Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}_X$-algebras cocomplete?
Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete?
2 Answers
Since you posted this on mathoverflow, I've answered it there:
If $X$ is a scheme (no a simple ringed or locally ringed space) the question is easy (by the usual Grothendieck isomorphism between q.c.modules and algebraic modules on a a affine schema)::
Let $X=\bigcup_{i\in I} U_i$ where $U_i$ is a affine open. If $(\mathcal{M_t})_{t\in T}$ is a diagram of of q.c-modules (quasi coherent modules sheaves). You have $ (\mathcal{M_t})_{|U_i}\cong \widetilde{(M_{t, i})} $ where $M_{t, i}$ a (algebraic) module on the ring $X(U)$, let $M_i:= \varinjlim_{t\in T} M_{t, i}$, now $(\widetilde{M_i})_{|U_j}\cong \varinjlim_{t\in T}\ \mathcal{M_t}_{|U_i\cap U_j}\cong (\widetilde{M_j})_{|U_i}$ and the isomorphisms are a $descent\ data$, then you "glue" these elements and get a module sheaf $\mathcal{M}$ such that $\mathcal{M}_{|U_i}\cong \widetilde{M_i}$ then $\mathcal{M}$ is q.c.module. The some proof if you use q.c.algebras instead modules.