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We analyse infinitesimal deformations of pairs 
$(X,{\mathcal{F}})$ with 
${\mathcal{F}}$ a coherent sheaf on a smooth projective variety 
$X$ over an algebraically 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.
