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

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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