The operators of linear elasticity can be arranged in complexes, and a goal is to construct subcomplexes consisting of finite element spaces. Such complexes are related to de Rham complexes through a diagram chase known as the BGG construction. The construction of finite element spaces makes appear cochains with coefficients in rigid motions. A de Rham theorem on cohomology groups can be proved in the case without curvature, whereas a Bianchi identity can be proved for the curvature. This is joint work with Kaibo Hu.