Definition:Affine Coordinate Ring
Jump to navigation
Jump to search
Definition
Let $k$ be a field.
Let $Y \subseteq k^n$ be an affine algebraic set.
Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.
Let $\map I Y \subseteq k \sqbrk {X_1, \ldots, X_n}$ be the vanishing ideal of $Y$.
Then the affine coordinate ring of $Y$ is defined as the quotient ring:
- $\ds \map A Y := k \sqbrk {X_1, \ldots, X_n}/\map I Y$
Sources
- 1977: Robin Hartshorne: Algebraic Geometry $\text{I}.1$ Affine Varieties