Practice problem

Commutative Algebra

Let $A$ be a ring and let $\mathfrak{q}$ be a prime ideal. Let $X=\operatorname{Spec}(A)$ be the spectrum of $A$ with the Zariski topology. Which of the following are equivalent interpretations of $V(\mathfrak{q})$ ?

Choose all correct answers

$\{\mathfrak{p}\in X: \mathfrak{q}\leq \mathfrak{p}\}$ (primes containing $\mathfrak{q}$ )

$\{\mathfrak{p}\in X: \mathfrak{q}\geq \mathfrak{p}\}$ (primes contained in $\mathfrak{q}$ )

$\overline{\{\mathfrak{q}\}}$ , the Zariski closure of the singleton $\{\mathfrak{q}\}$

The support of the quotient sheaf $\mathcal{O}_X/\mathfrak{q}\mathcal{O}_X$ , where $\mathfrak{q}\mathcal{O}_X = \widetilde{\mathfrak{q}}$ is the quasicoherent ideal sheaf corresponding to $\mathfrak{q}$

The set of points $\mathfrak{p}\in X$ such that the canonical morphism $\operatorname{Spec}(\mathcal{O}_{X,\mathfrak{q}}) \to X$ factors through $\operatorname{Spec}(\mathcal{O}_{X,\mathfrak{p}}) \to X$ .