110.60 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 95.15.5 and Criteria for Representability, Lemma 97.18.3. 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 104.6.1. 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 103.11.6. 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 110.60.1. 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 96.12.1) and satisfy the flat base change property (Cohomology of Stacks, Definition 103.7.1). 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 96.23.2, 96.16.1, Descent, Lemma 35.12.4, 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 21.23.10 with $d = 0$.
$\square$
Lemma 110.60.2. 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 110.60.1 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 96.12.2) 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 110.60.3. 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.
$\square$
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