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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 help me to fully understand the reasoning/analysis following the listing of some properties of $$\mathcal{I}$$ ... in particular the reasoning pertaining to property (10) ... ...
The relevant text from D&F page 661 is as follows:View attachment 4828In the above text we see the following statement by D&F:
" ... ... The last relation shows the maps $$\mathcal{Z}$$ and $$\mathcal{I}$$ act as inverses of each other provided one restricts to the collection of affine algebraic sets $$ V = \mathcal{Z} (I) \text{ in } \mathbb{A}^n$$ and to the set of ideals in $$k [ \mathbb{A}^n ]$$ of the form $$\mathcal{I} (V)$$ ... ... "I cannot see why we have to restrict to ideals in $$k [ \mathbb{A}^n ]$$ of the form $$\mathcal{I} (V)$$ when property 10 is stated in terms of ideals $$I = \mathcal{I} (A)$$ where $$A$$ is an arbitrary subset of $$\mathbb{A}^n$$ ... and hence NOT restricted to an affine algebraic set $$V$$ ... ... ?
Can someone please help me to clarify this issue/problem ... ?
Peter
***EDIT***
To ensure MHB readers understand the notation and context of the above post I am providing D&F's definitions of $$\mathcal{Z}$$ and $$\mathcal{I}$$, as follows:View attachment 4829
View attachment 4830View attachment 4831
At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...
I need someone to help me to fully understand the reasoning/analysis following the listing of some properties of $$\mathcal{I}$$ ... in particular the reasoning pertaining to property (10) ... ...
The relevant text from D&F page 661 is as follows:View attachment 4828In the above text we see the following statement by D&F:
" ... ... The last relation shows the maps $$\mathcal{Z}$$ and $$\mathcal{I}$$ act as inverses of each other provided one restricts to the collection of affine algebraic sets $$ V = \mathcal{Z} (I) \text{ in } \mathbb{A}^n$$ and to the set of ideals in $$k [ \mathbb{A}^n ]$$ of the form $$\mathcal{I} (V)$$ ... ... "I cannot see why we have to restrict to ideals in $$k [ \mathbb{A}^n ]$$ of the form $$\mathcal{I} (V)$$ when property 10 is stated in terms of ideals $$I = \mathcal{I} (A)$$ where $$A$$ is an arbitrary subset of $$\mathbb{A}^n$$ ... and hence NOT restricted to an affine algebraic set $$V$$ ... ... ?
Can someone please help me to clarify this issue/problem ... ?
Peter
***EDIT***
To ensure MHB readers understand the notation and context of the above post I am providing D&F's definitions of $$\mathcal{Z}$$ and $$\mathcal{I}$$, as follows:View attachment 4829
View attachment 4830View attachment 4831
Last edited: