stacks/stacks-project

Spectral sequence for a smooth morphism

aisejohan committed Sep 26, 2019
1 parent e0c8b04 commit cff9d55ebaf6d09612db75525b617a2c5095d271
Showing with 295 additions and 36 deletions.
1. +65 −0 cohomology.tex
2. +208 −1 derham.tex
3. +1 −1 injectives.tex
4. +21 −34 perfect.tex
 @@ -6341,6 +6341,22 @@ \section{Cohomology of unbounded complexes} coming from the counit $Lf^* \circ Rf_* \to \text{id}$. \end{remark} \section{Cohomology of filtered complexes} \label{section-cohomology-filtered-object} \noindent Filtered complexes of sheaves frequently come up in a natural fashion when studying cohomology of algebraic varieties, for example the de Rham complex comes with its Hodge filtration. In this sectionwe use the very general Injectives, Lemma \ref{injectives-lemma-K-injective-embedding-filtration} to find construct spectral sequences on cohomology and we relate these to previously constructed spectral sequences. \begin{lemma} \label{lemma-spectral-sequence-filtered-object} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a @@ -6444,6 +6460,55 @@ \section{Cohomology of unbounded complexes} (constructed using Cartan-Eilenberg resolutions). \end{example} \begin{example} \label{example-spectral-sequence-bis} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}_X$-modules. We can apply Lemma \ref{lemma-spectral-sequence-filtered-object} with $F^p\mathcal{F}^\bullet = \sigma_{\geq p}\mathcal{F}^\bullet$. (If $\mathcal{F}^\bullet$ is bounded below we can use Remark \ref{remark-spectral-sequence-filtered-object}.) Then we get a spectral sequence $$E_1^{p, q} = H^{p + q}(X, \mathcal{F}^p[-p]) = H^q(X, \mathcal{F}^p)$$ If $\mathcal{F}^\bullet$ is bounded below, then \begin{enumerate} \item we can use Remark \ref{remark-spectral-sequence-filtered-object} to construct this spectral sequence, \item the spectral sequence is bounded and converges to $H^{i + j}(X, \mathcal{F}^\bullet)$, and \item the spectral sequence is equal to the first spectral sequence of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor} (constructed using Cartan-Eilenberg resolutions). \end{enumerate} \end{example} \begin{lemma} \label{lemma-relative-spectral-sequence-filtered-object} Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet$ be a filtered complex of $\mathcal{O}_X$-modules. There exists a canonical spectral sequence $(E_r, \text{d}_r)_{r \geq 1}$ of bigraded $\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and $$E_1^{p, q} = R^{p + q}f_*\text{gr}^p\mathcal{F}^\bullet$$ If for every $n$ we have $$R^nf_*F^p\mathcal{F}^\bullet = 0 \text{ for }p \gg 0 \quad\text{and}\quad R^nf_*F^p\mathcal{F}^\bullet = R^nf_*\mathcal{F}^\bullet \text{ for }p \ll 0$$ then the spectral sequence is bounded and converges to $Rf_*\mathcal{F}^\bullet$. \end{lemma} \begin{proof} The proof is exactly the same as the proof of Lemma \ref{lemma-spectral-sequence-filtered-object}. \end{proof}
 @@ -172,6 +172,52 @@ \section{de Rham cohomology} This is a special case of Lemma \ref{lemma-coherence-relative}. \end{proof} \begin{lemma} \label{lemma-proper-smooth-de-Rham} Let $f : X \to S$ be a proper smooth morphism of schemes. Then $Rf_*\Omega^p_{X/S}$, $p \geq 0$ and $Rf_*\Omega^\bullet_{X/S}$ are perfect objects of $D(\mathcal{O}_S)$ whose formation commutes with arbitrary change of base. \end{lemma} \begin{proof} Since $f$ is smooth the modules $\Omega^p_{X/S}$ are finite locally free $\mathcal{O}_X$-modules, see Morphisms, Lemma \ref{morphisms-lemma-smooth-omega-finite-locally-free}. Their formation commutes with arbitrary change of base by Lemma \ref{lemma-base-change-de-rham}. Hence $Rf_*\Omega^p_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$ whose formation commutes with abitrary base change, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. This proves the first assertion of