Reference request for the dimension of intersection of affine varieties

Question

Let $X,Y \subset \mathbb{C}^n$ be two irreducible affine algebraic sets with nonempty intersection. I am looking for a reference where I can find the proof that $$\dim (X \cap Y) \geq \dim X+ \dim Y - n. $$ Any suggestion or a proof would be appreciated!

Answer 1

Here's the proof from Hartshorne (modulo a few minor editorial corrections):

Proposition I.7.1 (Affine dimension theorem): Let $Y,Z$ be varieties (respectively irreducible closed subschemes) of dimensions $r,s$ in $\Bbb A^n_k$ for $k$ an algebraically closed field (resp. any field). Then every irreducible component $W$ of $Y\cap Z$ has dimension $\geq r+s-n$.

Proof. We proceed in several steps. First, suppose that $Z$ is a hypersurface given by $V(f)$. If $Y\subset Z$, there is nothing to prove. If $Y\not\subset Z$, then we must show that each irreducible component of $Y\cap Z$ has dimension $r-1$. Let $A(Y)$ be the affine coordinate ring of $Y$. Then the irreducible components of $Y\cap Z$ correspond to the minimal primes $\mathfrak{p}$ lying over the ideal $(f)\subset A(Y)$. By Krull's principal ideal theorem, each such prime has height one, so by the dimension theorem, $A(Y)/\mathfrak{p}$ has dimension $r-1$. This shows that every irreducible component $W$ has dimension $r-1$.

Now for the general case. Consider the product $Y\times Z\subset \Bbb A^{2n}$, which is a variety of dimension $r+s$. Let $\Delta$ be the diagonal $\{P\times P \mid P\in\Bbb A^n\}$. Then $\Bbb A^n$ is isomorphic to $\Delta$ by the map $P\mapsto P\times P$, and under this isomorphism $Y\cap Z$ corresponds to $(Y\times Z)\cap \Delta$. Since $\Delta$ has dimension $n$, and since $r+s-n=(r+s)+n-2n$, we reduce to proving the result for the two varieties $Y\times Z$ and $\Delta$ in $\Bbb A^{2n}$. Now $\Delta$ is an intersection of exactly $n$ linear hypersurfaces, namely $x_i-y_i=0$ for $1\leq i \leq n$ where $x_1,\cdots,x_n,y_1,\cdots,y_n$ are the coordinates on $\Bbb A^{2n}$. Now applying the special case above $n$ times, we get the desired result.

Here are the relevant results referenced in this proof:

Krull's Principal ideal theorem: If $R$ is a noetherian ring and $I$ is a principal proper ideal, then all minimal primes over $I$ have height at most one.

Dimension theorem: Let $k$ be a field and let $B$ be an integral domain finitely generated as a $k$-algebra. Then for any prime ideal $\mathfrak{p}\subset B$, we have $\operatorname{height} \mathfrak{p} + \dim B/\mathfrak{p} = \dim B$.