Flat deformations form a Zariski sheaf of groupoids

Let $X^{\mathrm{\prime }}$ be a flat deformation of $X$ over $A^{\mathrm{\prime }}$ with base-change isomorphism $X^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong X$. The projection $\mathrm{Spec}A\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ is a closed immersion with ideal $I$ satisfying $I^2=0$, so its base change $\iota :X\to X^{\mathrm{\prime }}$ is a closed immersion with square-zero ideal sheaf; by [F1] the map $\iota$ is a homeomorphism, and by [F3], applied on affine charts, $V(J)=S^{\mathrm{\prime }}$ for the square-zero ideal, so $\mathrm{\mid }X\mathrm{\mid }\cong \mathrm{\mid }{X}^{\mathrm{\prime }}\mathrm{\mid }$ as topological spaces. For an open $V\subseteq X$ let $X^{\mathrm{\prime }}{\mathrm{\mid }}_V$ be the open subscheme of $X^{\mathrm{\prime }}$ whose underlying space is the image of $V$; it is a flat deformation of $V$ over $A^{\mathrm{\prime }}$ because flatness is checked on local rings ([F7]) and the base change of the open piece is $V$ by [F8]; the affine identification of the reduction is the computation $B^{\mathrm{\prime }}{\otimes }_{A^{\mathrm{\prime }}}A=B^{\mathrm{\prime }}\mathrm{/}I{B}^{\mathrm{\prime }}$ of [F2]. This makes restriction to opens a well-defined operation on deformations.

The restriction operation of step 1.1 is functorial: an isomorphism $u:X_1^{\mathrm{\prime }}\to X_2^{\mathrm{\prime }}$ of deformations of $X$ over $A^{\mathrm{\prime }}$ restricts to isomorphisms $X_1^{\mathrm{\prime }}{\mathrm{\mid }}_V\to X_2^{\mathrm{\prime }}{\mathrm{\mid }}_V$ of deformations of $V$ over $A^{\mathrm{\prime }}$, and composition and identities restrict to composition and identities. Hence any deformation $X^{\mathrm{\prime }}$ yields descent data $\mathrm{\Phi }(X^{\mathrm{\prime }})$ on the cover $\{U_i\}$: the objects are the deformations $X^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}$ of $U_i$, and on overlaps the canonical isomorphisms $X^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}{\mathrm{\mid }}_{U_i\cap U_j}\to X^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}\leftarrow X^{\mathrm{\prime }}{\mathrm{\mid }}_{U_j}{\mathrm{\mid }}_{U_i\cap U_j}$ coming from the identification of both sides with $X^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}$ have the same source and target, are isomorphisms over $A^{\mathrm{\prime }}$ inducing the identity on $U_i\cap U_j$, and satisfy the identity and cocycle conditions because they are induced by a single global object. Moreover an isomorphism $u$ of deformations restricts to a compatible family of isomorphisms, so $\mathrm{\Phi }$ is a functor from the groupoid of flat deformations of $X$ over $A^{\mathrm{\prime }}$ to the groupoid of descent data.

Conversely, let $(\{U_i^{\mathrm{\prime }}\},\{{\phi }_{ij}\})$ be descent data. Glue the locally ringed spaces $U_i^{\mathrm{\prime }}$ along their open subschemes $U_i^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}\cong U_j^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}$ using the isomorphisms ${\phi }_{ij}$; by [F4] this produces a locally ringed space $X^{\mathrm{\prime }}$ covered by open pieces isomorphic to the $U_i^{\mathrm{\prime }}$, with the cocycle condition ensuring the gluing is well defined on triple overlaps. Every point of $X^{\mathrm{\prime }}$ lies in some piece $U_i^{\mathrm{\prime }}$, which is a scheme, so it has an affine open neighbourhood inside that piece; by [F5] the glued locally ringed space $X^{\mathrm{\prime }}$ is a scheme.

