Some exact sequences of cohomology on picard schemes I'm looking for the answer of this question. $X$ is a variety over a field $k$, and $Art_k$ is the category of local artinian $k$-algebras whose residue field is $k$.
I consider the formal completion of the Picard functor of $X$, which is the functor $\hat{Pic_X}:Art_k\longrightarrow Set$ which send an algebra $A$ to $Ker(Pic(X_A)\rightarrow Pic(X)$) (alternatively, it is easy to see that $\hat{Pic_X}(A)=H^1(X,1+\mathcal{O}_X\otimes_k \mathfrak{m}_A)$).
Now let $f:A\rightarrow A'$ be a surjective morphism of $Art_k$ whose kernel is anihilated by $\mathfrak{m}_A$. I want to show that there are applications
$$H^1(X,\mathcal{O}_X)\otimes ker(f)\rightarrow \hat{Pic_X}(A)\rightarrow \hat{Pic_X}(A')\rightarrow H^2(X,\mathcal{O}_X)\otimes ker(f)$$
which should (as the hint says) come from the long exact sequences of cohomology arising from the two following short sequences :
$$1\rightarrow 1+\mathcal{O}_X\otimes \mathfrak{m}_A\rightarrow \mathcal{O}^{\ast}_{X_A}\rightarrow \mathcal{O}^{\ast}_X\rightarrow 1$$
$$1\rightarrow 1+\mathcal{O}_X\otimes I\rightarrow \mathcal{O}^{\ast}_{X_A}\rightarrow \mathcal{O}^{\ast}_{X_{A'}}\rightarrow 1$$
Unfortunately using these i only get applications
$$(3)~~~~~~~~~~~~~~~~H^1(X,1+\mathcal{O}_X\otimes ker(f))\rightarrow Pic(X_A)\rightarrow Pic(X_{A'})\rightarrow H^2(X,1+\mathcal{O}_X\otimes ker(f))$$
... if anybody would have some clue, i would be grateful. This should have connections with deformation theory.
 A: I think the point is simply that $1 + \mathcal O_X \otimes ker(f)$ is isomorphic to $\mathcal O_X \otimes ker(f)$ (a direct sum of copies of $\mathcal O_X$).  The isomorphism sends $1 + x$ to $x$, and takes multipliction to addition.  (This should be immediate to check, but you will have to use the fact that $ker(f)$ is killed by $\mathfrak m_A$, and the related fact that any element of $ker(f)$ has square equal to $0$.

Added:  The inclusion $k \hookrightarrow A$ and the surjection $A \to k$
induce a surjection Spec $A \to$ Spec $k$, and a section Spec $k \to $ Spec $A$,
which in turn induce a surjection $X_A \to X$ and a section $X \to X_A$.
These in turn induce a morphism Pic$(X) \to$ Pic$(X_A)$ (pull-back along
the surjection) and Pic$(X_A) \to$ Pic$(X)$ (restriction to the section).
The second of these is left-inverse to the first, and so we see that
Pic$(X)$ is naturally a summand of Pic$(X_A)$, with complement equal to
$\widehat{\mathrm{Pic}}_X(A)$.  Likewise with Pic$(X_{A'})$.
Thus in the long exact sequence, we can just remove these two Pic$(X)$ summands (which map isomorphically to one another) and so replace
Pic$(X_A)$ and Pic$(X_{A'})$ by $\widehat{\mathrm{Pic}}_X(A)$ and
$\widehat{\mathrm{Pic}}_X(A)$ without altering any other part of the sequence.
