Pseudorepresentations Not Arising from Genuine Representations

We show that a pseudorepresentation $\mathbf{D}\colon A[G] \rightarrow A$ of a (finite) group $G$ need not arise from a genuine representation, even if one is allowed to extend the ring $A$. This shows that a theorem of the ``embedding problem'' for residually multiplicity free pseudorepresentations in Bellaiche and Chenevier's work can not be extended to the general setting. We focus particularly on the case where $p=d=2$ with a trivial residual pseudorepresentation. The main idea is to explicitly compute the pseudodeformation ring and the trace subring of the framed deformation ring, demonstrating that these rings are different even for finite groups $G$.