Each structural morphism $U_i^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ is a morphism of schemes; on an overlap the two restrictions agree because the isomorphisms ${\phi }_{ij}$ are over $\mathrm{Spec}{A}^{\mathrm{\prime }}$ and agree with the identification of the pieces in the glued space. Therefore, by [F6], the continuous maps and the structure-sheaf maps glue to a morphism $X^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ which restricts to $U_i^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ on each piece; its stalk maps are the stalk maps of the pieces and hence local, so the glued locally ringed space is a scheme over $A^{\mathrm{\prime }}$. At every point the local ring of $X^{\mathrm{\prime }}$ equals the local ring of the piece containing it and the local ring map equals that of $U_i^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$, which is flat; by [F7] the morphism $X^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ is flat. If the pieces are locally of finite presentation over $\mathrm{Spec}{A}^{\mathrm{\prime }}$, then so is $X^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }}$ by [F9], the affine charts being taken inside the pieces; this proves the last claim.

The special fibre of $X^{\mathrm{\prime }}$ over $A$ is covered by the pieces $U_i^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong U_i$, with transition isomorphisms between the identifications on the pieces; the descent data prescribe that these identifications agree on overlaps, so the canonical morphisms $U_i\to X^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A$ glue by [F6] to a morphism $X\to X^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A$, and the same gluing of the identity maps in the other direction produces an inverse. Hence $X^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong X$ over $\mathrm{Spec}A$, so the glued object $X^{\mathrm{\prime }}$ is a flat deformation of $X$ over $A^{\mathrm{\prime }}$; its restrictions to the pieces are the given $U_i^{\mathrm{\prime }}$. This shows that $\mathrm{\Phi }$ is essentially surjective.

To prove that $\mathrm{\Phi }$ is fully faithful, let $X_1^{\mathrm{\prime }},X_2^{\mathrm{\prime }}$ be flat deformations of $X$ over $A^{\mathrm{\prime }}$ and let $v_1,v_2:X_1^{\mathrm{\prime }}\to X_2^{\mathrm{\prime }}$ be isomorphisms of deformations with $\mathrm{\Phi }(v_1)=\mathrm{\Phi }(v_2)$; by step 2.1 this means that $v_1$ and $v_2$ agree on every piece $X_1^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}$, and since the pieces cover $X_1^{\mathrm{\prime }}$ the two morphisms agree as continuous maps and as structure-sheaf maps, so $v_1=v_2$. Conversely, given a morphism of descent data $(u_i)$ from $\mathrm{\Phi }(X_1^{\mathrm{\prime }})$ to $\mathrm{\Phi }(X_2^{\mathrm{\prime }})$, the morphisms $u_i:X_1^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}\to X_2^{\mathrm{\prime }}$ agree on overlaps because the descent data morphism is compatible with the gluing isomorphisms, so by [F6] they glue to a morphism $u:X_1^{\mathrm{\prime }}\to X_2^{\mathrm{\prime }}$ over $A^{\mathrm{\prime }}$; its restriction to each piece is $u_i$, hence its base change to $\mathrm{Spec}A$ is the identity on $X$, so $u$ is an isomorphism of deformations and $\mathrm{\Phi }(u)=(u_i)$. Thus $\mathrm{\Phi }$ is fully faithful.

By steps 5.1 and 6.1 the functor $\mathrm{\Phi }$ is an equivalence of groupoids, which is the asserted effective Zariski descent. The same argument applies to a first-order thickening $S\hookrightarrow S^{\mathrm{\prime }}$ and a flat $S$-scheme $X$ by considering flat $S^{\mathrm{\prime }}$-schemes $X^{\mathrm{\prime }}$ equipped with $X^{\mathrm{\prime }}{\times }_{S^{\mathrm{\prime }}}S\cong X$; [F1] applies to this specified flat lift, without choosing or asserting a retraction $S^{\mathrm{\prime }}\to S$. The proof never uses separatedness, quasi-finiteness, or the local finite presentation convention. The Axiom of Choice is used only through the cited gluing and fibre-product suppliers.