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I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ...
At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...
I need someone to confirm some thoughts on the correspondence between D&F's definition of $$\mathcal{I}$$ and the ideals of $$k[ \mathbb{A}^n ]$$ ...
On page 660 (in Section 15.1) of D&F we find the following text:View attachment 4737In the above text we find the following text:
" ... ... It is immediate that $$\mathcal{I} (A)$$ is an ideal, and is the unique largest ideal of functions that are identically zero on $$A$$. This defines a correspondence
$$\mathcal{I} \ : \ \{ \text{ subsets in } k[ \mathbb{A}^n ] \} \ \rightarrow \ \{ \text{ ideals of } k[ \mathbb{A}^n ] \}$$. ... ... "I am thinking about the above correspondence and how we can take any arbitrary set of points $$A$$ in $$\mathbb{A}^n $$and find a set of functions that are identically zero on $$A$$ ... ... I thought this may be impossible for many sets, maybe infinitely many sets $$A$$ ... but then realized that in these cases the set of functions would be the zero function and the corresponding ideal would be $$\{ z \}$$ where $$z$$ is the zero function ...
Can someone please confirm that my thinking/reflections above are on the right track ... I am also worrying a bit about why $$\mathcal{I} (A)$$ is the unique and largest such ideal ... ...
Hope someone can help with these issues ... simple though they may be ...
Peter
At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...
I need someone to confirm some thoughts on the correspondence between D&F's definition of $$\mathcal{I}$$ and the ideals of $$k[ \mathbb{A}^n ]$$ ...
On page 660 (in Section 15.1) of D&F we find the following text:View attachment 4737In the above text we find the following text:
" ... ... It is immediate that $$\mathcal{I} (A)$$ is an ideal, and is the unique largest ideal of functions that are identically zero on $$A$$. This defines a correspondence
$$\mathcal{I} \ : \ \{ \text{ subsets in } k[ \mathbb{A}^n ] \} \ \rightarrow \ \{ \text{ ideals of } k[ \mathbb{A}^n ] \}$$. ... ... "I am thinking about the above correspondence and how we can take any arbitrary set of points $$A$$ in $$\mathbb{A}^n $$and find a set of functions that are identically zero on $$A$$ ... ... I thought this may be impossible for many sets, maybe infinitely many sets $$A$$ ... but then realized that in these cases the set of functions would be the zero function and the corresponding ideal would be $$\{ z \}$$ where $$z$$ is the zero function ...
Can someone please confirm that my thinking/reflections above are on the right track ... I am also worrying a bit about why $$\mathcal{I} (A)$$ is the unique and largest such ideal ... ...
Hope someone can help with these issues ... simple though they may be ...
Peter