I studied tenison's Sheaf Theory and Atiyah-Macdonad's Introduction to Commutative Algebra. Now I am looking for some problems in commutative algebra and their solution with sheaf theory to understand the application of sheaves for solving problems. (For example we know that if $I$ is an ideal of commutative ring $R$ and $ I_1, \dots, I_n$ are pairwise comaximal ideals of $R$ such that $\sqrt{I}=I_1\dots I_n$, then there exist pairwise comaximal radical ideals $I^{'}_1, \dots, I^{'}_n$ of $R$ such that ${I}= I^{'}_1 \dots, I^{'}_n$ with $I^{'}_i \subseteq I_i$ and $V(I_i)= V(I^{'}_i)$ for $i=1,\dots, n$, where $V(J)=\{\text{the set of all prime ideal of $R$ contaning $J$}\}$ for each ideal $J$ of $R$. How can we prove this result by sheaf theory and gluing conditions?)
Note: I saw Pierce's sheaf and some theorem that are proved by Pierce's sheaf, but these theorems can be proved with commutative algebra techniques.