The Stacks project

92.25 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 92.24.3. In this section we give a description of $L_{X/\Lambda }$ similar to Lemma 92.4.3. 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

92.25.0.1
\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 92.4.

Lemma 92.25.1. 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

92.25.1.1
\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 21.38.1. 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 92.25.2. 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 92.5.5 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 21.39.7. 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 21.40.2 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 92.25.3. 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 92.25.2 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 92.25.2. Now Remark 92.18.5 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 92.24.3. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08V7. Beware of the difference between the letter 'O' and the digit '0'.



View Section 92.25 as pdf