Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 3 / b / Solution

Choose finite generating sets $I=(g_1,\dots ,g_s)$ and $J=(f_1,\dots ,f_t)$, using the Hilbert basis theorem. The assumed inclusion says that each $f_j$ vanishes on $V_{\mathbb{C}}(I)$. By the Strong Hilbert Nullstellensatz, for every $j$ there is an exponent $N_j$ such that

The coefficients in an expression $f_j^{N_j}={\sum }_i{q}_{ij}{g}_i$ solve a finite system of linear equations with rational coefficients. Since it has a complex solution, Gaussian elimination gives a rational solution. Clearing the finitely many denominators produces a nonzero integer $D$ such thatfor every $j$.

For any prime $p\nmid D$, reduce these identities modulo $p$. At a common zero of ${\pi }_p(I)$ in the algebraic closure ${\overline{\mathbb{F}}}_p$, they give ${\pi }_p(f_j{)}^{N_j}=0$, hence ${\pi }_p(f_j)=0$, for all $j$. Thereforefor every prime except the finitely many divisors of $D$. This is the spreading out of an affine zero-set inclusion.