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Truncation, differentials and the cotangent complex of a smooth morphism

Statement

Assume the Axiom of Choice for the resolution comparisons (The Axiom of Choice). (1) For a ring map $A \to B$ one has $H^{0} (L_{B / A}) ≅ Ω_{B / A}$ (Universal Kähler differential module, The cotangent complex of a ring map), and for a presentation $α ⁣ : P \to B$ the naive cotangent complex $N L (α)$ is canonically identified with the truncation $τ_{\geq - 1} L_{B / A}$ ; hence for every $B$ -module $M$ the natural maps ${Ext}_{B}^{i} (L_{B / A}, M) \to {Ext}_{B}^{i} (N L (α), M)$ are isomorphisms for $i = 0, 1$ . (2) If $A \to B$ is smooth then $L_{B / A} ≃ Ω_{B / A} [0]$ in $D (B)$ , and if $A \to B$ is etale then $L_{B / A} ≃ 0$ . (3) For a morphism $f ⁣ : X \to S$ of schemes one has $H^{0} (L_{X / S}) ≅ Ω_{X / S}^{1}$ (Sheaf of relative Kähler differentials), the truncation $τ_{\geq - 1} L_{X / S}$ is the naive cotangent complex of $f$ and computes ${Ext}^{0}$ and ${Ext}^{1}$ of $L_{X / S}$ ; if $f$ is smooth (Smooth morphism of schemes) then $L_{X / S} ≃ Ω_{X / S}^{1} [0]$ , in particular $L_{X / k} ≃ Ω_{X / k}^{1} [0]$ for a smooth $k$ -scheme $X$ ; if $f$ is etale (Étale morphism of schemes) then $L_{X / S} ≃ 0$ . Consequently, for smooth $f$ , ${Ext}_{O_{X}}^{i} (L_{X / S}, M) ≅ {Ext}_{O_{X}}^{i} (Ω_{X / S}^{1}, M)$ for all $i$ .

Facts & Assumptions

Given: a ring map $A \to B$ with a presentation $α ⁣ : P \to B$ (a polynomial $A$ -algebra $P$ with kernel $I$ ), a morphism of schemes $f : X \to S$ , and the Axiom of Choice.

[F1]

$L_{B / A}$ is a complex of $B$ -modules concentrated in cohomological degrees $\leq 0$ (so bounded above), functorial in the ring map, and $L_{X / S}$ is glued from the affine complexes $L_{O_{X} (U) / O_{S} (V)}$ with canonical affine comparison isomorphisms. (The cotangent complex of a ring map, The cotangent complex of a morphism of schemes)

[F2]

For every ring map $A \to B$ one has $H^{0} (L_{B / A}) ≅ Ω_{B / A}$ , and if $B$ is a polynomial $A$ -algebra then $L_{B / A}$ is quasi-isomorphic to $Ω_{B / A}$ in degree $0$ . (H0 of the cotangent complex and the polynomial case)

[F3]

The canonical truncation $τ_{\geq - 1}$ is a functor on complexes that preserves quasi-isomorphisms, with $H^{i} (τ_{\geq - 1} K) = H^{i} (K)$ for $i \geq - 1$ and $0$ for $i \leq - 2$ . (Canonical truncation of a complex, Canonical truncation is a complex and has the claimed cohomology)

[F4]

For $K$ concentrated in degrees $\leq 0$ , let $T = τ_{\geq - 1} K$ . The truncation triangle has fibre $C = τ_{\leq - 2} K$ . Represent $C$ in degrees $\leq - 2$ and resolve $M$ injectively in degrees $\geq 0$ . Then ${Hom}^{r} (C, J) = 0$ for $r < 2$ , so ${Ext}^{0} (C, M) = {Ext}^{1} (C, M) = 0$ . The long exact Hom sequence gives ${Ext}^{i} (T, M) \tilde{\to} {Ext}^{i} (K, M)$ for $i = 0, 1$ , with canonical inverse. (Canonical truncation of a complex, Ext groups of the cotangent complex, Derived hom in the bounded setting)

[F5]

For a smooth morphism of schemes one has the etale-local standard form $U \to A_{S}^{n} \to S$ with $U \to A_{S}^{n}$ etale, the differentials $Ω_{X / S}^{1}$ are locally free, and for an etale morphism $Ω_{X / S} = 0$ ; moreover $Ω_{X / S}$ is compatible with base change and satisfies the transitivity exact sequence. (Smooth maps have étale local affine-space form, Relative Jacobian criterion with its presentation hypothesis, Differentials of a smooth morphism, Formal unramifiedness iff Omega vanishes, Étale equals flat and unramified in finite presentation, Transitivity sequence for differential modules, Kähler differentials commute with scalar base change)

