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*}

21.39 Calculating derived lower shriek

In this section we apply the results from Section 21.38 to compute $L\pi _!$ in Situation 21.37.1 and $Lg_!$ in Situation 21.37.3.

Lemma 21.39.1. Assumptions and notation as in Situation 21.37.1. For $\mathcal{F}$ in $\textit{PAb}(\mathcal{C})$ and $n \geq 0$ consider the abelian sheaf $L_ n(\mathcal{F})$ on $\mathcal{D}$ which is the sheaf associated to the presheaf

\[ V \longmapsto H_ n(\mathcal{C}_ V, \mathcal{F}|_{\mathcal{C}_ V}) \]

with restriction maps as indicated in the proof. Then $L_ n(\mathcal{F}) = L_ n(\mathcal{F}^\# )$.

Proof. For a morphism $h : V' \to V$ of $\mathcal{D}$ there is a pullback functor $h^* : \mathcal{C}_ V \to \mathcal{C}_{V'}$ of fibre categories (Categories, Definition 4.32.6). Moreover for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ V)$ there is a strongly cartesian morphism $h^*U \to U$ covering $h$. Restriction along these strongly cartesian morphisms defines a transformation of functors

\[ \mathcal{F}|_{\mathcal{C}_ V} \longrightarrow \mathcal{F}|_{\mathcal{C}_{V'}} \circ h^*. \]

By Example 21.38.3 we obtain the desired restriction map

\[ H_ n(\mathcal{C}_ V, \mathcal{F}|_{\mathcal{C}_ V}) \longrightarrow H_ n(\mathcal{C}_{V'}, \mathcal{F}|_{\mathcal{C}_{V'}}) \]

Let us denote $L_{n, p}(\mathcal{F})$ this presheaf, so that $L_ n(\mathcal{F}) = L_{n, p}(\mathcal{F})^\# $. The canonical map $\gamma : \mathcal{F} \to \mathcal{F}^+$ (Sites, Theorem 7.10.10) defines a canonical map $L_{n, p}(\mathcal{F}) \to L_{n, p}(\mathcal{F}^+)$. We have to prove this map becomes an isomorphism after sheafification.

Let us use the computation of homology given in Example 21.38.2. Denote $K_\bullet (\mathcal{F}|_{\mathcal{C}_ V})$ the complex associated to the restriction of $\mathcal{F}$ to the fibre category $\mathcal{C}_ V$. By the remarks above we obtain a presheaf $K_\bullet (\mathcal{F})$ of complexes

\[ V \longmapsto K_\bullet (\mathcal{F}|_{\mathcal{C}_ V}) \]

whose cohomology presheaves are the presheaves $L_{n, p}(\mathcal{F})$. Thus it suffices to show that

\[ K_\bullet (\mathcal{F}) \longrightarrow K_\bullet (\mathcal{F}^+) \]

becomes an isomorphism on sheafification.

Injectivity. Let $V$ be an object of $\mathcal{D}$ and let $\xi \in K_ n(\mathcal{F})(V)$ be an element which maps to zero in $K_ n(\mathcal{F}^+)(V)$. We have to show there exists a covering $\{ V_ j \to V\} $ such that $\xi |_{V_ j}$ is zero in $K_ n(\mathcal{F})(V_ j)$. We write

\[ \xi = \sum (U_{i, n + 1} \to \ldots \to U_{i, 0}, \sigma _ i) \]

with $\sigma _ i \in \mathcal{F}(U_{i, 0})$. We arrange it so that each sequence of morphisms $U_ n \to \ldots \to U_0$ of $\mathcal{C}_ V$ occurs are most once. Since the sums in the definition of the complex $K_\bullet $ are direct sums, the only way this can map to zero in $K_\bullet (\mathcal{F}^+)(V)$ is if all $\sigma _ i$ map to zero in $\mathcal{F}^+(U_{i, 0})$. By construction of $\mathcal{F}^+$ there exist coverings $\{ U_{i, 0, j} \to U_{i, 0}\} $ such that $\sigma _ i|_{U_{i, 0, j}}$ is zero. By our construction of the topology on $\mathcal{C}$ we can write $U_{i, 0, j} \to U_{i, 0}$ as the pullback (Categories, Definition 4.32.6) of some morphisms $V_{i, j} \to V$ and moreover each $\{ V_{i, j} \to V\} $ is a covering. Choose a covering $\{ V_ j \to V\} $ dominating each of the coverings $\{ V_{i, j} \to V\} $. Then it is clear that $\xi |_{V_ j} = 0$.

Surjectivity. Proof omitted. Hint: Argue as in the proof of injectivity. $\square$

Lemma 21.39.2. Assumptions and notation as in Situation 21.37.1. For $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$ and $n \geq 0$ the sheaf $L_ n\pi _!(\mathcal{F})$ is equal to the sheaf $L_ n(\mathcal{F})$ constructed in Lemma 21.39.1.

