Spreading out of an affine zero-set inclusion
= Spreading out of an affine zero-set inclusion
Let $I,J\subseteq\mathbb Z[x_1,\ldots,x_n]$. If $V_{\mathbb C}(I)\subseteq V_{\mathbb C}(J)$, then
$$
V_{\overline{\mathbb F}_p}(I\bmod p)\subseteq V_{\overline{\mathbb F}_p}(J\bmod p)
$$
for all but finitely many primes $p$. The Nullstellensatz puts a power of each generator of $J$ in $I\mathbb Q[x_1,\ldots,x_n]$; clearing denominators leaves only finitely many exceptional characteristics.