the lemma. \medskip\noindent To prove that $Rf_*\Omega^\bullet_{X/S}$ is perfect on $S$ we may work locally on $S$. Thus we may assume $S$ is quasi-compact. This means we may assume that $\Omega^n_{X/S}$ is zero for $n$ large enough. For every $p \geq 0$ we claim that $Rf_*\sigma_{\geq p}\Omega^\bullet_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$ whose formation commutes with arbitrary change of base. By the above we see that this is true for $p \gg 0$. Suppose the claim holds for $p$ and consider the distinguished triangle $$\sigma_{\geq p}\Omega^\bullet_{X/S} \to \sigma_{\geq p - 1}\Omega^\bullet_{X/S} \to \Omega^{p - 1}_{X/S}[-(p - 1)] \to (\sigma_{\geq p}\Omega^\bullet_{X/S})$$ in $D(f^{-1}\mathcal{O}_S)$. Applying the exact functor $Rf_*$ we obtain a distinguished triangle in $D(\mathcal{O}_S)$. Since we have the 2-out-of-3 property for being perfect (Cohomology, Lemma \ref{cohomology-lemma-two-out-of-three-perfect}) we conclude $Rf_*\sigma_{\geq p - 1}\Omega^\bullet_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$. Similarly for the commutation with arbitrary base change. \end{proof} @@ -513,7 +559,7 @@ \section{K\"unneth formula} \end{lemma} \begin{proof} By Derived Categories of Schemes, Equation (\ref{perfect-equation-silly}) By Derived Categories of Schemes, Remark \ref{perfect-remark-silly} we have $$p^{-1}\Omega^i_{X/S} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\Omega^j_{Y/S} = @@ -1053,6 +1099,167 @@ \section{de Rham cohomology of projective space} \section{The spectral sequence for a smooth morphism} \label{section-relative-spectral-sequence} \noindent Let S be a scheme. Let f : X \to Y be a quasi-compact and quasi-separated smooth morphism of schemes over S. Then we obtain a locally split short exact sequence$$ 0 \to f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0 $$by Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials-smooth}. Let us think of this as a descending filtration on \Omega_{X/S}. Then for every p we obtain an induced filtration on \Omega^p_{X/S} whose succesive quotients are$$ f^*\Omega^k_{Y/S} \otimes_{\mathcal{O}_X} \Omega^{p - k}_{X/Y} = f^{-1}\Omega^k_{Y/S} \otimes_{f^{-1}\mathcal{O}_Y} \Omega^{p - k}_{X/Y} $$for k = 0, \ldots, p. In fact, these filtrations give \Omega^\bullet_{X/S} the structure of a filtered complex. We can also directly define the filtration by setting$$ F^k\Omega^\bullet_{X/S} = \Im\left(\wedge : \text{Tot}( f^{-1}\sigma_{\geq k}\Omega^\bullet_{Y/S} \otimes_{f^{-1}\mathcal{O}_Y} \Omega^\bullet_{X/S}) \longrightarrow \Omega^\bullet_{X/S}\right) $$as complexes on X. Thus we have$$ \Omega^\bullet_{X/S} = F^0\Omega^\bullet_{X/S} \supset F^1\Omega^\bullet_{X/S} \supset F^2\Omega^\bullet_{X/S} \supset \ldots $$and we have the identification of complexes$$ \text{gr}^k\Omega^\bullet_{X/S} = f^{-1}\Omega^k_{Y/S}[-k] \otimes_{f^{-1}\mathcal{O}_Y} \Omega^\bullet_{X/Y} $$by the discussion above. Let us study the spectral sequence associated to this filtered complex of Cohomology, Lemma \ref{cohomology-lemma-relative-spectral-sequence-filtered-object}. By Lemma \ref{lemma-cohomology-de-rham-base-change} we have$$ Rf_*\text{gr}^k\Omega^\bullet_{X/S} = \Omega^k_{Y/S}[-k] \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y} $$All in all we conclude that there is a spectral sequence$$ E_1^{p, q} = H^q(\Omega^p_{Y/S} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y}) $$converging to R^{p + q}f_*\Omega^\bullet_{X/S}. \begin{lemma} \label{lemma-cohomology-de-rham-base-change} Let f : X \to Y be a quasi-compact and quasi-separated morphism of schemes. Let \mathcal{F}^\bullet be a locally bounded complex of f^{-1}\mathcal{O}_Y-modules. Assume for all n \in \mathbf{Z} the sheaf \mathcal{F}^n is a flat f^{-1}\mathcal{O}_Y-module and \mathcal{F}^n has the structure of a quasi-coherent \mathcal{O}_X-module compatible with the given p^{-1}\mathcal{O}_Y-module structure (but the differentials in the complex \mathcal{F}^\bullet need not be \mathcal{O}_X-linear). Then$$ \mathcal{G} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\mathcal{F}^\bullet = Rf_*(f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{F}^\bullet) $$for any quasi-coherent \mathcal{O}_Y-module \mathcal{G}. \end{lemma} \begin{proof} This lemma is a variant of Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} and we urge the reader to read the proof of that lemma first. Denote f' : (X, f^{-1}\mathcal{O}_Y) \to (Y, \mathcal{O}_Y) the obvious flat morphism of ringed spaces. We view \mathcal{F}^\bullet as a complex of modules on the ringed space (X, f^{-1}\mathcal{O}_Y). Since \mathcal{F}^\bullet is a locally bounded complex of flat f^{-1}\mathcal{O}_Y-modules we see that the complex f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{F}^\bullet represents L(f')^*\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet in D(f^{-1}\mathcal{O}_Y). Hence the statement of the lemma is really that$$ \mathcal{G} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet = Rf'_*(L(f')^*\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet) $$Formulated in this manner we can generalize the statement to the statement that the canonical, functorial arrow Cohomology, Equation (\ref{cohomology-equation-projection-formula-map})$$ G \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet \longrightarrow Rf'_*(L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet) $$is an isomorphism for all G in D_\QCoh(\mathcal{O}_Y). Formulated in this manner the problem is local on Y and we may assume Y is affine. Moreover, the source and target of the arrow are exact functors D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_Y) of triangulated categories. Hence if G_1 \to G_2 \to G_3 \to G_1 is a distinguished triangle in D_\QCoh(\mathcal{O}_Y) and the result holds for two out of G_1, G_2, G_3, then the result holds for the third. Finally, both sides of the arrow commute with arbitrary direct sums (see below for the right hand side). Thus, exactly as in the proof of Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} it suffices to prove that$$ \mathcal{O}_Y \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet \longrightarrow Rf'_*(L(f')^*\mathcal{O}_Y \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet) $$which is obvious. \medskip\noindent We will have to show that G \mapsto Rf'_*(L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet) commutes with direct sums on D_\QCoh(\mathcal{O}_Y). This is where we will use \mathcal{F}^n has the structure of a quasi-coherent \mathcal{O}_X-module. First, observe that G \mapsto L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet commutes with arbitrary direct sums. Next, if \mathcal{F}^\bullet consists of a single quasi-coherent \mathcal{O}_X-module \mathcal{F}^\bullet = \mathcal{F}^n[-n] then we have L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet = Lf^*G \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}^n[-n], see Cohomology, Lemma \ref{cohomology-lemma-variant-derived-pullback}. Hence in this case the commutation with direct sums follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-pushforward-direct-sums}. Now, in general, since Y is affine and \mathcal{F}^\bullet is locally bounded, we see that$$ \mathcal{F}^\bullet = (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) $$is bounded. Arguing by induction on b - a and considering the distinguished triangle$$ \mathcal{F}^b[-b] \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^b[-b + 1] $$the proof is finished. Some details omitted. \end{proof} \section{Comparing sheaves of differential forms} \label{section-quasi-finite-syntomic}  @@ -2147,7 +2147,7 @@ \section{Additional remarks on Grothendieck abelian categories}$$ such that $\alpha^p \circ \alpha^{p' p} = \alpha^{p'}$ and $\alpha^{p'p''} \circ \alpha^{pp'} = \alpha^{pp''}$. The problem is that the maps $\alpha^P : I^{p, \bullet} \to I^\bullet$ The problem is that the maps $\alpha^p : I^{p, \bullet} \to I^\bullet$ need not be injective. For each $p$ we choose an injection $t^p : I^{p, \bullet} \to J^{p, \bullet}$ into an acyclic K-injective complex $J^{p, \bullet}$ whose terms are injective objects of $\mathcal{A}$
