If $V, W$ are algebraic subsets of $\mathbb A^n(k)$. Show that $I(V∩W)=\sqrt{I(V)+I(W)}$
The ideal of the intersection of two algebraic subsets
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algebraic-geometry
commutative-algebra
1 Answers
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Claim: $V(I(V)+I(W))=V \cap W$.
Prove the claim, and apply Hilbert's Nullstellensatz in the form $I(V(J)) = \sqrt{J}$ for every ideal $J$.
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0@Miguel: I've merged your old account into your new one. I recommend that you register your account which will, among other things, allow you to accept answers and comment on your own questions and answers thereof. – 2012-11-07