Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 1 / b

Choose an open cover of $X$ by affine opens $U_i=\mathrm{Spec}{B}_i$. Since $X$ is a quasi-compact topological space, finitely many suffice. Let $f_i,a_i\in B_i$ be the restrictions of $f,a$. Part (a) identifies $U_i\cap X_f$ with $D(f_i)$, whose ring of regular functions is the localization of a ring $(B_i{)}_{f_i}$. The vanishing of $a$ there meansso $f_i^{n_i}{a}_i=0$ for some $n_i\ge 0$. Taking $n>0$ at least as large as every $n_i$ gives $(f^{n}a){|}_{U_i}=0$ on every member of the finite cover. The local identity axiom for the sheaf of rings ${\mathcal{O}}_X$ therefore gives