The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

17.26 The naive cotangent complex

This section is the analogue of Algebra, Section 10.132 for morphisms of ringed spaces. We urge the reader to read that section first.

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. In this section, for any sheaf of sets $\mathcal{E}$ on $X$ we denote $\mathcal{A}[\mathcal{E}]$ the sheafification of the presheaf $U \mapsto \mathcal{A}(U)[\mathcal{E}(U)]$. Here $\mathcal{A}(U)[\mathcal{E}(U)]$ denotes the polynomial algebra over $\mathcal{A}(U)$ whose variables correspond to the elements of $\mathcal{E}(U)$. We denote $[e] \in \mathcal{A}(U)[\mathcal{E}(U)]$ the variable corresponding to $e \in \mathcal{E}(U)$. There is a canonical surjection of $\mathcal{A}$-algebras

17.26.0.1
\begin{equation} \label{modules-equation-canonical-presentation} \mathcal{A}[\mathcal{B}] \longrightarrow \mathcal{B},\quad [b] \longmapsto b \end{equation}

whose kernel we denote $\mathcal{I} \subset \mathcal{A}[\mathcal{B}]$. It is a simple observation that $\mathcal{I}$ is generated by the local sections $[b][b'] - [bb']$ and $[a] - a$. According to Lemma 17.25.9 there is a canonical map

17.26.0.2
\begin{equation} \label{modules-equation-naive-cotangent-complex} \mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega _{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{B}]} \mathcal{B} \end{equation}

whose cokernel is canonically isomorphic to $\Omega _{\mathcal{B}/\mathcal{A}}$.

Definition 17.26.1. Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. The naive cotangent complex $\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}$ is the chain complex (17.26.0.2)

\[ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} = \left(\mathcal{I}/\mathcal{I}^2 \longrightarrow \Omega _{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{B}]} \mathcal{B}\right) \]

with $\mathcal{I}/\mathcal{I}^2$ placed in degree $-1$ and $\Omega _{\mathcal{A}[\mathcal{B}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{B}]} \mathcal{B}$ placed in degree $0$.

This construction satisfies a functoriality similar to that discussed in Lemma 17.25.8 for modules of differentials. Namely, given a commutative diagram

17.26.1.1
\begin{equation} \label{modules-equation-commutative-square-sheaves} \vcenter { \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } } \end{equation}

of sheaves of rings on $X$ there is a canonical $\mathcal{B}$-linear map of complexes

\[ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \longrightarrow \mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'} \]

Namely, the maps in the commutative diagram give rise to a canonical map $\mathcal{A}[\mathcal{B}] \to \mathcal{A}'[\mathcal{B}']$ which maps $\mathcal{I}$ into $\mathcal{I}' = \mathop{\mathrm{Ker}}(\mathcal{A}'[\mathcal{B}'] \to \mathcal{B}')$. Thus a map $\mathcal{I}/\mathcal{I}^2 \to \mathcal{I}'/(\mathcal{I}')^2$ and a map between modules of differentials, which together give the desired map between the naive cotangent complexes. The map is compatible with compositions in the following sense: given a commutative diagram

\[ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \ar[r] & \mathcal{B}'' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] \ar[r] & \mathcal{A}'' \ar[u] } \]

of sheaves of rings then the composition

\[ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \longrightarrow \mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'} \longrightarrow \mathop{N\! L}\nolimits _{\mathcal{B}''/\mathcal{A}''} \]

is the map for the outer rectangle.

We can choose a different presentation of $\mathcal{B}$ as a quotient of a polynomial algebra over $\mathcal{A}$ and still obtain the same object of $D(\mathcal{B})$. To explain this, suppose that $\mathcal{E}$ is a sheaves of sets on $X$ and $\alpha : \mathcal{E} \to \mathcal{B}$ a map of sheaves of sets. Then we obtain an $\mathcal{A}$-algebra homomorphism $\mathcal{A}[\mathcal{E}] \to \mathcal{B}$. If this map is surjective, i.e., if $\alpha (\mathcal{E})$ generates $\mathcal{B}$ as an $\mathcal{A}$-algebra, then we set

\[ \mathop{N\! L}\nolimits (\alpha ) = \left( \mathcal{J}/\mathcal{J}^2 \longrightarrow \Omega _{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes _{\mathcal{A}[\mathcal{E}]} \mathcal{B}\right) \]

where $\mathcal{J} \subset \mathcal{A}[\mathcal{E}]$ is the kernel of the surjection $\mathcal{A}[\mathcal{E}] \to \mathcal{B}$. Here is the result.

