Summary: We prove that irreducible residual crystalline representations of the absolute Galois group of an unramified extension of \(\mathbb{Q}_p\) have smooth representable crystalline framed deformation problems, provided that the Hodge-Tate weights lie in the Fontaine-Laffaille range. We then extend this result to the flagged lifting problem associated to any Fontaine-Laffaille upper triangular representation whose flag is of maximal length. We calculate the relative dimension of these various crystalline lifting functors in terms of the underlying Hodge-Tate weight structures, and also apply these results to give an alternative proof of the fact that every such residual representation admits a so-called “universally twistable lift”. Finally we give some brief indications as to the various directions in which these results might be generalised.