Finite-stage descent of finitely presented quasi-coherent sheaves

Affine Hom lemma. Let $B_i$ be a filtered system of rings with , let $M_i$ be a finitely presented $B_i$-module and $N_i$ any $B_i$-module, and write $M_j,N_j,M,N$ for their base changes. The natural map is bijective. Indeed, choose a presentation $B_i^r\stackrel{D}{\to }{B}_i^s\to M_i\to 0$. A homomorphism $M\to N$ is a tuple of $s$ elements of $N$ annihilating the $r$ relations given by $D$. Lift the tuple to one $N_j$ and then, by [F3], enlarge until its finitely many relation values vanish. This produces a stage map. If two stage maps become equal over $B$, their values on the $s$ generators become equal at one later stage by [F3]. Localization at one element preserves the presentation and the same argument. In particular, an isomorphism and its inverse descend as maps, and their two inverse identities become true after a common stage.

Affine presentations. Choose a finite affine cover $X_i={\bigcup }_{a=1}^m{U}_{a,i}$ with $U_{a,i}=\mathrm{Spec}{B}_{a,i}$. On $U_a=U_{a,i}{\times }_{S_i}S$, [F1] and [F2] identify $\mathcal{F}{\mathrm{\mid }}_{U_a}$ with a finitely presented $B_a$-module $M_a$. Choose a finite matrix presentation $B_a^{r_a}\to B_a^{s_a}\to M_a\to 0$. All matrix entries occur at a common stage by [F3]. Their stage cokernels $M_{a,j}$ are finitely presented and pull back to $M_a$, since tensoring preserves cokernels. Take one stage for the finitely many charts.

Overlap isomorphisms. For every pair $a,b$, the stage overlap $W_{ab,j}=U_{a,j}\cap U_{b,j}$ is quasi-compact by [F4]; choose a finite principal affine cover of it inside $U_{a,j}$. Its base change covers the limit overlap. The two stage sheaves defined by $M_{a,j},M_{b,j}$ restrict to finitely presented modules on every one of these principal affines by [F1] and [F2]. Their limit restrictions are canonically isomorphic, since both represent $\mathcal{F}$ there. Apply step 1.1 on each principal affine to descend the isomorphism and its inverse. On pairwise intersections of these principal affines, step 1.1 makes the local maps agree at a later stage; thus they glue to an overlap isomorphism. Impose both inverse equations by the same eventual-equality argument. There are finitely many overlaps, so one stage handles them all.

On every triple overlap $U_a\cap U_b\cap U_c$, the two composites of the overlap isomorphisms agree because they are both the identity identification through $\mathcal{F}$. A triple overlap is quasi-compact by [F4]; cover it by finitely many principal affines inside $U_a$ and use the equality part of step 1.1 to impose every triple cocycle equation at one later stage. The stage modules therefore glue to a quasi-coherent sheaf ${\mathcal{F}}_k$ by [F4]. It is finitely presented, since this is affine-local and its restrictions to the finite affine cover are the modules $M_{a,k}$. Its pullback is $\mathcal{F}$ with the chosen chart identifications.

Morphisms and equality. Let ${\mathcal{F}}_i,{\mathcal{G}}_i$ be fixed finitely presented stage sheaves. A map of their pullbacks is, on the finite affine cover, a map of finitely presented modules. Step 1.1 descends all the chart maps. Their agreements on the finitely many quasi-compact pairwise overlaps are equalities on finite principal covers, so step 1.1 makes them hold at a common stage. The chart maps glue to the desired stage sheaf map. If two stage maps become equal over $X$, their restrictions to every affine chart become equal at a common stage by step 1.1, hence the sheaf maps are equal there.

The empty source has its unique empty sheaf at every stage. The zero sheaf is presented by the zero matrix data and is carried by zero stage modules. All enlargements above are finite in number, so directedness gives one common stage. The declared AC supports the finite chart and presentation selections; no flatness of the ring maps was used. Stacks Tag 01ZR is source evidence for the argument and is not a proof premise.