Definition 76.21.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The naive cotangent complex of $f$ is the complex defined in Modules on Sites, Definition 18.35.4 for the morphism of ringed topoi $f_{small}$ between the small étale sites of $X$ and $Y$, see Properties of Spaces, Lemma 66.21.3. Notation: $\mathop{N\! L}\nolimits _ f$ or $\mathop{N\! L}\nolimits _{X/Y}$.
76.21 The naive cotangent complex
This section is the continuation of Modules on Sites, Section 18.35 which in turn continues the discussion in Algebra, Section 10.134.
The next lemmas show this definition is compatible with the definition for ring maps and for schemes and that $\mathop{N\! L}\nolimits _{X/Y}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.
Lemma 76.21.2. Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ étale. Then there is a canonical identification $\mathop{N\! L}\nolimits _{X/Y}|_{U_{\acute{e}tale}} = \mathop{N\! L}\nolimits _{U/V}$ in $D(\mathcal{O}_ U)$.
\[ \xymatrix{ U \ar[d]_ p \ar[r]_ g & V \ar[d]^ q \\ X \ar[r]^ f & Y } \]
Proof. Formation of the naive cotangent complex commutes with pullback (Modules on Sites, Lemma 18.35.3) and we have $p_{small}^{-1}\mathcal{O}_ X = \mathcal{O}_ U$ and $g_{small}^{-1}\mathcal{O}_{V_{\acute{e}tale}} = p_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}$ because $q_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \mathcal{O}_{V_{\acute{e}tale}}$ by Properties of Spaces, Lemma 66.26.1. Tracing through the definitions we conclude that $\mathop{N\! L}\nolimits _{X/Y}|_{U_{\acute{e}tale}} = \mathop{N\! L}\nolimits _{U/V}$. $\square$
Lemma 76.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$. Then there is a canonical identification $\mathop{N\! L}\nolimits _{X/Y} = \epsilon ^*\mathop{N\! L}\nolimits _{X_0/Y_0}$ in $D(\mathcal{O}_ X)$ where $\epsilon $ is as in Derived Categories of Spaces, Section 75.4 and $\mathop{N\! L}\nolimits _{X_0/Y_0}$ is as in More on Morphisms, Definition 37.13.1.
Proof. Let $f_0 : X_0 \to Y_0$ be the morphism of schemes corresponding to $f$. There is a canonical map $\epsilon ^{-1}f_0^{-1}\mathcal{O}_{Y_0} \to f_{small}^{-1}\mathcal{O}_ Y$ compatible with $\epsilon ^\sharp : \epsilon ^{-1}\mathcal{O}_{X_0} \to \mathcal{O}_ X$ because there is a commutative diagram
\[ \xymatrix{ X_{0, Zar} \ar[d]_{f_0} & X_{\acute{e}tale}\ar[l]^\epsilon \ar[d]^ f \\ Y_{0, Zar} & Y_{\acute{e}tale}\ar[l]_\epsilon } \]
see Derived Categories of Spaces, Remark 75.6.3. Thus we obtain a canonical map
\[ \epsilon ^{-1}\mathop{N\! L}\nolimits _{X_0/Y_0} = \epsilon ^{-1}\mathop{N\! L}\nolimits _{\mathcal{O}_{X_0}/f_0^{-1}\mathcal{O}_{Y_0}} = \mathop{N\! L}\nolimits _{\epsilon ^{-1}\mathcal{O}_{X_0}/\epsilon ^{-1}f_0^{-1}\mathcal{O}_{Y_0}} \to \mathop{N\! L}\nolimits _{\mathcal{O}_ X/f^{-1}_{small}\mathcal{O}_ Y} = \mathop{N\! L}\nolimits _{X/Y} \]
by functoriality of the naive cotangent complex. To see that the induced map $\epsilon ^*\mathop{N\! L}\nolimits _{X_0/Y_0} \to \mathop{N\! L}\nolimits _{X/Y}$ is an isomorphism in $D(\mathcal{O}_ X)$ we may check on stalks at geometric points (Properties of Spaces, Theorem 66.19.12). Let $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X_0$ be a geometric point lying over $x \in X_0$, with $\overline{y} = f \circ \overline{x}$ lying over $y \in Y_0$. Then
\[ \mathop{N\! L}\nolimits _{X/Y, \overline{x}} = \mathop{N\! L}\nolimits _{\mathcal{O}_{X, \overline{x}}/\mathcal{O}_{Y, \overline{y}}} \]
This is true because taking stalks at $\overline{x}$ is the same as taking inverse image via $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ and we may apply Modules on Sites, Lemma 18.35.3. On the other hand we have
