Criteria for pseudo-representations to arise from genuine representations

In application to modularity, one often seeks for an R=T theorem, that is, the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudo-representations, which is a generalization of the notion of traces of representations, is invented to get around this issue. We will introduce the notion of pseudo-representations and discuss the criteria for pseudo representations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudo-deformation rings for finitely presented groups, which leads to the discovery of a counterexample.