This talk is about joint work with Ted Chinburg and Luc Illusie. I will first recall a generalization of Mazur's deformations of Galois representations to deformations of complexes in a derived category. I will then show how one can define obstructions to deforming such complexes by using spectral sequences associated to certain double complexes. This construction relies on ideas by O. Gabber. This talk is sponsored by a grant from the Fund to Encourage Women (FEW) of the Trustees' Council of Penn Women.