 @@ -4650,46 +4650,35 @@ \section{K\"unneth formula} and the cup product constructed in Cohomology, Section \ref{cohomology-section-cup-product}. \medskip\noindent This cup product has an application to the de Rham cohomology. In this case (as in the next lemma) the sheaves $\mathcal{F}^n$ have the structure of quasi-coherent $\mathcal{O}_X$-modules compatible with the given $p^{-1}\mathcal{O}_S$-module structures (but the differentials in the complex $\mathcal{F}^\bullet$ need not be $\mathcal{O}_X$-linear) and the sheaves $\mathcal{G}^m$ have the structure of a quasi-coherent $\mathcal{O}_Y$-modules compatible with the given $q^{-1}\mathcal{O}_S$-module structures (but the differentials in the complex $\mathcal{G}^\bullet$ need not be $\mathcal{O}_Y$-linear). If so then we have \begin{equation} \label{equation-silly} \begin{aligned} & p^{-1}\mathcal{F}^n \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^m \\ \begin{remark} \label{remark-silly} In the situation above suppose we have an $\mathcal{O}_X$-module $\mathcal{F}$ and an $\mathcal{O}_Y$-module $\mathcal{G}$. Then we have \begin{align*} & p^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G} \\ & = p^{-1}(\mathcal{F}^n \otimes_{\mathcal{O}_X} \mathcal{O}_X) p^{-1}(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X) \otimes_{f^{-1}\mathcal{O}_S} q^{-1}(\mathcal{O}_Y \otimes_{\mathcal{O}_Y} \mathcal{G}^m) \\ q^{-1}(\mathcal{O}_Y \otimes_{\mathcal{O}_Y} \mathcal{G}) \\ & = p^{-1}\mathcal{F}^n \otimes_{p^{-1}\mathcal{O}_X} p^{-1}\mathcal{O}_X p^{-1}\mathcal{F} \otimes_{p^{-1}\mathcal{O}_X} p^{-1}\mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{O}_Y \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\ q^{-1}\mathcal{O}_Y \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F}^n \otimes_{q^{-1}\mathcal{O}_X} p^{-1}\mathcal{F} \otimes_{q^{-1}\mathcal{O}_X} \mathcal{O}_{X \times_S Y} \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\ \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F}^n \otimes_{q^{-1}\mathcal{O}_X} p^{-1}\mathcal{F} \otimes_{q^{-1}\mathcal{O}_X} \mathcal{O}_{X \times_S Y} \otimes_{\mathcal{O}_{X \times_S Y}} \mathcal{O}_{X \times_S Y} \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\ \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\ & = p^*\mathcal{F}^n \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}^m \end{aligned} \end{equation} The target of the cup product (\ref{equation-de-rham-kunneth}) is $R\Gamma(X \times_S Y, \text{Tot}(p^*\mathcal{F}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}^\bullet))$ in this case. p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G} \end{align*} \end{remark} \begin{lemma} \label{lemma-kunneth-special} @@ -4906,13 +4895,11 @@ \section{K\"unneth formula} \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}))$represents the derived tensor product$R\Gamma(X, \mathcal{F}^\bullet) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet)$as claimed. Finally, we have$R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G})$as claimed. Finally, we have$p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G} = p^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}\$ by (\ref{equation-silly}). by Remark \ref{remark-silly}. \medskip\noindent We still have to show that (\ref{equation-kunneth-on-cech})