The Stacks project

108.56 Derived pushforward of quasi-coherent modules

Let $k$ be a field of characteristic $p > 0$. Let $S = \mathop{\mathrm{Spec}}(k[x])$. Let $G = \mathbf{Z}/p\mathbf{Z}$ viewed either as an abstract group or as a constant group scheme over $S$. Consider the algebraic stack $\mathcal{X} = [S/G]$ where $G$ acts trivially on $S$, see Examples of Stacks, Remark 93.15.5 and Criteria for Representability, Lemma 95.18.3. Consider the structure morphism

\[ f : \mathcal{X} \longrightarrow S \]

This morphism is quasi-compact and quasi-separated. Hence we get a functor

\[ Rf_{\mathit{QCoh}, *} : D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow D^{+}_\mathit{QCoh}(\mathcal{O}_ S), \]

see Derived Categories of Stacks, Proposition 102.5.1. Let's compute $Rf_{\mathit{QCoh}, *}\mathcal{O}_\mathcal {X}$. Since $D_\mathit{QCoh}(\mathcal{O}_ S)$ is equivalent to the derived category of $k[x]$-modules (see Derived Categories of Schemes, Lemma 36.3.5) this is equivalent to computing $R\Gamma (\mathcal{X}, \mathcal{O}_\mathcal {X})$. For this we can use the covering $S \to \mathcal{X}$ and the spectral sequence

\[ H^ q(S \times _\mathcal {X} \ldots \times _\mathcal {X} S, O) \Rightarrow H^{p + q}(\mathcal{X}, \mathcal{O}_\mathcal {X}) \]

see Cohomology of Stacks, Proposition 101.10.4. Note that

\[ S \times _\mathcal {X} \ldots \times _\mathcal {X} S = S \times G^ p \]

which is affine. Thus the complex

\[ k[x] \to \text{Map}(G, k[x]) \to \text{Map}(G^2, k[x]) \to \ldots \]

computes $R\Gamma (\mathcal{X}, \mathcal{O}_\mathcal {X})$. Here for $\varphi \in \text{Map}(G^{p - 1}, k[x])$ its differential is the map which sends $(g_1, \ldots , g_ p)$ to

\[ \varphi (g_2, \ldots , g_ p) + \sum \nolimits _{i = 1}^{p - 1} (-1)^ i\varphi (g_1, \ldots , g_ i + g_{i + 1}, \ldots , g_ p) + (-1)^ p\varphi (g_1, \ldots , g_{p - 1}). \]

This is just the complex computing the group cohomology of $G$ acting trivially on $k[x]$ (insert future reference here). The cohomology of the cyclic group $G$ on $k[x]$ is exactly one copy of $k[x]$ in each cohomological degree $\geq 0$ (insert future reference here). We conclude that

\[ Rf_*\mathcal{O}_\mathcal {X} = \bigoplus \nolimits _{n \geq 0} \mathcal{O}_ S[-n] \]

Now, consider the complex

\[ E = \bigoplus \nolimits _{m \geq 0} \mathcal{O}_\mathcal {X}[m] \]

This is an object of $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. We interrupt the discussion for a general result.

Lemma 108.56.1. Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\mathcal{O}_\mathcal {X})$ whose cohomology sheaves are locally quasi-coherent (Sheaves on Stacks, Definition 94.11.5) and satisfy the flat base change property (Cohomology of Stacks, Definition 101.7.1). Then there exists a distinguished triangle

\[ K \to \prod \nolimits _{n \geq 0} \tau _{\geq -n} K \to \prod \nolimits _{n \geq 0} \tau _{\geq -n} K \to K[1] \]

in $D(\mathcal{O}_\mathcal {X})$. In other words, $K$ is the derived limit of its canonical truncations.

Proof. Recall that we work on the “big fppf site” $\mathcal{X}_{fppf}$ of $\mathcal{X}$ (by our conventions for sheaves of $\mathcal{O}_\mathcal {X}$-modules in the chapters Sheaves on Stacks and Cohomology on Stacks). Let $\mathcal{B}$ be the set of objects $x$ of $\mathcal{X}_{fppf}$ which lie over an affine scheme $U$. Combining Sheaves on Stacks, Lemmas 94.22.2, 94.15.1, Descent, Lemma 35.9.4, and Cohomology of Schemes, Lemma 30.2.2 we see that $H^ p(x, \mathcal{F}) = 0$ if $\mathcal{F}$ is locally quasi-coherent and $x \in \mathcal{B}$. Now the claim follows from Cohomology on Sites, Lemma 21.22.10 with $d = 0$. $\square$

Lemma 108.56.2. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}_ n$ is a collection of locally quasi-coherent sheaves with the flat base change property on $\mathcal{X}$, then $\oplus _ n \mathcal{F}_ n[n] \to \prod _ n \mathcal{F}_ n[n]$ is an isomorphism in $D(\mathcal{O}_\mathcal {X})$.

Proof. This is true because by Lemma 108.56.1 we see that the direct sum is isomorphic to the product. $\square$

We continue our discussion. Since a quasi-coherent module is locally quasi-coherent and satisfies the flat base change property (Sheaves on Stacks, Lemma 94.11.6) we get

\[ E = \prod \nolimits _{m \geq 0} \mathcal{O}_\mathcal {X}[m] \]

Since cohomology commutes with limits we see that

\[ Rf_*E = \prod \nolimits _{m \geq 0} \left(\bigoplus \nolimits _{n \geq 0} \mathcal{O}_ S[m - n]\right) \]

Note that this complex is not an object of $D_\mathit{QCoh}(\mathcal{O}_ S)$ because the cohomology sheaf in degree $0$ is an infinite product of copies of $\mathcal{O}_ S$ which is not even a locally quasi-coherent $\mathcal{O}_ S$-module.

Lemma 108.56.3. A quasi-compact and quasi-separated morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks need not induce a functor $Rf_* : D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$.

Proof. See discussion above. $\square$

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 07DC. Beware of the difference between the letter 'O' and the digit '0'.