Global cotangent-complex obstruction and lifting torsor

Write $C=\mathbf{L}{f}^{\ast }{L}_{S\mathrm{/}{S}^{\mathrm{\prime }}}$, $K=L_{X\mathrm{/}{S}^{\mathrm{\prime }}}$, and $L=L_{X\mathrm{/}S}$. The conormal presentation of the surjection $A^{\mathrm{\prime }}\to A$ identifies ${\tau }_{\ge -1}{L}_{S\mathrm{/}{S}^{\mathrm{\prime }}}$ with $\tilde{I}[1]$. Since $f$ is flat, $C$ lies in degrees at most $-1$ and ${\mathcal{H}}^{-1}(C)=M$. Hence ${\mathrm{Ext}}_{{\mathcal{O}}_X}^0(C,M)=0$ and ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(C,M)={\mathrm{Hom}}_{{\mathcal{O}}_X}(M,M)$. This is a global calculation: the part in degrees at most $-2$ has derived Hom into a degree-zero module zero in degrees below two, so truncation gives these identities without replacing global Ext by local sections of a sheaf Ext. Let $\gamma$ be the identity of $M$, equivalently the map induced by $c:i\mapsto 1\otimes i$.

By [F1], ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(K,M)$ classifies square-zero $f^{-1}{\mathcal{O}}_{S^{\mathrm{\prime }}}$-algebra sheaves $0\to M\to E\to {\mathcal{O}}_X\to 0$. Restriction to the base followed by coefficient change is the map to ${\mathrm{Ext}}_{{\mathcal{O}}_X}^1(C,M)$. In the conormal presentation of $A^{\mathrm{\prime }}\to A$ it sends an extension to its structural multiplication map $I{\otimes }_A{\mathcal{O}}_X\to M$: evaluating the relation $i\in I$ gives exactly its structural image in $E$. The sheaf-generator proof and naturality in [F1] show this equality globally, so it includes the required gluing of algebra data. Thus the fiber over $\gamma$ is exactly the marked sheaf-ring lifting problem with the prescribed kernel map. No overlap choices or triple-overlap obstructions have been suppressed by an appeal to affine descent.

The canonical triangle $C\to K\to L\to C[1]$ gives the exact sequence ${\mathrm{Ext}}^1(K,M)\to {\mathrm{Ext}}^1(C,M)\stackrel{\mathrm{\partial }}{\to }{\mathrm{Ext}}^2(L,M)$, all groups taken globally over ${\mathcal{O}}_X$. Define $o(X,A^{\mathrm{\prime }})=\mathrm{\partial }\gamma$. It vanishes exactly when the fiber in step 2.1 is nonempty. The preceding exact sequence, beginning with ${\mathrm{Ext}}^0(C,M)=0$, injects ${\mathrm{Ext}}^1(L,M)$ into ${\mathrm{Ext}}^1(K,M)$ with image the kernel of restriction. Each nonempty fiber is therefore a torsor for ${\mathrm{Ext}}^1(L,M)$. The boundary is canonical because both the global sheaf-ring cotangent triangle and the class $\gamma$ are canonical; [F1]'s presentation comparisons prove that constructing the fiber with different sheaves of generators gives the same class and action.

The module $M$ is quasi-coherent, and in every extension in this fiber the structural multiplication map onto its kernel is the identity of $M$. Apply [F2]: $(\mathrm{\mid }X\mathrm{\mid },E)$ is a scheme $X^{\mathrm{\prime }}$ flat and locally finitely presented over $A^{\mathrm{\prime }}$, with $I{X}^{\mathrm{\prime }}=M$ at the sheaf level and $X^{\mathrm{\prime }}{\times }_{A^{\mathrm{\prime }}}A=X$ with exactly its given marking. Conversely every flat lift has kernel canonically $I{\otimes }_A{\mathcal{O}}_X$ and structural map $c$, so gives precisely an object of this fiber. Isomorphisms in the two problems agree by [F2]. This proves the existence criterion and torsor for scheme lifts, rather than merely for abstract ringed-space extensions. Composing the unchanged $X$-marking with the specified marking of $X_0$ preserves the latter automatically.

The automorphism calculation in [F1] gives global derivations ${\mathcal{O}}_X\to M$ over $A^{\mathrm{\prime }}$, which are the same as those over $A$ because $A^{\mathrm{\prime }}\twoheadrightarrow A$. They are ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathrm{\Omega }}_{X\mathrm{/}A}^1,M)={\mathrm{Ext}}_{{\mathcal{O}}_X}^0(L,M)$ by [F4], with additive composition as in [F1]. By step 4.1 these are exactly the scheme-lift automorphisms inducing the identity on $X$; their action on the kernel is forced to be the identity because that kernel is generated by $I$. On affine restrictions the boundary, the kernel map and the automorphism calculation coincide with [F5]'s affine theorem by the affine comparison and the evaluation calculation of step 2.1. Compatible such restrictions glue as in [F5]; this checks agreement of the two interfaces without using compatible-lift descent as the construction of $o$.

If $X\mathrm{/}A$ is smooth, [F4] gives $L\simeq {\mathrm{\Omega }}_{X\mathrm{/}A}^1[0]$ with ${\mathrm{\Omega }}_{X\mathrm{/}A}^1$ finite locally free. Internal Hom from this module is exact, so an injective resolution computes global derived Hom as $\mathbf{R}\mathrm{\Gamma }(X,\mathcal{H}om({\mathrm{\Omega }}_{X\mathrm{/}A}^1,M))$. Locally choosing a finite basis identifies this internal Hom canonically with $T_{X\mathrm{/}A}{\otimes }_{A}I$; these basis-independent evaluation maps agree on overlaps. Thus ${\mathrm{Ext}}^2(L,M)=H^2(X,T_{X\mathrm{/}A}{\otimes }_{A}I)$ and the already constructed obstruction lies in the group in the Statement. This makes no claim that the group vanishes on a general $X$. The zero-kernel case has $M=0$, the zero obstruction and the unique trivial marked lift. All required conclusions follow.