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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 definition of an affine algebraic set ... ...
D&F explain and define an affine algebraic set in the following text ... ...View attachment 4750
View attachment 4751My question regarding the above definition is this:
Does any arbitrary subset (that is any subset) of the affine plane $$\mathbb{A}^n$$ qualify as an affine algebraic set ...
I was thinking maybe that it does because ... ...
If we have a set $$A_1$$, say that has no non-zero functions with zeros at each of its points, then could we regard the function corresponding to the zero polynomial as having zeros at each of the points of $$A_1$$ ... ... so that the set of functions {$$z$$} where $$z$$ is the zero function is zero on all points of $$A_1$$ ... and hence $$A_1$$ qualifies as an affine algebraic set ... ... I suspect that something is wrong with my analysis above ... but what exactly ...
Can someone clarify this issue for me please ... ...
Peter
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 definition of an affine algebraic set ... ...
D&F explain and define an affine algebraic set in the following text ... ...View attachment 4750
View attachment 4751My question regarding the above definition is this:
Does any arbitrary subset (that is any subset) of the affine plane $$\mathbb{A}^n$$ qualify as an affine algebraic set ...
I was thinking maybe that it does because ... ...
If we have a set $$A_1$$, say that has no non-zero functions with zeros at each of its points, then could we regard the function corresponding to the zero polynomial as having zeros at each of the points of $$A_1$$ ... ... so that the set of functions {$$z$$} where $$z$$ is the zero function is zero on all points of $$A_1$$ ... and hence $$A_1$$ qualifies as an affine algebraic set ... ... I suspect that something is wrong with my analysis above ... but what exactly ...
Can someone clarify this issue for me please ... ...
Peter