Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 3 - page 660

[Math Amateur]

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment)

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Let $$V = \mathcal{Z}(x^3 - y^2)$$ in $$\ \ \mathbb{A}^2$$.

If $$(a, b) \in \mathbb{A}^2$$ is an element of V, then $$a^3 = b^2$$.

If $$a \ne 0$$, then also $$b \ne 0$$ and we can write$$a = (b/a)^2, \ b = (b/a)^3$$.

It follows that V is the set $$\{ (a^2, a^3) \ | \ a \in k \}$$.

For any polynomial $$f(x,y) \in k[x,y]$$. we can write $$f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$$

... ... ... etc etc

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I cannot follow the line of reasoning:

"For any polynomial $$f(x,y) \in k[x,y]$$. we can write $$f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$$"

Can anyone clarify why this is true and why D&F are taking this step?

Peter

[This has also been posted on MHF]

 

[jvicens]

Hello Peter,

Thank you for sharing your confusion with us. I am also familiar with Dummit and Foote's book and I can help clarify this step for you.

Firstly, let's understand what the notation \mathcal{Z}(x^3 - y^2) means. This notation represents the set of all points in \mathbb{A}^2 where the polynomial x^3 - y^2 is equal to zero. In other words, it represents the set of solutions to the equation x^3 - y^2 = 0.

Now, let's consider the polynomial f(x,y) \in k[x,y]. This polynomial can be written as a linear combination of monomials in x and y, with coefficients in the field k. For example, f(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + ... where a_i \in k.

Now, if we substitute x^3 - y^2 = 0 into this polynomial, we get f(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + ... = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y) where f_0(x) and f_1(x) are polynomials in x with coefficients in k, and g(x,y) is a polynomial in x and y with coefficients in k.

So essentially, by substituting x^3 - y^2 = 0, we are saying that any polynomial in k[x,y] can be written as a linear combination of polynomials in x and y, where one of the terms is (x^3 - y^2)g(x,y). This is what Dummit and Foote mean when they say "we can write f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)".

I hope this explanation helps clarify the reasoning behind this step. Let me know if you have any further questions.