Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ be the inclusion of the generic point of $C$.
Question:
Is it true that the natural morphism $H^2(C,J)\to H^2(C, j_* j^* J)$ is injective? (Here we are working with étale or fppf cohomologies.)
Background:
I was reading on a paper related to the Hitchin fibration.
In this context, we have a reductive group $G$ over $k$ with Lie algebra $\mathfrak{g}$. $G$ acts on $\mathfrak{g}$ by conjugation. Given any point $x$ in $\mathfrak{g}$, let $G_x$ be the stabilizer.
When $x$ is regular, that is when $G_x$ has dimension as small as possible, we have that $G_x$ is commutative.
For example, when $x$ is regular semisimple, $G_x$ is a maximal torus in $G$; when $x$ is regular nilpotent and $G=\operatorname{SL}_2$, we have that $G_x\cong \mu_2\times\mathbb{G}_a$.
The $J/C$ in the question is the neutral component of a group scheme of regular centralizers. In particular, over each point $c$ of $C$, each fiber $J_c$ is isomorphic to the neutral component $G_x^0$ of a $G_x$ for some regular $x$ in $\mathfrak{g}$. Note that the fibers can vary, for example, it may happen that a general fiber is of multiplicative type while a special fiber is unipotent.
In the paper that I am reading, the authors prove that $H^2(\eta, j^* J)=0$ and then conclude that $H^2(C, J)=0$ without saying why we can pass from $\eta$ to $C$.
I think something is missing, namely, let $Q$ be the quotient of $J\to j_* j^* J$, we need to show that $H^1(C, Q)=0$. However, I do not know what this $Q$ is.
In the very special case where $J=\mathbb{G}_m $, we know that $Q$ is the sheaf of Cartier divisors.
However, in general I don't know what this $Q$ looks like.
Question 2: Is $Q$ always a direct sum of skyscraper sheaves on codimension 1 points?
Any help is greatly appreciated!
Best Answer
Question 2: Yes.
First, a criterion for a sheaf to be a direct sum of skyscraper sheaves on closed points: It suffices for every section to be supported on finitely many closed points. Indeed, for any sheaf $\mathcal F$ on a space $X$ and point $x$ there is a map from the skyscraper sheaf at $x$ associated to the group of sections of $\mathcal F$ supported only at $x$ to $\mathcal F$, hence a map from the direct sum over closed points $x$ of the skyscraper sheaf of sections supported at that point to $\mathcal F$. This map is always injective and is surjective if every section is supported at finitely many points, as then by the sheaf condition it is a sum of sections supported only at one point.
Second, every section of $Q$ is supported at finitely many points because every section of $Q$ locally comes from a section of $j_* j^* J$ which corresponds to a section of $J$ defined over the field of fractions, whose coordinates are finitely many fractions, and away from the points where those fractions have poles, the section arises from a section of $J$ and thus vanishes in $Q$.
I guess this implies a positive answer to Question 1 too by the compatibility of cohomology of Noetherian spaces with infinite direct sums and the vanishing of cohomology of skyscraper sheaves since we're over an algebraically closed field.