On 2-adic deformations

We compute the versal deformation ring of a split generic $2$-dimensional representation ${\chi }_1\oplus {\chi }_2$ of the absolute Galois group of ${\mathbb{Q}}_p$. As an application, we show that the Breuil–M'ezard conjecture for both non-split extensions of ${\chi }_1$ by ${\chi }_2$ and ${\chi }_2$ by ${\chi }_1$ implies the Breuil–M'ezard conjecture for ${\chi }_1\oplus {\chi }_2$. The result is new for $p=2$, the proof works for all primes.