Lemma 17.26.2. In the situation above there is a canonical isomorphism $\mathop{N\! L}\nolimits (\alpha ) = \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}$ in $D(\mathcal{B})$.

Proof. Observe that $\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} = \mathop{N\! L}\nolimits (\text{id}_\mathcal {B})$. Thus it suffices to show that given two maps $\alpha _ i : \mathcal{E}_ i \to \mathcal{B}$ as above, there is a canonical quasi-isomorphism $\mathop{N\! L}\nolimits (\alpha _1) = \mathop{N\! L}\nolimits (\alpha _2)$ in $D(\mathcal{B})$. To see this set $\mathcal{E} = \mathcal{E}_1 \amalg \mathcal{E}_2$ and $\alpha = \alpha _1 \amalg \alpha _2 : \mathcal{E} \to \mathcal{B}$. Set $\mathcal{J}_ i = \mathop{\mathrm{Ker}}(\mathcal{A}[\mathcal{E}_ i] \to \mathcal{B})$ and $\mathcal{J} = \mathop{\mathrm{Ker}}(\mathcal{A}[\mathcal{E}] \to \mathcal{B})$. We obtain maps $\mathcal{A}[\mathcal{E}_ i] \to \mathcal{A}[\mathcal{E}]$ which send $\mathcal{J}_ i$ into $\mathcal{J}$. Thus we obtain canonical maps of complexes

\[ \mathop{N\! L}\nolimits (\alpha _ i) \longrightarrow \mathop{N\! L}\nolimits (\alpha ) \]

and it suffices to show these maps are quasi-isomorphism. To see this it suffices to check on stalks (Lemma 17.3.1). If $x \in X$ then the stalk of $\mathop{N\! L}\nolimits (\alpha )$ is the complex $\mathop{N\! L}\nolimits (\alpha _ x)$ of Algebra, Section 10.132 associated to the presentation $\mathcal{A}_ x[\mathcal{E}_ x] \to \mathcal{B}_ x$ coming from the map $\alpha _ x : \mathcal{E}_ x \to \mathcal{B}_ x$. (Some details omitted; use Lemma 17.25.7 to see compatibility of forming differentials and taking stalks.) We conclude the result holds by Algebra, Lemma 10.132.2. $\square$

Lemma 17.26.3. Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $Y$. Then $f^{-1}\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} = \mathop{N\! L}\nolimits _{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.

Proof. Omitted. Hint: Use Lemma 17.25.6. $\square$

Lemma 17.26.4. Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}, x} = \mathop{N\! L}\nolimits _{\mathcal{B}_ x/\mathcal{A}_ x}$.

Proof. This is a special case of Lemma 17.26.3 for the inclusion map $\{ x\} \to X$. $\square$

Lemma 17.26.5. Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be maps of sheaves of rings. Let $C$ be the cone (Derived Categories, Definition 13.9.1) of the map of complexes $\mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{A}} \to \mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{B}}$. There is a canonical map

\[ c : \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B} \mathcal{C} \longrightarrow C[-1] \]

of complexes of $\mathcal{C}$-modules which produces a canonical six term exact sequence

\[ \xymatrix{ H^0(\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B} \mathcal{C}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{A}}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{B}}) \ar[r] & 0 \\ H^{-1}(\mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B} \mathcal{C}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{A}}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{B}}) \ar[llu] } \]

of cohomology sheaves.

