I'm doing exercises in Hartshorne. I have a detail I want to ask.
Say $X$ is a scheme over $k$, which by definition means that there is a morphism of schemes $$(f,f^\sharp):(X,\mathcal{O}_X)\to (\text{Spec } k, \mathcal{O}_{\text{Spec } k} )$$Also given that $$(g,g^\sharp):(\text{Spec } k[x]/(x^2), \mathcal{O}_{\text{Spec } k[x]/(x^2)} )\to (X,\mathcal{O}_X)$$ is a $k$-morphism. Then $\text{Spec } k[x]$ must also be a scheme over $k$, right? Hence we must have a morphism $$(\text{Spec } k[x]/(x^2), \mathcal{O}_{\text{Spec } k[x]/(x^2)} )\to (\text{Spec } k, \mathcal{O}_{\text{Spec }k } )$$ By the definition of a $k$-morphism, it's safe to choose such a morphism to be the composition of $(f,f^\sharp)$ and $(g,g^\sharp)$. Then this map induces a map on global sections $k\to k[x]/(x^2) $. My question is, can we explicitly describe this map?
It's just a mess when I tried to use the definition because the sheaves are not easy to describe.