Obstructions to lift of representations Let $G$ be a profinite group. Let $\Lambda$ be a discrete valuation ring over $\mathbb{Z}_p$. Let $A_0,A_1$ be artinian ring over $\Lambda$. Let $\alpha : A_1 \rightarrow A_0$ is a small map that is $\mathfrak{m}_{A_1}.ker(\alpha) = 0$. Assume that a representation $\rho : G \rightarrow GL_n(A_0)$ is given. I would like to know when can a representation be lifted to $GL_n(A_1)$. 
I vaguely(and perhaps incorrect) know that the obstruction lies in $H^2(G, Ad\rho)$. What is the cocycle class associated to $\rho$ which must vanish so that $\rho$ can be lifted?
Any references/hint/solution are welcome.
Thanks!
 A: Something like what you want is true, though it needs to be formulated slightly differently.  Let $k$ be the residue field of $A_0$, and let $\overline{\rho}$ denote the ``residual'' representation $\rho \otimes_{A_0} k$.
Let $I$ denote the kernel of the map from $A_1$ to $A_0$; then $I$ is a $k$-vector space.  We can associate to $\rho$ a natural ``obstruction class'' that lives in $H^2(G, Ad \overline{\rho}) \otimes_k I$.  This class vanishes if, and only if, $\rho$ lifts to $GL_n(A_1)$.  (Moreover, if this is the case, then the set of such lifts is a torsor over $H^1(G, Ad \overline{\rho}) \otimes_k I$.
Here's a brief sketch of the construction of this obstruction class (ignoring some issues with continuity that are routine but slightly annoying): 
For each $g \in G$, choose a lift $x_g$ of $\rho(g)$ to $GL_n(A_1)$.  Of course the map sending $g$ to $x_g$ need not be a group homomorphism, but for any $g,h$ in $G$, we have:
$$x_{gh}^{-1} x_g x_h = 1 + y_{g,h},$$
where $y_{g,h}$ lies in $M_n(I)$.  The group $GL_n(A_1)$ acts on $M_n(I)$ by conjugation; this action factors through the quotient $GL_n(k)$.  We can thus regard $G$ as acting on $M_n(I)$ via $\overline{\rho}$; this identifies $M_n(I)$ with $(Ad \overline{\rho}) \otimes_k I$.
With these identifications it is easy to see that $y_{g,h}$ is a $2$-cocycle with values in $(Ad \overline{\rho}) \otimes_k I$.  This cocycle vanishes if, and only if, our chosen set of lifts $x_g$ define a homomorphism of $G$ into $GL_n(A_1)$ lifting $\rho$.  Moreover if we replace our chosen lifts $x_g$ with another set of lifts $x'_g$, then this changes $y_{g,h}$ by a coboundary.
Thus there exists some lift of $\rho$ if, and only if, $y_{g,h}$ is a coboundary, completing the argument.
