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Maximilian Keßler 2022-02-16 02:01:45 +01:00
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@ -1844,9 +1844,9 @@ Let $R = \mathfrak{k}[X_1,\ldots,X_n]$ and $I \subseteq R$ an ideal.
\begin{corollary}\label{codimintersection} \begin{corollary}\label{codimintersection}
Let $A$ and $B$ be irreducible subsets of $\mathfrak{k}^n$. If $C$ is an irreducible component of $A \cap B$, then $\codim(C, \mathfrak{k}^n) \le \codim(A, \mathfrak{k}^n) + \codim(B, \mathfrak{k}^n)$. Let $A$ and $B$ be irreducible subsets of $\mathfrak{k}^n$. If $C$ is an irreducible component of $A \cap B$, then $\codim(C, \mathfrak{k}^n) \le \codim(A, \mathfrak{k}^n) + \codim(B, \mathfrak{k}^n)$.
\end{corollary} \end{corollary}
\begin{dremark} \begin{remark}+
Equivalently, $\dim(C) \ge \dim(A) + \dim(B)-n$. Equivalently, $\dim(C) \ge \dim(A) + \dim(B)-n$.
\end{dremark} \end{remark}
\begin{proof} \begin{proof}
Let $X = A \times B \subseteq \mathfrak{k}^{2n}$, where we use $(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ as coordinates of $\mathfrak{k}^{2n}$. Let $X = A \times B \subseteq \mathfrak{k}^{2n}$, where we use $(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ as coordinates of $\mathfrak{k}^{2n}$.
Let $\Delta \coloneqq \{(x_1,\ldots,x_n,x_1,\ldots,x_n) | x \in \mathfrak{k}^n\} $ be the diagonal in $\mathfrak{k}^n \times \mathfrak{k}^n$. Let $\Delta \coloneqq \{(x_1,\ldots,x_n,x_1,\ldots,x_n) | x \in \mathfrak{k}^n\} $ be the diagonal in $\mathfrak{k}^n \times \mathfrak{k}^n$.