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:
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).
There are two easy components.
An image of an irreducible set under a continuous map is irreducible.
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