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Ext of a locally free cotangent sheaf via sheaf cohomology

Statement

Assume the Axiom of Choice (it supplies the Dependent Choice of the derived Hom, The Axiom of Choice, AC implies DC implies countable choice). Let $X$ be a scheme, let $F$ be a locally free $O_{X}$ -module of finite rank (Locally free sheaves of finite rank) regarded as a complex in degree $0$ , and let $M$ be a quasi-coherent $O_{X}$ -module with $T = H o m_{O_{X}} (F, M) ≅ F^{\lor} \otimes_{O_{X}} M$ (Internal Hom of module sheaves, Dual and base change for finite locally free sheaves, Invertible sheaves). Then for every $i$ , ${Ext}_{O_{X}}^{i} (F, M) ≅ H^{i} (X, H o m_{O_{X}} (F, M)) = H^{i} (X, F^{\lor} \otimes M),$ so in particular ${Ext}^{0} = H^{0} (X, T)$ , ${Ext}^{1} = H^{1} (X, T)$ and ${Ext}^{2} = H^{2} (X, T)$ . Applying this to $F = Ω_{X / S}^{1}$ for $S$ -smooth $X$ gives the classical deformation cohomology groups $H^{i} (X, T_{X / S} \otimes M)$ with tangent sheaf $T_{X / S} = H o m (Ω_{X / S}^{1}, O_{X})$ .

Facts & Assumptions

Given: a scheme $X$ , a locally free finite-rank $O_{X}$ -module $F$ , a quasi-coherent $O_{X}$ -module $M$ , and the Axiom of Choice.

[F1]

${Ext}_{O_{X}}^{i} (K, N) = H^{i} (R {Hom}_{O_{X}} (K, N))$ for a bounded-above complex $K$ and a module $N$ in degree $0$ , and a bounded-below injective resolution $N \to J^{∙}$ computes this derived Hom by the global Hom complex ${Hom}_{O_{X}} (K, J^{∙})$ . (Ext groups of the cotangent complex, Derived hom in the bounded setting)

[F2]

For $O_{X}$ -modules there is the tensor-Hom adjunction ${Hom}_{O_{X}} (F \otimes_{O_{X}} N, N^{'}) = {Hom}_{O_{X}} (N, H o m_{O_{X}} (F, N^{'}))$ , and if $F$ is locally free then $H o m_{O_{X}} (F, -)$ is exact and $H o m (F, N)$ is again a sheaf of $O_{X}$ -modules. (Internal Hom of module sheaves, The internal Hom sheaf of two module sheaves)

[F3]

For every sheaf $G$ of $O_{X}$ -modules, the functor ${Hom}_{O_{X}} (O_{X}, G)$ is the global-sections functor $Γ (X, G)$ , and its right derived functors are the cohomology groups $H^{i} (X, G)$ . (Sheaf cohomology as right derived global sections, Injective modules are flasque and Ext from the structure sheaf is cohomology)

[F4]

$F^{\lor} = H o m_{O_{X}} (F, O_{X})$ is finite locally free, and for finite locally free $F$ there is a canonical isomorphism $H o m_{O_{X}} (F, M) ≅ F^{\lor} \otimes_{O_{X}} M$ . (Dual and base change for finite locally free sheaves, Invertible sheaves)

[F5]

If $f : X \to S$ is smooth then $L_{X / S} ≃ Ω_{X / S}^{1} [0]$ and $Ω_{X / S}^{1}$ is locally free of finite rank. (Truncation, differentials and the cotangent complex of a smooth morphism, Differentials of a smooth morphism, Sheaf of relative Kähler differentials, Smooth morphism of schemes)

Proof

technique · replace the derived Hom out of a locally free sheaf by global sections of its Hom sheaf using exactness and the tensor-Hom adjunction, then specialize along the smooth case of the cotangent complex

1.1F1F2F3given

Choose an injective resolution $M \to J^{∙}$ in sheaves of $O_{X}$ -modules, as in [F1]. The sheaf functor $H o m (F, -)$ is exact: on an open where $F ≅ O_{X}^{r}$ it is the finite-product functor $(-)^{r}$ . It also preserves injectives. Indeed, for an injective $J$ , the adjunction $Hom (N, H o m (F, J)) ≅ Hom (F \otimes N, J)$ shows that the left side is exact in $N$ , since $F \otimes -$ is exact by the same local freeness argument. Thus $H o m (F, M) \to H o m (F, J^{∙})$ is an injective resolution. The complexes $Hom (F, J^{∙})$ and $Γ (X, H o m (F, J^{∙}))$ agree degreewise by [F2]. The first computes ${Ext}^{i} (F, M)$ by [F1], and the second computes $H^{i} (X, H o m (F, M))$ by [F3]: module-injectives are flasque as abelian sheaves, so their resolution computes the stipulated abelian-sheaf cohomology. This gives the claimed natural identification.

1.2F4

By [F4] the Hom sheaf is the tensor product $H o m (F, M) ≅ F^{\lor} \otimes M$ , so the displayed isomorphisms give the formula of the Statement; the cases $i = 0, 1, 2$ are the specialization to those degrees.

2.1F1F5given∎

For the smooth specialization, [F5] gives $L_{X / S} ≃ Ω_{X / S}^{1} [0]$ with $Ω_{X / S}^{1}$ locally free of finite rank; applying step 1.1 with $F = Ω_{X / S}^{1}$ and using the definition ${Ext}^{i} (L_{X / S}, M) = H^{i} (R Hom (L_{X / S}, M))$ together with the quasi-isomorphism gives ${Ext}_{O_{X}}^{i} (L_{X / S}, M) ≅ H^{i} (X, H o m (Ω_{X / S}^{1}, M)) = H^{i} (X, T_{X / S} \otimes M)$ , which is the classical deformation cohomology. The Axiom of Choice is inherited from the derived-Hom and injective-resolution/cohomology suppliers.

Source application. The smooth specialization uses the smooth-cotangent comparison of the declared supplier, proved there by the exact Stacks tag 08R5 application. The general finite locally free Ext formula above is proved with injective resolutions.

Depends on

Used by

Dependency tree · two levels

88 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, The Cotangent Complex, complete chapter (Chapter 92) (standard reference, not scraped)
The Stacks Project, Cohomology of Schemes, complete chapter (Chapter 30) (standard reference, not scraped)