The Stacks project

Exercise 111.64.6 (Two vectors). Let $A = \mathbf{Z}[a_1, a_2, a_3, b_1, b_2, b3]$. Set $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ in $A^{\oplus 3}$. Consider the set

\[ Z = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(A) \mid a, b \text{ map to linearly dependent vectors of } \kappa (\mathfrak p)^{\oplus 3}\} \]

  1. Prove the $Z$ is a closed subset of $\mathop{\mathrm{Spec}}(A)$.

  2. What is the dimension $\dim (Z)$ of $Z$?

  3. What would happen to $\dim (Z)$ if we replaced $\mathbf{Z}$ by a field?

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View Exercise 111.64.6 as pdf