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August 2nd, 2017, 06:30 AM  #1 
Member Joined: Oct 2013 Posts: 36 Thanks: 0  Ideal of the empty Variety
So in AtiyahMacdonald in exercise 5.17 they ask you to show if $I(X) \neq (1)$ then $X$ is nonempty where $X$ is an affine algebraic variety in $k^n$, $k$ is an algebraically closed field and $I(X)$ denotes $\{f \in k[t_1,...,t_n] : f(x) = 0 \forall x \in X\}$. They then go on to provide a hint using the previous (very complicated exercise) and all of the solutions I've seen seem to use the hint. However why can't you just say that if $X$ is empty then vacuously $I(X) =k[t_1,...,t_n]$ which gives the result? 
August 2nd, 2017, 06:50 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 119 Thanks: 38 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
Looks good to me


Tags 
commutative algebra, empty, ideal, variety 
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