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*}

14.25 Dold-Kan for cosimplicial objects

Let $\mathcal{A}$ be an abelian category. According to Homology, Lemma 12.5.2 also $\mathcal{A}^{opp}$ is abelian. It follows formally from the definitions that

\[ \text{CoSimp}(\mathcal{A}) = \text{Simp}(\mathcal{A}^{opp})^{opp}. \]

Thus Dold-Kan (Theorem 14.24.3) implies that $\text{CoSimp}(\mathcal{A})$ is equivalent to the category $\text{Ch}_{\geq 0}(\mathcal{A}^{opp})^{opp}$. And it follows formally from the definitions that

\[ \text{CoCh}_{\geq 0}(\mathcal{A}) = \text{Ch}_{\geq 0}(\mathcal{A}^{opp})^{opp}. \]

Putting these arrows together we obtain an equivalence

\[ Q : \text{CoSimp}(\mathcal{A}) \longrightarrow \text{CoCh}_{\geq 0}(\mathcal{A}). \]

In this section we describe $Q$.

First we define the cochain complex $s(U)$ associated to a cosimplicial object $U$. It is the cochain complex with terms zero in negative degrees, and $s(U)^ n = U_ n$ for $n \geq 0$. As differentials we use the maps $d^ n : s(U)^ n \to s(U)^{n + 1}$ defined by $d^ n = \sum _{i = 0}^{n + 1} (-1)^ i \delta ^{n + 1}_ i$. In other words the complex $s(U)$ looks like

\[ \xymatrix{ 0 \ar[r] & U_0 \ar[rr]^{\delta ^1_0 - \delta ^1_1} & & U_1 \ar[rr]^{\delta ^2_0 - \delta ^2_1 + \delta ^2_2} & & U_2 \ar[r] & \ldots } \]

This is sometimes also called the Moore complex associated to $U$.

On the other hand, given a cosimplicial object $U$ of $\mathcal{A}$ set $Q(U)^0 = U_0$ and

\[ Q(U)^ n = \mathop{\mathrm{Coker}}( \xymatrix{ \bigoplus _{i = 0}^{n - 1} U_{n - 1} \ar[r]^-{\delta ^ n_ i} & U_ n }). \]

The differential $d^ n : Q(U)^ n \to Q(U)^{n + 1}$ is induced by $(-1)^{n + 1}\delta ^{n + 1}_{n + 1}$, i.e., by fitting the morphism $(-1)^{n + 1}\delta ^{n + 1}_{n + 1}$ into a commutative diagram

\[ \xymatrix{ U_ n \ar[rr]_{(-1)^{n + 1}\delta ^{n + 1}_{n + 1}} \ar[d] & & U_{n + 1} \ar[d] \\ Q(U)^ n \ar[rr]^{d_ n} & & Q(U)^{n + 1}. } \]

We leave it to the reader to show that this diagram makes sense, i.e., that the image of $\delta ^ n_ i$ maps into the kernel of the right vertical arrow for $i = 0, \ldots , n - 1$. (This is dual to Lemma 14.18.8.) Thus our cochain complex $Q(U)$ looks like this

\[ 0 \to Q(U)^0 \to Q(U)^1 \to Q(U)^2 \to \ldots \]

This is called the normalized cochain complex associated to $U$. The dual to the Dold-Kan Theorem 14.24.3 is the following.

Lemma 14.25.1. Let $\mathcal{A}$ be an abelian category.

  1. The functor $s : \text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$ is exact.

  2. The maps $s(U)^ n \to Q(U)^ n$ define a morphism of cochain complexes.

  3. There exists a functorial direct sum decomposition $s(U) = A(U) \oplus Q(U)$ in $\text{CoCh}_{\geq 0}(\mathcal{A})$.

  4. The functor $Q$ is exact.

  5. The morphism of complexes $s(U) \to Q(U)$ is a quasi-isomorphism.

  6. The functor $U \mapsto Q(U)^\bullet $ defines an equivalence of categories $\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$.


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 019H. Beware of the difference between the letter 'O' and the digit '0'.