.
