Let $f : X \to \operatorname{Spec}(R)$ be a flat, projective morphism with reduced fibers, where $R =\mathbb{C}[[t]]$. One may assume that the dualizing sheaf of the special fiber is trivial. Let $\mathcal{F}$ be a coherent sheaf on $X$, not necessarily flat over $R$. Let $\mathcal{G}$ be the sheaf of $t$-torsion sections of $\mathcal{F}$, i.e $\mathcal{G} = \operatorname{ker}(\mathcal{F} \xrightarrow{t} \mathcal{F})$.


Does $H^1(X,\mathcal{F}) = 0$ imply that $H^1(X,\mathcal{G})= 0$?



Cohomology of $t$-torsion subsheaf
$\def\C{\mathbb{C}}\def\OO{\mathcal{O}}\def\cL{\mathcal{E}}\def\cF{\mathcal{F}}$Let $X\_0$ be a variety over $\C$ and $X=X\_0\times\_{\C}R$. Denote by $i:X\_0\to X$ the obvious closed immersion.


Let $\cL$ be a coherent sheaf on $X\_0$. We will construct an extension $0\to i\_\*\cL\to \cF\xrightarrow{p} i\_\*\OO\to 0$. Such extensions are classified by $$Hom\_{D(\OO\_X)}(i\_\*\OO\_{X\_0},i\_\*\cL[1])=Hom\_{D(\OO\_{X\_0})}(Li^\*i\_\*\OO\_{X\_0},\cL[1])=\\ Hom\_{D(\OO\_{X\_0})}(Li^\*(cone(\OO\_{X}\xrightarrow{t}\OO\_{X})),\cL[1])=Hom\_{D(\OO\_{X\_0})}(\OO\_{X\_0}[1]\oplus \OO\_{X\_0},\cL[1])$$
Given such extension, there is a long exact sequence $$0\to H^0(X\_0,\cL)\to H^0(X,\cF)\to H^0(X\_0,\OO\_{X\_0})\xrightarrow{a} H^1(X\_0,\cL)\to H^1(X,\cF)\to H^1(X\_0,\OO\_{X\_0})$$
The map $a$ carries the section $1\in H^0(X\_0,\OO\_{X\_0})$ to the class in $H^1(X\_0,\cL)=Hom\_{D(X\_0)}(\OO\_{X\_0},\cL[1])$ which is the projection of the class in $Hom\_{D(\OO\_{X\_0})}(\OO\_{X\_0}[1]\oplus \OO\_{X\_0},\cL[1])$ corresponding to our extension.


Let's now analyze $\cF$ itself. Since it is an extension of sheaves killed by $t$, it is killed by $t^2$. Multiplication by $t$ on $\cF$ kills $i\_\*\cL$ and defines a map of sheaves $f:i\_\*\OO\_{X\_0}\to i\_\*\cL$ which is, when considered as an element of $Hom\_{D(\OO\_{X\_0})}(\OO\_{X\_0}[1],\cL[1])$ is again another projection of the class corresponding to the exntension. Thus, $\ker(\cF\xrightarrow{t}\cF)=i\_\*\cL+p^{-1}(\ker(\OO\_{X\_0}\to \cL))$. Note that $\cF$ is usually not going to be flat over $\C[t]/t^2$ since $\mathrm{im}(\cF\xrightarrow{t}\cF)=\mathrm{im}(i\_\*\OO\_{X\_0}\to i\_\*\cL)$ and for a flat sheaf $\ker t$ and $\mathrm{im}\, t $ should be equal(they are going to be equall iff $f$ is an isomorphism).


So, let's pick an irreducible $X\_0$ and $\cL$ such that $H^0(X\_0,\cL)\neq 0, H^1(X\_0,\cL)=\C, H^1(X\_0,\OO\_{X\_0})=0$, e.g. $X\_0=\mathbb{P}^1,\cL=\OO\oplus\OO(-2)$(probably, one can also find such situattion on a variety with trivial dualizing sheaf, say, on a K3 surface). Using a non-zero section $\OO\_{X\_0}\to \cL$ and a non-zero cohomology class $\OO\_{X\_0}\to \cL[1]$, construct a class in $Hom\_{D(\OO\_{X\_0})}(\OO\_{X\_0}[1]\oplus \OO\_{X\_0},\cL[1])$ and take the corresponding extension $\cF$. Since the map $f:i\_{\*}\OO\_{X\_0}\to i\_\*\cL$ is injective(it corresponds to a non-zero section), $\mathcal{G}=i\_\*\cL$. Since we took a non-zero cohomology class, which, moreover, generates $H^1(X\_0,\cL)$, the connecting homomorphism $a$ is surjective so $H^1(X,\cF)=0$, but $H^1(X,\mathcal{G})=H^1(X\_0,i\_\*\cL)\neq 0$.


