The Stacks project

Exercise 111.53.2. Prime avoidance.

  1. Let $A$ be a ring. Let $I \subset A$ be an ideal and let $\mathfrak q_1$, $\mathfrak q_2$ be prime ideals such that $I \not\subset \mathfrak q_ i$. Show that $I \not\subset \mathfrak q_1 \cup \mathfrak q_2$.

  2. What is a geometric interpretation of (1)?

  3. Let $X = \text{Proj}(S)$ for some graded ring $S$. Let $x_1, x_2 \in X$. Show that there exists a standard open $D_{+}(F)$ which contains both $x_1$ and $x_2$.

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View Exercise 111.53.2 as pdf