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The affine dimension formula on a plane curve domain

Example

Let

A = k [x, y] / (y - x^{2}) .

Then $A ≅ k [x]$ is a one-dimensional affine domain. For the prime ideals $(0)$ and $m = (\overset{ˉ}{x}, \overset{ˉ}{y})$ , the affine dimension formula reads

ht (p) + \dim (A / p) = 1.

Facts & Assumptions

Given: A field $k$ , the domain $A = k [x, y] / (y - x^{2})$ , and the primes $(0)$ and $m = (\overset{ˉ}{x}, \overset{ˉ}{y})$ .

[L1]

The affine dimension formula identifies $ht (p) + {trdeg}_{k} Frac (A / p) = {trdeg}_{k} Frac (A)$ (The dimension formula for affine domains).

[L2]

Maximal ideals of an affine domain have full height, and the quotient-dimension reformulation is also available (Maximal ideals of an affine domain have full height, Height plus quotient dimension equals ambient dimension in an affine domain).

Verification

technique · direct computation

1.1

The ring $A ≅ k [x]$ has dimension $1$ , and [L1] at the zero prime reads $0 + {trdeg}_{k} Frac (A) = 1$ , which is true because $Frac (A) ≅ k (x)$ .

L1given

2.1

For the maximal ideal $m$ , the quotient is $A / m ≅ k$ , so its quotient dimension is $0$ . Then [L2] gives $ht (m) = 1$ , and the formula becomes $1 + 0 = 1$ .

L2step 1.1

3.1

Thus the affine dimension formula is verified term by term on this plane curve domain.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §§18, 21 (standard reference, not scraped)
The Stacks Project, Section 10.116: Dimension of finite type algebras over fields, reprise (standard reference, not scraped)