I was wondering if I could verify this with someone. I could only find it for when everything is projective but I am guessing it is true for affine as well. Let $X,Y\subset \mathbb A^n_k$ be algebraic sets. Does it then follows that
$$ \operatorname {codim} (X\cap Y) \leq \operatorname {codim} X+\operatorname {codim}Y.$$
Dimension of intersections of affine algebraic sets
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algebraic-geometry
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1Take $X$ to be $x=0$, where $x$ is a co-ordinate function and $Y$ to be $x=1$. Then both codimensions are one, but the codimension of $X\cap Y=\emptyset$ is not less than or equal to 2. – 2017-02-22
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0Does it hold as long as the intersection is non-empty? – 2017-02-22
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1It does and follows from Krull's principal ideal theorem and the lovely diagonal trick of Serre. – 2017-02-22
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0Would you happen to know where I could reference this fact by any chance? – 2017-02-23
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1A proof can be found in Serre's Local Algebra. – 2017-02-23