In this post we will see definitions of the following terms
- sheaf of \(\mathcal{O}_X\) module.
- Tensor product of two sheaves.
- Direct image sheaf \(\mathcal{O}_X\) module.
- Inverse image sheaf \(\mathcal{O}_X\) module.
Definition : Let \((X,\mathcal{O}_X)\) be a ringed space. A sheaf of \(\mathcal{O}_X\) modules is a sheaf \(\mathcal{F}\) on \(X\) such that for each open \(U\subseteq X\), \(\mathcal{F}(U)\) is an \(\mathcal{O}_X(U)\) module and for each inclusion \(V\subseteq U\) we have compatibility of restriction maps with module structure i.e., following diagram is commuatative
