AG dictionary
| Geometry | Algebra |
|---|---|
| Affine algebraic set $X$ | f.g. $K$-algebra without nilpotents $A=K[X]$. |
| Embedding $X\subset\mathbb{A}^n$ | Choice of generators of $A$ |
| $f:X\to Y$ | $f^\ast: K[Y]\to K[X]$ |
| $X$ is irreducible (an affine variety) | $A=K[X]$ is a domain |
| $f$ is dominant (\overline{f(X)}=Y) | $f^\ast$ is injective |
| $f$ is a closed embedding | $f^\ast$ is surjective |
(Closed embedding means $f$ is an isomorphism onto its image and $f(X)$ is closed in $Y$.)