# Product of monomial ideals and simplicial complexes

Let $I = (f_1, \ldots, f_n)$ and $J = (g_1,\ldots,g_m)$ be two ideals generated by regular sequences of monomials in the polynomial ring $R = k[x_1, x_2, \ldots , x_u]$

Show that $$\Delta_{p(IJ)} = \Delta_I \cup \Delta_J,$$

where $p(IJ)$ is the polarization of $IJ$, $\Delta_I$ is the simplicial complex corresponding to the squarefree monomial ideal $I$, and $\Delta_J$ is the simplicial complex corresponding to the squarefree monomial ideal $I$.

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