Infinitesimal neighborhoods of an affine bundle Let $S$ be an $\mathbb{A}^1$-bundle over $\mathbb{P}^1_{\mathbb{C}}$.
We denote by $\Delta$ the diagonal of $S \times S$ (so, $\Delta \simeq S$).
Let us consider the $k$-th infinitesimal neighborhoods $\Delta_{(k)}$ of $\Delta$ in $S \times S$.
Then, my question is

Are there any quasi-coherent sheaf $\mathcal{F}$ on $\Delta_{(k)}$ such that $$ H^2(\Delta_{(k)}, \mathcal{F}) \neq 0 ?$$

 A: I think it is still true that $\mathrm{H}^{2}(\Delta_{(k)},\mathcal{F}) = 0$ for all quasi-coherent $\mathcal{O}_{\Delta_{(k)}}$-modules $\mathcal{F}$, the point being that $\Delta_{(k)}$ is (1) separated and (2) can be covered by two affine open subschemes. Given (1) and (2), we can use the fact that derived functor cohomology can be computed with Cech cohomology for quasi-coherent sheaves for (semi-)separated Noetherian schemes (see either Hartshorne, Algebraic Geometry, III, Theorem 4.5 or SP Tag 01ET, etc).
Let $f : S \to \mathbb{P}_{\mathbb{C}}^{1}$ be the projection, and let $\alpha_{k} : S \to \Delta_{(k)}$ be the nilpotent closed immersion. We have $\Delta_{(0)} \simeq S$. Since $f$ is an $\mathbb{A}^{1}$-bundle morphism, it is in particular an affine morphism.
For (1), note that $f : S \to \mathbb{P}_{\mathbb{C}}^{1}$ and $\mathbb{P}_{\mathbb{C}}^{1} \to \operatorname{Spec} \mathbb{C}$ are separated so their composition $S \to \operatorname{Spec} \mathbb{C}$ is separated; thus the fiber product $S \times_{\mathbb{C}} S \to \operatorname{Spec} \mathbb{C}$ is separated, thus $\Delta_{(k)} \to \operatorname{Spec} \mathbb{C}$ is separated.
For (2), let $U_{0},U_{1} \subseteq \mathbb{P}_{\mathbb{C}}^{1}$ be the standard affine open covering of $\mathbb{P}_{\mathbb{C}}^{1}$, and set $V_{i} := f^{-1}(U_{i})$; as noted above, $f$ is an affine morphism, so each $V_{i}$ is affine. Since $\alpha_{k}$ is a nilpotent closed immersion, the underlying map of $\alpha_{k}$ is a homeomorphism; let $V_{i,(k)} \subset \Delta_{(k)}$ denote the unique open subscheme such that $\alpha_{k}^{-1}(V_{i,(k)}) = V_{i}$. Then $V_{0,(k)},V_{1,(k)}$ are open subschemes of $\Delta_{(k)}$ whose preimages in $S$ are affine; hence they themselves are affine by e.g. SP Tag 06AD.