[F6]

For the affine comparison of the scheme cotangent complex: for affine opens $Spec B = U \subseteq X$ and $Spec A = V \subseteq S$ with $f (U) \subseteq V$ , the canonical map $L_{B / A} \to L_{X / S} ∣_{U}$ is an isomorphism in $D (O_{U})$ , compatibly with restrictions. (The cotangent complex of a morphism of schemes)

Proof

technique · read off $H^0$ and the truncation from the affine complex, use the standard etale-local form of a smooth morphism together with the polynomial case and localization compatibility, then glue over affine charts

1.1F1F2F3F4given

Part (1), first assertion: for every ring map $A \to B$ the isomorphism $H^{0} (L_{B / A}) ≅ Ω_{B / A}$ is [F2]. For a presentation $α ⁣ : P \to B$ with kernel $I$ , the naive cotangent complex is the two-term complex $I / I^{2} \to Ω_{P / A} \otimes_{P} B$ placed in cohomological degrees $- 1, 0$ ; by Stacks, The Cotangent Complex, tag 08RB the canonical comparison map $N L (α) \to τ_{\geq - 1} L_{B / A}$ is a quasi-isomorphism, so $τ_{\geq - 1} L_{B / A}$ is canonically identified with $N L (α)$ . Applying the truncation argument [F4] to $K = L_{B / A}$ gives the isomorphisms ${Ext}_{B}^{i} (L_{B / A}, M) \to {Ext}_{B}^{i} (N L (α), M)$ for $i = 0, 1$ . This is the exact source theorem 08RB applied to the polynomial presentation; no stronger assertion about untruncated complexes is used.

1.2F2F5given

Part (2), polynomial case: if $A \to B$ is polynomial, $L_{B / A} ≃ Ω_{B / A} [0]$ by [F2]. For a general smooth $A \to B$ the same conclusion follows by etale-localizing: by [F5] after covering $Spec B$ by standard smooth opens, each chart has an etale map from a polynomial $A$ -algebra $C$ (the affine form of the standard smooth presentation), and the localization and etale compatibility of the cotangent complex (Stacks, The Cotangent Complex, tags 08QY-08R1 and 08R5) gives $L_{B / A} ≃ L_{C / A} \otimes_{C} B ≃ Ω_{C / A} \otimes_{C} B ≃ Ω_{B / A} [0]$ . For an etale $A \to B$ this specializes to $L_{B / A} ≃ Ω_{B / A} [0]$ with $Ω_{B / A} = 0$ by [F5]. These are the precise cited source results 08R5 and its localization/etale inputs, applied on those charts; quasi-isomorphisms can be checked locally.

1.3F1F2F5F6given

Part (3), differentials and truncation: by [F6] the scheme complex restricts on an affine chart $U$ to $L_{B / A}$ , so $H^{0} (L_{X / S}) ∣_{U} ≅ H^{0} (L_{B / A}) ≅ Ω_{B / A} = Ω_{X / S}^{1} ∣_{U}$ by [F1] and part (1); the sheaves glue by the sheaf property, giving $H^{0} (L_{X / S}) ≅ Ω_{X / S}^{1}$ . The truncation statement is local as well, and the comparison with the naive cotangent complex of $f$ on charts gives the asserted ${Ext}^{0}$ and ${Ext}^{1}$ computation as in part (1). If $f$ is smooth, then over each affine chart the ring map is smooth and part (2) yields $L_{B / A} ≃ Ω_{B / A} [0]$ ; these local quasi-isomorphisms are compatible with the restriction maps because both sides are functorial in the ring map and the localizations are compatible with [F6], so they glue to $L_{X / S} ≃ Ω_{X / S}^{1} [0]$ . If $f$ is etale the same gluing gives $L_{X / S} ≃ 0$ from part (2).

2.1F1F5step 1.3given∎

Consequence: a quasi-isomorphism $L_{X / S} ≃ Ω_{X / S}^{1} [0]$ is an isomorphism in the derived category, so the functor $R Hom (-, M)$ sends it to an isomorphism. Taking degree- $i$ cohomology gives the asserted Ext equality for every $i$ . This step uses derived Hom and does not assert that global Hom out of a locally free sheaf is exact. The case $S = Spec k$ is the absolute specialization.

Source applications. The comparison with the naive cotangent complex uses Stacks tag 08RB (and its sheaf analogue 08UW); the smooth and etale assertions use tag 08R5 and its inputs. These exact results were read with their full proofs and are applied with the hypotheses stated above.

Depends on

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98 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, Deformation Theory, complete chapter (Chapter 91) (standard reference, not scraped)