global section of vector bundle and reduction

Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ the smooth projective curve from mod $p$ reduction of $C_1$ , and $H_0$ the vector bundle from the reduction of $H_1$ . Is the following statment true?

(1) If $H^0(C_0,H_0)\neq 0$, then $H^0(C_1,H_1)\neq0$.

If (1) is ture, I may go on asking:

(2) The map $H^0(C_1,H_1)\to H^0(C_0,H_0)$ is surjective.

Thank you!

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(1) is true because you have a monomorphism of sheaves $p\cdot H_1=H_0\to H_1$ and the functor of global sections is left exact. –  Damian Rössler Nov 25 '12 at 21:30
Note that the reason that (2) fails in general is, loosely speaking, the semicontinuity theorem. If you assume that $H^0$ and $H^1$ are locally free (over $W_2(k)$ in this case), then (2) holds. –  Damian Rössler Nov 26 '12 at 9:30
What does mean $H^0$ and $H^1$ are locally free? –  TOM Nov 26 '12 at 10:02
Or are there any condition so that the map in (2) is not a zero map ? –  TOM Nov 26 '12 at 10:10
I really mean $H^0=H^0(C_1,H_1)$ and $H^1=H^1(C_1,H_1)$ locally free (not $H_1$). –  Damian Rössler Nov 26 '12 at 12:01

Here is a counterexample to (2).

Let $H_1$ be the lifting of the trivial line bundle on $C_0$ and suppose that $H_1\not\simeq{\cal O}_{C_1}$. Examples of such line bundles $H_1$ may be produced using the Picard scheme of $C_1$ over $W_2(k)$. I contend that the morphism $H^0(C_1,H_1)\to H^0(C_0,H_0)$ vanishes. To see this, let $\sigma_1\in H^0(C_1,H_1)$. This corresponds to a morphism of sheaves $\sigma_1:{\cal O}_{C_1}\to H_1$. Let $K_1$ be the kernel of $\sigma_1$ and ${\rm CK}_1$ be the cokernel of $\sigma_1$. Let let $K_0$ (resp. ${\rm CK}_0$) be the reduction mod. $p$ of $K_1$ (resp. ${\rm CK}_1$). The reduction mod. $p$ of $\sigma_1$ gives a morphism $\sigma_0:{\cal O}_{C_0}\to H_0\simeq{\cal O}_{C_0}$. We want to show that $\sigma_0=0$. To get a contradiction, suppose that $\sigma_0\not=0$. Then $\sigma_0$ is an isomorphism, since $C_0$ is proper over $k$ and the source and target of $\sigma_0$ are trivial. Since the tensor product is right-exact, we deduce that ${\rm CK}_0$ vanishes; but this implies that ${\rm CK}_1$ vanishes. Now using the fact that $H_1$ is locally free, we deduce likewise that $K_0$ vanishes and hence that $K_1$ vanishes. This shows that $\sigma_1$ is an isomorphism, which contradicts the assumption on $H_1$. Hence $\sigma_0=0$, which is what we wanted.

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