Proof. Consider the sequence of functors $\mathcal{F} \mapsto L_ n(\mathcal{F})$ from $\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$. Since for each $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ the sequence of functors $H_ n(\mathcal{C}_ V, - )$ forms a $\delta $-functor so do the functors $\mathcal{F} \mapsto L_ n(\mathcal{F})$. Our goal is to show these form a universal $\delta $-functor. In order to do this we construct some abelian presheaves on which these functors vanish.

For $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ consider the abelian presheaf $\mathcal{F}_{U'} = j_{U'!}^{\textit{PAb}}\mathbf{Z}_{U'}$ (Modules on Sites, Remark 18.19.7). Recall that

\[ \mathcal{F}_{U'}(U) = \bigoplus \nolimits _{\mathop{Mor}\nolimits _\mathcal {C}(U, U')} \mathbf{Z} \]

If $U$ lies over $V = p(U)$ in $\mathcal{D})$ and $U'$ lies over $V' = p(U')$ then any morphism $a : U \to U'$ factors uniquely as $U \to h^*U' \to U'$ where $h = p(a) : V \to V'$ (see Categories, Definition 4.32.6). Hence we see that

\[ \mathcal{F}_{U'}|_{\mathcal{C}_ V} = \bigoplus \nolimits _{h \in \mathop{Mor}\nolimits _\mathcal {D}(V, V')} j_{h^*U'!}\mathbf{Z}_{h^*U'} \]

where $j_{h^*U'} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ V/h^*U') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ V)$ is the localization morphism. The sheaves $j_{h^*U'!}\mathbf{Z}_{h^*U'}$ have vanishing higher homology groups (see Example 21.38.2). We conclude that $L_ n(\mathcal{F}_{U'}) = 0$ for all $n > 0$ and all $U'$. It follows that any abelian presheaf $\mathcal{F}$ is a quotient of an abelian presheaf $\mathcal{G}$ with $L_ n(\mathcal{G}) = 0$ for all $n > 0$ (Modules on Sites, Lemma 18.28.7). Since $L_ n(\mathcal{F}) = L_ n(\mathcal{F}^\# )$ we see that the same thing is true for abelian sheaves. Thus the sequence of functors $L_ n(-)$ is a universal delta functor on $\textit{Ab}(\mathcal{C})$ (Homology, Lemma 12.11.4). Since we have agreement with $H^{-n}(L\pi _!(-))$ for $n = 0$ by Lemma 21.37.8 we conclude by uniqueness of universal $\delta $-functors (Homology, Lemma 12.11.5) and Derived Categories, Lemma 13.17.6. $\square$

Lemma 21.39.3. Assumptions and notation as in Situation 21.37.3. For an abelian sheaf $\mathcal{F}'$ on $\mathcal{C}'$ the sheaf $L_ ng_!(\mathcal{F}')$ is the sheaf associated to the presheaf

\[ U \longmapsto H_ n(\mathcal{I}_ U, \mathcal{F}'_ U) \]

For notation and restriction maps see proof.

Proof. Say $p(U) = V$. The category $\mathcal{I}_ U$ is the category of pairs $(U', \varphi )$ where $\varphi : U \to u(U')$ is a morphism of $\mathcal{C}$ with $p(\varphi ) = \text{id}_ V$, i.e., $\varphi $ is a morphism of the fibre category $\mathcal{C}_ V$. Morphisms $(U'_1, \varphi _1) \to (U'_2, \varphi _2)$ are given by morphisms $a : U'_1 \to U'_2$ of the fibre category $\mathcal{C}'_ V$ such that $\varphi _2 = u(a) \circ \varphi _1$. The presheaf $\mathcal{F}'_ U$ sends $(U', \varphi )$ to $\mathcal{F}'(U')$. We will construct the restriction mappings below.

Choose a factorization

\[ \xymatrix{ \mathcal{C}' \ar@<1ex>[r]^{u'} & \mathcal{C}'' \ar[r]^{u''} \ar@<1ex>[l]^ w & \mathcal{C} } \]

of $u$ as in Categories, Lemma 4.32.14. Then $g_! = g''_! \circ g'_!$ and similarly for derived functors. On the other hand, the functor $g'_!$ is exact, see Modules on Sites, Lemma 18.16.6. Thus we get $Lg_!(\mathcal{F}') = Lg''_!(\mathcal{F}'')$ where $\mathcal{F}'' = g'_!\mathcal{F}'$. Note that $\mathcal{F}'' = h^{-1}\mathcal{F}'$ where $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is the morphism of topoi associated to $w$, see Sites, Lemma 7.23.1. The functor $u''$ turns $\mathcal{C}''$ into a fibred category over $\mathcal{C}$, hence Lemma 21.39.2 applies to the computation of $L_ ng''_!$. The result follows as the construction of $\mathcal{C}''$ in the proof of Categories, Lemma 4.32.14 shows that the fibre category $\mathcal{C}''_ U$ is equal to $\mathcal{I}_ U$. Moreover, $h^{-1}\mathcal{F}'|_{\mathcal{C}''_ U}$ is given by the rule described above (as $w$ is continuous and cocontinuous by Stacks, Lemma 8.10.3 so we may apply Sites, Lemma 7.21.5). $\square$


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 08P7. Beware of the difference between the letter 'O' and the digit '0'.