Proof. To give the map $c$ we have to give a map $c_1 : \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B} \mathcal{C} \to \mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{A}}$ and an explicity homotopy between the composition

\[ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B} \mathcal{C} \to \mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{A}} \to \mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{B}} \]

and the zero map, see Derived Categories, Lemma 13.9.3. For $c_1$ we use the functoriality described above for the obvious diagram. For the homotopy we use the map

\[ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}^0 \otimes _\mathcal {B} \mathcal{C} \longrightarrow \mathop{N\! L}\nolimits _{\mathcal{C}/\mathcal{B}}^{-1},\quad \text{d}[b] \otimes 1 \longmapsto [\varphi (b)] - b[1] \]

where $\varphi : \mathcal{B} \to \mathcal{C}$ is the given map. Please compare with Algebra, Remark 10.132.5. To see the consequence for cohomology sheaves, it suffices to show that $H^0(c)$ is an isomorphism and $H^{-1}(c)$ surjective. To see this we can look at stalks, see Lemma 17.26.4, and then we can use the corresponding result in commutative algebra, see Algebra, Lemma 10.132.4. Some details omitted. $\square$

The cotangent complex of a morphism of ringed spaces is defined in terms of the cotangent complex we defined above.

Definition 17.26.6. The naive cotangent complex $\mathop{N\! L}\nolimits _ f = \mathop{N\! L}\nolimits _{X/Y}$ of a morphism of ringed spaces $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ is $\mathop{N\! L}\nolimits _{\mathcal{O}_ X/f^{-1}\mathcal{O}_ Y}$.

Given a commutative diagram

\[ \xymatrix{ X' \ar[r]_ g \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ h & Y } \]

of ringed spaces, there is a canonical map $c : g^*\mathop{N\! L}\nolimits _{X/Y} \to \mathop{N\! L}\nolimits _{X'/Y'}$. Namely, it is the map

\[ g^*\mathop{N\! L}\nolimits _{X/Y} = \mathcal{O}_{X'} \otimes _{g^{-1}\mathcal{O}_ X} \mathop{N\! L}\nolimits _{g^{-1}\mathcal{O}_ X/g^{-1}f^{-1}\mathcal{O}_ Y} \longrightarrow \mathop{N\! L}\nolimits _{\mathcal{O}_{X'}/(f')^{-1}\mathcal{O}_{Y'}} = \mathop{N\! L}\nolimits _{X'/Y'} \]

where the arrow comes from the commutative diagram of sheaves of rings

\[ \xymatrix{ g^{-1}\mathcal{O}_ X \ar[r]_{g^\sharp } & \mathcal{O}_{X'} \\ g^{-1}f^{-1}\mathcal{O}_ Y \ar[r]^{g^{-1}h^\sharp } \ar[u]^{g^{-1}f^\sharp } & (f')^{-1}\mathcal{O}_{Y'} \ar[u]_{(f')^\sharp } } \]

as in (17.26.1.1) above. Given a second such diagram

\[ \xymatrix{ X'' \ar[r]_{g'} \ar[d] & X' \ar[d] \\ Y'' \ar[r] & Y' } \]

the composition of $(g')^*c$ and the map $c' : (g')^*\mathop{N\! L}\nolimits _{X'/Y'} \to \mathop{N\! L}\nolimits _{X''/Y''}$ is the map $(g \circ g')^*\mathop{N\! L}\nolimits _{X''/Y''} \to \mathop{N\! L}\nolimits _{X/Y}$.

Lemma 17.26.7. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $C$ be the cone of the map $\mathop{N\! L}\nolimits _{X/Z} \to \mathop{N\! L}\nolimits _{X/Y}$ of complexes of $\mathcal{O}_ X$-modules. There is a canonical map

\[ f^*\mathop{N\! L}\nolimits _{Y/Z} \to C[-1] \]

which produces a canonical six term exact sequence

\[ \xymatrix{ H^0(f^*\mathop{N\! L}\nolimits _{Y/Z}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{X/Z}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{X/Y}) \ar[r] & 0 \\ H^{-1}(f^*\mathop{N\! L}\nolimits _{Y/Z}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{X/Z}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) \ar[llu] } \]

of cohomology sheaves.

Proof. Consider the maps of sheaves rings

\[ (g \circ f)^{-1}\mathcal{O}_ Z \to f^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X \]

and apply Lemma 17.26.5. $\square$


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 08TG. Beware of the difference between the letter 'O' and the digit '0'.