The Irrelevant Ideal

[Tsakanikas Nickos]

Let \( \displaystyle B = \bigoplus_{d \geq 0} B_{d} \) be a graded ring and let \( \mathfrak{a} \) be a homogeneous ideal of \( B \). Show that the following are equivalent:

1. \(V_{+}(\mathfrak{a}) = \emptyset \)
2. \(B_{+} \subset \sqrt{\mathfrak{a}}\)
3. If \( \mathfrak{a} = (\alpha_{i}) \), where \( \alpha_{i} \) are homogeneous, then \( \displaystyle \bigcup D_{+}(\alpha_{i}) = Proj(B) \).