The Stacks project

90.25 The cotangent complex of a scheme over a ring

Let $\Lambda $ be a ring and let $X$ be a scheme over $\Lambda $. Write $L_{X/\mathop{\mathrm{Spec}}(\Lambda )} = L_{X/\Lambda }$ which is justified by Lemma 90.24.3. In this section we give a description of $L_{X/\Lambda }$ similar to Lemma 90.4.3. Namely, we construct a category $\mathcal{C}_{X/\Lambda }$ fibred over $X_{Zar}$ and endow it with a sheaf of (polynomial) $\Lambda $-algebras $\mathcal{O}$ such that

\[ L_{X/\Lambda } = L\pi _!(\Omega _{\mathcal{O}/\underline{\Lambda }} \otimes _\mathcal {O} \underline{\mathcal{O}}_ X). \]

We will later use the category $\mathcal{C}_{X/\Lambda }$ to construct a naive obstruction theory for the stack of coherent sheaves.

Let $\Lambda $ be a ring. Let $X$ be a scheme over $\Lambda $. Let $\mathcal{C}_{X/\Lambda }$ be the category whose objects are commutative diagrams
\begin{equation} \label{cotangent-equation-object} \vcenter { \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ \mathop{\mathrm{Spec}}(\Lambda ) & \mathbf{A} \ar[l] } } \end{equation}

of schemes where

  1. $U$ is an open subscheme of $X$,

  2. there exists an isomorphism $\mathbf{A} = \mathop{\mathrm{Spec}}(P)$ where $P$ is a polynomial algebra over $\Lambda $ (on some set of variables).

In other words, $\mathbf{A}$ is an (infinite dimensional) affine space over $\mathop{\mathrm{Spec}}(\Lambda )$. Morphisms are given by commutative diagrams. Recall that $X_{Zar}$ denotes the small Zariski site $X$. There is a forgetful functor

\[ u : \mathcal{C}_{X/\Lambda } \to X_{Zar},\ (U \to \mathbf{A}) \mapsto U \]

Observe that the fibre category over $U$ is canonically equivalent to the category $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ introduced in Section 90.4.

Lemma 90.25.1. In the situation above the category $\mathcal{C}_{X/\Lambda }$ is fibred over $X_{Zar}$.

Proof. Given an object $U \to \mathbf{A}$ of $\mathcal{C}_{X/\Lambda }$ and a morphism $U' \to U$ of $X_{Zar}$ consider the object $U' \to \mathbf{A}$ of $\mathcal{C}_{X/\Lambda }$ where $U' \to \mathbf{A}$ is the composition of $U \to \mathbf{A}$ and $U' \to U$. The morphism $(U' \to \mathbf{A}) \to (U \to \mathbf{A})$ of $\mathcal{C}_{X/\Lambda }$ is strongly cartesian over $X_{Zar}$. $\square$

We endow $\mathcal{C}_{X/\Lambda }$ with the topology inherited from $X_{Zar}$ (see Stacks, Section 8.10). The functor $u$ defines a morphism of topoi $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/\Lambda }) \to \mathop{\mathit{Sh}}\nolimits (X_{Zar})$. The site $\mathcal{C}_{X/\Lambda }$ comes with several sheaves of rings.

  1. The sheaf $\mathcal{O}$ given by the rule $(U \to \mathbf{A}) \mapsto \Gamma (\mathbf{A}, \mathcal{O}_\mathbf {A})$.

  2. The sheaf $\underline{\mathcal{O}}_ X = \pi ^{-1}\mathcal{O}_ X$ given by the rule $(U \to \mathbf{A}) \mapsto \mathcal{O}_ X(U)$.

