  In this paper we study classical deformations of diagrams of commutative
algebras over a field of characteristic 0. In particular we determine several
homotopy classes of DG-Lie algebras, each one of them controlling this above
deformation problem: the first homotopy type is described in terms of the
projective model structure on the category of diagrams of differential graded
algebras, the others in terms of the Reedy model structure on truncated
Bousfield-Kan approximations. The first half of the paper contains an
elementary introduction to the projective model structure on the category of
commutative differential graded algebras, while the second half is devoted to
the main results.
