Let $X$ be a smooth scheme over $k[t]/(t^2),$ where $k$ is a field of characteristic 0 (the case when $X$ is a projective curve is already interesting). Let $X\_{0} \to X$ denote $X$ with the reduced induced structure, i.e. just the fiber product of $X$ with $k$ over $k[t]/(t^2).$ Let $\beta \in H^{0}(X\_{0},\Omega^{1} \_{X\_{0}/k}).$ 


Is there a smooth $Y\_0/k$ and a map $f:X\to Y\_{0}$ as schemes over $k$ and an element $\alpha \in H^{0}(Y\_{0},\Omega^{1} \_{X\_{0}/k}) $ such that $f^\* \alpha =\beta. $



Maps from a scheme over the dual numbers to constant schemes
Welcome new contributor. Your post still has typos, so it is not completely clear what you mean. However, my best guess of your meaning has a **negative** answer. The simplest examples have $X\_0$ a smooth projective curve of genus $g=2$ or a non-hyperelliptic curve of genus $g>2$. This follows from the proof (rather than the statement) of the infinitesimal Torelli theorem for nonhyperelliptic, smooth, projective curves $X\_0$ of genus $g> 2$.


Denote by $D$ the $k$-algebra $k[t]/\langle t^2\rangle$. Thus, the $k$-vector space $\Omega\_{D/k}$ is $1$-dimensional. The sheaf of relative differentials $\Omega\_{X/k}$ is a coherent $\mathcal{O}\_X$-module. By functoriality / transitivity for differentials, for every $k$-morphism $f:X\to Y\_0$, there are induced morphisms of $k$-vector spaces, $$H^0(Y\_0,\Omega\_{Y\_0/k}) \to H^0(X,\Omega\_{X/k}) \to H^0(X\_0,\Omega\_{X\_0/k}).$$ Thus, if there exists $f$ and $\beta$ as you state, then there exists a section $f^\*\beta\in H^0(X,\Omega\_{X/k})$ such that the image in $H^0(X\_0,\Omega\_{X\_0/k})$ equals $\alpha$. To understand this better, consider the short exact sequence of locally free $\mathcal{O}\_{X\_0}$-modules,
$$0 \to \Omega\_{D/k}\otimes\_k \mathcal{O}\_{X\_0} \to \Omega\_{X/k}\otimes\_D (D/tD) \to \Omega\_{X\_0/k} \to 0.$$ The long exact sequence of cohomology gives a connecting map
$$
\alpha(\xi):H^0(X\_0,\Omega\_{X\_0/k}) \to \Omega\_{D/k}\otimes\_k H^1(X\_0,\mathcal{O}\_{X\_0}).
$$


Now assume that the Kuranishi map of $X/S$ is nonzero, $$\kappa:k \to \text{Ext}^1\_{\mathcal{O}\_{X\_0}}(\Omega\_{X\_0/k},\mathcal{O}\_{X\_0}).$$ Also assume that the Torelli map is an immersion at $X\_0$. This holds if $X\_0$ has genus $g=2$ or if $X\_0$ is a non-hyperelliptic curve of genus $g>2$. Then it follows by the proof of the infinitesimal Torelli theorem that there exists $\alpha\in H^0(X\_0,\Omega\_{X\_0/k})$ that does not lift to a global section in $H^0(X,\Omega\_{X/S})$. For instance, confer Corollary 1.9, p. 8 of the following.


Chris Peters.   

Lectures on Torelli Theorems. Spring School Rennes 2014.   

<https://www.lebesgue.fr/sites/default/files/attach/hodge-peters.pdf>


