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I am reading this article and in page 6 there is an exercise to show that the ideals $(x,y)$ and $(x,y)^2$ define the same blowup of the affine space $A^2$ when taken as a center.

I have two questions:

What exactly is the ideal $(x,y)^2$? I would like to understand who are the elements of this ideal.

Also, I would appreciate some hint about how to solve this exercise. Maybe considering that the two of them define the same zero locus?

Thanks in advance.

user26857
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yuuu
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1 Answers1

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The product of two ideals $I$ and $J$ is the ideal $IJ$ generated by all products $ab$, where $a \in I$ and $b \in J$.

The square of an ideal $I$ is the ideal $I^2 = II$. So it is generated by all products $ab$, where $a,b \in I$.

The ideal $(x,y)$ in the ring $K[x,y]$ is generated by $x$ and $y$. So $(x,y)^2$ is generated by $x^2$, $xy$, and $y^2$.

Matthew Leingang
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