Dimension of Affine Algebraic Set is Dimension of Affine Coordinate Ring

Theorem

Let $k$ be a field.

Let $Y \subseteq k^n$ be an affine algebraic set.

Let $\map A Y$ be the affine coordinate ring.

Then:

$\map \dim Y = \map \dim {\map A Y}$

where:

$\map \dim Y$ is the Krull dimension of $Y$ with respect to Zariski topology

$\map \dim {\map A Y}$ is the Krull dimension of $\map A Y$

Proof

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Sources

• 1977: Robin Hartshorne: Algebraic Geometry $\text{I}.1$ Affine Varieties