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A triangular change makes a bivariate relation monic

Example

Let $k$ be an infinite field of characteristic not equal to $2$ , and let

A = \frac{k [x, y]}{(x^{2} + y^{2})} .

Put $u = x - y$ . Then the triangular change $x = u + y$ makes the defining relation monic in $y$ , and $A$ is module-finite over the polynomial subring $k [\overset{ˉ}{u}]$ .

Facts & Assumptions

Given: An infinite field $k$ with $2 \neq 0$ and the quotient $A = k [x, y] / (x^{2} + y^{2})$ .

[L1]

Over an infinite field, a triangular change can make a nonzero polynomial monic in one variable (Over an infinite field, a triangular change makes a nonzero polynomial monic).

[L2]

A monic relation makes the last generator integral over the subalgebra on the earlier generators (A monic relation makes the last generator integral over the earlier ones).

Verification

technique · direct

1.1

Introduce the triangular coordinate $u = x - y$ , so $x = u + y$ . Then $(u + y)^{2} + y^{2} = u^{2} + 2 u y + 2 y^{2} .$ Multiplying by $2^{- 1}$ gives the monic polynomial $y^{2} + u y + \frac{1}{2} u^{2}$ in the variable $y$ . This is the concrete instance of [L1].

L1given

2.1

In the quotient algebra, $\overset{ˉ}{u} = \overset{ˉ}{x} - \overset{ˉ}{y}$ and the class $\overset{ˉ}{y}$ satisfies ${\overset{ˉ}{y}}^{2} + \overset{ˉ}{u} \overset{ˉ}{y} + \frac{1}{2} {\overset{ˉ}{u}}^{2} = 0$ over $k [\overset{ˉ}{u}]$ . By [L2], $\overset{ˉ}{y}$ is integral over $k [\overset{ˉ}{u}]$ . Since $A = k [\overset{ˉ}{u}, \overset{ˉ}{y}]$ , the algebra $A$ is module-finite over $k [\overset{ˉ}{u}]$ .

L2step 1.1∎

Depends on

Used by

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Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §15 (standard reference, not scraped)