I'm having trouble understanding the definition of globally generated quasi-coherent sheaves that is given in my lecture notes:
Let $X$ be a scheme, M a quasi-coherent sheaf, then M is globally generated if it is generated by global sections, i.e. if there exist sections $s_i\in M(X),\,i\in I$ with $I$ a set, such that the map $\bigoplus_{i\in I}O_X\to M,\,e_i\mapsto s_i$ is surjective.
In particular, I don't understand the following: How is the direct sum of sheaves of rings defined and why does it even exist? What are the $e_i$? The stacks project seems to be using the notation $\bigoplus_{i\in I}O_X$ as well but I haven't found a definition so far.
Best Answer
The direct sum is always defined by its universal property: $\bigoplus_i A_i$ is the universal recipient of a family of maps, one from each $A_i$, in the appropriate category. In this case the category in question is the category of (sheaves of) $\mathcal O_X$-modules. (Not rings--the same universal property in a category of rings defines the tensor product.)
Of course this doesn't answer the question of whether the direct sum exists; we still need to construct it. This can be done as follows. First, the direct sum of a family of presheaves is the direct sum on sections; i.e. $(\bigoplus_i \mathcal F_i)(U) = \bigoplus_i (\mathcal F_i(U))$ for each open set $U$. The direct sum of a family of sheaves is the sheafification of the direct sum of the underlying presheaves.
This construction is justified by a general fact from category theory: left adjoints commute with colimits. The direct sum is defined as a colimit, and sheafification is left-adjoint to the inclusion functor $\mathrm{Sh} \hookrightarrow \mathrm{PSh}$, so "sheaf-direct-sum of sheafifications equals sheafification of presheaf-direct-sum".
Note also that a finite direct sum in an abelian category agrees with the direct product, which is a limit. In this case the dual statement (right adjoints commute with limits) implies that the direct product of presheaves is already a sheaf: "presheaf-direct-product of underlying presheaves equals underlying presheaf of sheaf-direct-product".