While reading through Algebraic Geometry I by Görtz and Wedhorn (2nd version) I came across the following remark on p. 352 (between Proposition 12.66 and Corollary 12.67):
If $Y$ is an affine $k$-scheme, we can embed it into affine space $\mathbb{A}^{(I)}_k$ ($I$ some index set).
This remark is not further explained. Why is it true? For me it seems non-trivial, if $Y$ is not of finite type.
Best Answer
An affine $k$-scheme $X$ is given by the spectrum of a $k$-algebra $A$. We can pick generators $\{a_i\}_{i\in I}$ for $A$ as a $k$-algebra, which gives a surjective morphism from the polynomial ring $k[x_i]_{i\in I}\to A$ given by sending $x_i\mapsto a_i$. The spectrum of this map is exactly the closed immersion of $X$ in to $\Bbb A^{(I)}_k$. This proof does not care whether $I$ is finite or not.