I'm reading Eisenbud and Harris's The Geometry of Schemes, and in particular I'm trying to make sense of the example in section II.1.2 when they explain how the scheme $X = \operatorname{Spec} K[x]_{(x)}$ can be viewed as the germ of $\mathbb{A}^1_K$ at the origin. The natural map $K[x] \to K[x]_{(x)}$ gives a morphism of schemes $X \to \mathbb{A}^1_K$, and they claim that this allows one to view $X$ as a subscheme of $\mathbb{A}^1_K$. My questions are
- How does one give $X \subset \mathbb{A}^1_K$ the structure of a subscheme even if its not closed nor open in $\mathbb{A}^1_K$, and
- How would one check that the scheme structures on $X = \operatorname{Spec} K[x]_{(x)}$ and whatever subscheme structure $X$ inherits from $\mathbb{A}^1_K$ are the same?