The Stacks project

Lemma 104.8.3. Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Then $Lg_!K$ satisfies the following property: for any morphism $x \to x'$ of $\mathcal{X}_{affine}$ the map

\[ R\Gamma (x', Lg_!K) \otimes _{\mathcal{O}(x')}^\mathbf {L} \mathcal{O}(x) \longrightarrow R\Gamma (x, Lg_!K) \]

is a quasi-isomorphism.

Proof. By Lemma 104.5.3 part (2)(c) the object $Lg_!K$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. It follows readily from this that the map displayed in the lemma is an isomorphism if $\mathcal{O}(x') \to \mathcal{O}(x)$ is a flat ring map; we omit the details.

In this paragraph we argue that the question is local for the étale topology. Let $x \to x'$ be a general morphism of $\mathcal{X}_{affine}$. Let $\{ x'_ i \to x'\} $ be a covering in $\mathcal{X}_{affine, {\acute{e}tale}}$. Set $x_ i = x \times _{x'} x'_ i$ so that $\{ x_ i \to x\} $ is a covering of $\mathcal{X}_{affine, {\acute{e}tale}}$ too. Then $\mathcal{O}(x') \to \prod \mathcal{O}(x'_ i)$ is a faithfully flat étale ring map and

\[ \prod \mathcal{O}(x_ i) = \mathcal{O}(x) \otimes _{\mathcal{O}(x')} \left(\prod \mathcal{O}(x'_ i)\right) \]

Thus a simple algebra argument we omit shows that it suffices to prove the result in the statement of the lemma holds for each of the morphisms $x_ i \to x'_ i$ in $\mathcal{X}_{affine}$. In other words, the problem is local in the étale topology.

Choose a scheme $X$ and a surjective smooth morphism $f : X \to \mathcal{X}$. We may view $f$ as an object of $\mathcal{X}$ (by our abuse of notation) and then $(\mathit{Sch}/X)_{fppf} = \mathcal{X}/f$, see Sheaves on Stacks, Section 96.9. By Sheaves on Stacks, Lemma 96.19.10 for example, there exist an étale covering $\{ x'_ i \to x'\} $ such that $x'_ i : U'_ i = p(x'_ i) \to \mathcal{X}$ factors through $f$. By the result of the previous paragraph, we may assume that $x \to x'$ is a morphism which is the image of a morphism $U \to U'$ of $(\textit{Aff}/X)_{fppf}$ by the functor $(\mathit{Sch}/X)_{fppf} \to \mathcal{X}$. At this point we see use that the restriction to $(\mathit{Sch}/X)_{fppf}$ of $Lg_!K$ is equal to $f^*Lg_!K = L(g')_!(f')^*K$ by Lemma 104.3.2. This reduces us to the case discussed in the next paragraph.

Assume $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ and $x \to x'$ corresponds to the morphism of affine schemes $U \to U'$. We may still work étale (or Zariski) locally on $U'$ and hence we may assume $U' \to X$ factors through some affine open of $X$. This reduces us to the case discussed in the next paragraph.

Assume $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ where $X = \mathop{\mathrm{Spec}}(R)$ is an affine scheme and $x \to x'$ corresponds to the morphism of affine schemes $U \to U'$. Let $M^\bullet $ be a complex of $R$-modules representing $R\Gamma (X, K)$. By the construction in More on Algebra, Lemma 15.59.10 we may assume $M^\bullet = \mathop{\mathrm{colim}}\nolimits P_ n^\bullet $ where each $P_ n^\bullet $ is a bounded above complex of free $R$-modules. Details omitted; see also More on Algebra, Remark 15.59.11. Consider the complex of modules $M^\bullet _{flat, fppf}$ on $X_{flat, fppf} = (\mathit{Sch}/X)_{flat, fppf}$ given by the rule

\[ U \longmapsto \Gamma (U, M^\bullet \otimes _ R \mathcal{O}_ U) \]

This is a complex of sheaves by the discussion in Descent, Section 35.8. There is a canonical map $M^\bullet _{flat, fppf} \to K$ which by our initial remarks of the proof produces an isomorphism on sections over the affine objects of $X_{flat, fppf}$. Since every object of $X_{flat, fppf}$ has a covering by affine objects we see that $M^\bullet _{flat, fppf}$ agrees with $K$.

Let $M^\bullet _{fppf}$ be the complex of modules on $X_{fppf}$ given by the same formula as displayed above. Recall that $Lg_!\mathcal{O} = g_!\mathcal{O} = \mathcal{O}$. Since $Lg_!$ is the left derived functor of $g_!$ we conclude that $Lg_!P_{n, flat, fppf}^\bullet = P_{n, fppf}^\bullet $. Since the functor $Lg_!$ commutes with homotopy colimits (or by its construction in Cohomology on Sites, Lemma 21.37.2) and since $M^\bullet = \mathop{\mathrm{colim}}\nolimits P_ n^\bullet $ we conclude that $Lg_!M^\bullet _{flat, fppf} = M^\bullet _{fppf}$. Say $U = \mathop{\mathrm{Spec}}(A)$, $U' = \mathop{\mathrm{Spec}}(A')$ and $U \to U'$ corresponds to the ring map $A' \to A$. From the above we see that

\[ R\Gamma (U, Lg_!K) = M^\bullet \otimes _ R A \quad \text{and}\quad R\Gamma (U', Lg_!K) = M^\bullet \otimes _ R A' \]

Since $M^\bullet $ is a K-flat complex of $R$-modules, by transitivity of tensor product it follows that

\[ R\Gamma (U', Lg_!K) \otimes _{A'}^\mathbf {L} A \longrightarrow R\Gamma (U, Lg_!K) \]

is a quasi-isomorphism as desired. $\square$


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 0H15. Beware of the difference between the letter 'O' and the digit '0'.