  3. The constant sheaf $\underline{\Lambda }$.

We obtain morphisms of ringed topoi
\begin{equation} \label{cotangent-equation-pi-schemes} \vcenter { \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/\Lambda }), \underline{\mathcal{O}}_ X) \ar[r]_ i \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/\Lambda }), \mathcal{O}) \\ (\mathop{\mathit{Sh}}\nolimits (X_{Zar}), \mathcal{O}_ X) } } \end{equation}

The morphism $i$ is the identity on underlying topoi and $i^\sharp : \mathcal{O} \to \underline{\mathcal{O}}_ X$ is the obvious map. The map $\pi $ is a special case of Cohomology on Sites, Situation 21.37.1. An important role will be played in the following by the derived functors $ Li^* : D(\mathcal{O}) \longrightarrow D(\underline{\mathcal{O}}_ X) $ left adjoint to $Ri_* = i_* : D(\underline{\mathcal{O}}_ X) \to D(\mathcal{O})$ and $ L\pi _! : D(\underline{\mathcal{O}}_ X) \longrightarrow D(\mathcal{O}_ X) $ left adjoint to $\pi ^* = \pi ^{-1} : D(\mathcal{O}_ X) \to D(\underline{\mathcal{O}}_ X)$. We can compute $L\pi _!$ thanks to our earlier work.

Remark 90.25.2. In the situation above, for every $U \subset X$ open let $P_{\bullet , U}$ be the standard resolution of $\mathcal{O}_ X(U)$ over $\Lambda $. Set $\mathbf{A}_{n, U} = \mathop{\mathrm{Spec}}(P_{n, U})$. Then $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of the fibre category $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ of $\mathcal{C}_{X/\Lambda }$ over $U$. Moreover, as discussed in Remark 90.5.5 we have that $\mathbf{A}_{\bullet , U}$ is a cosimplicial object of $\mathcal{C}_{\mathcal{O}_ X(U)/\Lambda }$ as in Cohomology on Sites, Lemma 21.38.7. Since the construction $U \mapsto \mathbf{A}_{\bullet , U}$ is functorial in $U$, given any (abelian) sheaf $\mathcal{F}$ on $\mathcal{C}_{X/\Lambda }$ we obtain a complex of presheaves

\[ U \longmapsto \mathcal{F}(\mathbf{A}_{\bullet , U}) \]

whose cohomology groups compute the homology of $\mathcal{F}$ on the fibre category. We conclude by Cohomology on Sites, Lemma 21.39.2 that the sheafification computes $L_ n\pi _!(\mathcal{F})$. In other words, the complex of sheaves whose term in degree $-n$ is the sheafification of $U \mapsto \mathcal{F}(\mathbf{A}_{n, U})$ computes $L\pi _!(\mathcal{F})$.

With this remark out of the way we can state the main result of this section.

Lemma 90.25.3. In the situation above there is a canonical isomorphism

\[ L_{X/\Lambda } = L\pi _!(Li^*\Omega _{\mathcal{O}/\underline{\Lambda }}) = L\pi _!(i^*\Omega _{\mathcal{O}/\underline{\Lambda }}) = L\pi _!(\Omega _{\mathcal{O}/\underline{\Lambda }} \otimes _\mathcal {O} \underline{\mathcal{O}}_ X) \]

in $D(\mathcal{O}_ X)$.

Proof. We first observe that for any object $(U \to \mathbf{A})$ of $\mathcal{C}_{X/\Lambda }$ the value of the sheaf $\mathcal{O}$ is a polynomial algebra over $\Lambda $. Hence $\Omega _{\mathcal{O}/\underline{\Lambda }}$ is a flat $\mathcal{O}$-module and we conclude the second and third equalities of the statement of the lemma hold.

By Remark 90.25.2 the object $L\pi _!(\Omega _{\mathcal{O}/\underline{\Lambda }} \otimes _\mathcal {O} \underline{\mathcal{O}}_ X)$ is computed as the sheafification of the complex of presheaves

\[ U \mapsto \left(\Omega _{\mathcal{O}/\underline{\Lambda }} \otimes _\mathcal {O} \underline{\mathcal{O}}_ X\right)(\mathbf{A}_{\bullet , U}) = \Omega _{P_{\bullet , U}/\Lambda } \otimes _{P_{\bullet , U}} \mathcal{O}_ X(U) = L_{\mathcal{O}_ X(U)/\Lambda } \]

using notation as in Remark 90.25.2. Now Remark 90.18.5 shows that $L\pi _!(\Omega _{\mathcal{O}/\underline{\Lambda }} \otimes _\mathcal {O} \underline{\mathcal{O}}_ X)$ computes the cotangent complex of the map of rings $\underline{\Lambda } \to \mathcal{O}_ X$ on $X$. This is what we want by Lemma 90.24.3. $\square$

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