Why are Grothendieck's and Hartshorne's definitions of quasi-coherence equivalent?

Solution 1:

This is not so hard to prove, using the basic properties of sheafifying modules:

Hartshorne to Grothendieck:

If we choose a presentation $A_i^I \to A_i^J \to M_i$ and then sheafify, we obtain a presentation of $\mathcal F_{| U_i}$ as a cokernel of a morphism $\mathcal O_{U_i}^I \to \mathcal O_{U_i}^J,$ i.e. as the cokernel of a morphism of free $\mathcal O_{U_i}$-modules.

Grothendieck to Hartshorne:

Shrinking $U$ around $x$ if necessary, we may assume that $U = \mathrm{Spec} \, A$ is affine. By assumption, we have an exact sequence $$\mathcal O_U^I \to \mathcal O_U^J \to \mathcal F_{|U} \to 0.$$ Now passing to global sections with respect to the first map, we obtain a morphism of $A$-modules $$A^I \to A^J;$$ let $M$ be the cokernel of this map. Sheafifying the right-exact sequence $$A^I \to A^J \to M \to 0,$$ we find that $\widetilde{M}$ is identified with the cokernel of $\mathcal O_U^I \to \mathcal O_U^J,$ and hence is isomorphic to $\mathcal F_{| U}$.

Summary: both directions use the fact that the global sections of $\mathcal O_U$ on $U = $ Spec $A$ are naturally isomorphic to $A$, and that sheafification of $A$-modules is exact, but nothing more.