\[ (\epsilon ^*\mathop{N\! L}\nolimits _{X_0/Y_0})_{\overline{x}} = \mathop{N\! L}\nolimits _{X_0/Y_0, x} \otimes _{\mathcal{O}_{X_0, x}} \mathcal{O}_{X, \overline{x}} = \mathop{N\! L}\nolimits _{\mathcal{O}_{X_0, x}/\mathcal{O}_{Y_0, y}} \otimes _{\mathcal{O}_{X_0, x}} \mathcal{O}_{X, \overline{x}} \]
Some details omitted (hint: use that the stalk of a pullback is the stalk at the image point, see Sites, Lemma 7.34.2, as well as the corresponding result for modules, see Modules on Sites, Lemma 18.36.4). Observe that $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{X_0, x}$ and similarly for $\mathcal{O}_{Y, \overline{y}}$ (Properties of Spaces, Lemma 66.22.1). Thus the result follows from More on Algebra, Lemma 15.33.8. $\square$
Lemma 76.21.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The cohomology sheaves of the complex $\mathop{N\! L}\nolimits _{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\Omega _{X/Y}$ in degree $0$.
Proof. By construction of the naive cotangent complex in Modules on Sites, Section 18.35 we have that $\mathop{N\! L}\nolimits _{X/Y}$ is a complex sitting in degrees $-1$, $0$ and that its cohomology in degree $0$ is $\Omega _{X/Y}$ (by our construction of $\Omega _{X/Y}$ in Section 76.7). The sheaf of differentials is quasi-coherent (by Lemma 76.7.4). To finish the proof it suffices to show that $H^{-1}(\mathop{N\! L}\nolimits _{X/Y})$ is quasi-coherent. This follows by checking étale locally (allowed by Lemma 76.21.2 and Properties of Spaces, Lemma 66.29.6) reducing to the case of schemes (Lemma 76.21.3) and finally using the result in the case of schemes (More on Morphisms, Lemma 37.13.3). $\square$
Lemma 76.21.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\mathop{N\! L}\nolimits _{X/Y}$ is étale locally on $X$ quasi-isomorphic to a complex of quasi-coherent $\mathcal{O}_ X$-modules with $\mathcal{F}^0$ of finite presentation and $\mathcal{F}^{-1}$ of finite type.
\[ \ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots \]
Proof. Formation of the naive cotangent complex commutes with étale localization by Lemma 76.21.2. This reduces us to the case of schemes by Lemma 76.21.3. The result in the case of schemes is More on Morphisms, Lemma 37.13.4. $\square$
Lemma 76.21.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent
$f$ is formally smooth,
$H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$ and $H^0(\mathop{N\! L}\nolimits _{X/Y}) = \Omega _{X/Y}$ is locally projective.
Proof. This follows from Lemma 76.19.10, Lemma 76.21.2, Lemma 76.21.3 and the case of schemes which is More on Morphisms, Lemma 37.13.5. $\square$
Lemma 76.21.7. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent
$f$ is formally étale,
$H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = H^0(\mathop{N\! L}\nolimits _{X/Y}) = 0$.
Proof. Assume (1). A formally étale morphism is a formally smooth morphism. Thus $H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$ by Lemma 76.21.6. On the other hand, a formally étale morphism if formally unramified hence we have $\Omega _{X/Y} = 0$ by Lemma 76.14.6. Conversely, if (2) holds, then $f$ is formally smooth by Lemma 76.21.6 and formally unramified by Lemma 76.14.6 and hence formally étale by Lemmas 76.19.4. $\square$
Lemma 76.21.8. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent
$f$ is smooth, and
$f$ is locally of finite presentation, $H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) = 0$, and $H^0(\mathop{N\! L}\nolimits _{X/Y}) = \Omega _{X/Y}$ is finite locally free.
Proof. This follows from Lemma 76.19.10, Lemma 76.21.2, Lemma 76.21.3 and the case of schemes which is More on Morphisms, Lemma 37.13.7. $\square$
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