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Exercise 111.16.5. Let $A = {\mathbf C}[x_{11}, x_{12}, x_{21}, x_{22}, y_{11}, y_{12}, y_{21}, y_{22}]$. Let $I$ be the ideal of $A$ generated by the entries of the matrix $XY$, with

\[ X = \left( \begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix} \right) \quad \text{and}\quad Y = \left( \begin{matrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{matrix} \right). \]

Find the irreducible components of the closed subset $V(I)$ of $\mathop{\mathrm{Spec}}(A)$. (I mean describe them and give equations for each of them. You do not have to prove that the equations you write down define prime ideals.) Hints:

  1. You may use the Hilbert Nullstellensatz, and it suffices to find irreducible locally closed subsets which cover the set of closed points of $V(I)$.

  2. There are two easy components.

  3. An image of an irreducible set under a continuous map is irreducible.

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View Exercise 111.16.5 as pdf