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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.

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1 Answer 1

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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.

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excuse me but could you elaborate a little more? With what you say and my sequence 3) we get a map $H^1(X,\mathcal{O}_X\otimes ker(f))\rightarrow Pic(X_A)$, but i would like to derive from that a map $H^1(X,\mathcal{O}_X)\otimes ker(f)\rightarrow \hat{Pic_X}(A)$. –  Louis La Brocante Apr 22 '13 at 8:16
    
@LouisLaBrocante: Dear Louis, See my edit. Regards, –  Matt E Apr 22 '13 at 17:36

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