# stacks/stacks-project

Complex of forms with log poles along a divisor

Another one bites the dust!
 @@ -81,14 +81,38 @@ \section{The de Rham complex} $$be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms f^*\Omega^p_{Y/X} \to \Omega^p_{Y'/X'}. f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}. \end{lemma} \begin{proof} Combine Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials} with the fact that formation of exterior power commutes with base change. \end{proof} \begin{lemma} \label{lemma-etale} Consider a commutative diagram of schemss$$ \xymatrix{ X' \ar[r]_f \ar[d] & X \ar[d] \\ S' \ar[r] & S } $$If X' \to X and S' \to S are \'etale, then the maps discussed above induce isomorphisms f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}. \end{lemma} \begin{proof} We have \Omega_{S'/S} = 0 and \Omega_{X'/X} = 0, see for example Morphisms, Lemma \ref{morphisms-lemma-etale-at-point}. Then by the short exact sequences of Morphisms, Lemmas \ref{morphisms-lemma-triangle-differentials} and \ref{morphisms-lemma-triangle-differentials-smooth} we see that \Omega_{X'/S'} = \Omega_{X'/S} = f^*\Omega_{X/S}. Taking exterior powers we conclude. \end{proof} @@ -1499,7 +1523,9 @@ \section{The spectral sequence for a smooth morphism} \item K^\bullet is a complex of R-modules whose terms are A-modules, \item K^\bullet represents R\Gamma(X, \Omega^\bullet_{X/S}) in D(R) (Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes}), (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes}), \item there is a natural map \Omega^\bullet_{A/R} \to K^\bullet of complexes of R-modules which is A-linear on terms and induces the pullback map H^*_{dR}(Y/S) \to H^*_{dR}(X/S) @@ -1637,6 +1663,327 @@ \section{Projective space bundle formula} \section{Log poles along a divisor} \label{section-divisor} \noindent Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. If X \'etale locally along Y looks like Y \times \mathbf{A}^1, then there is a canonical short exact sequence of complexes$$ 0 \to \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}(\log Y) \to \Omega^\bullet_{Y/S}[-1] \to 0 $$having many good properties we will discuss in this section. \begin{definition} \label{definition-local-product} Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. We say the {\it de Rham complex of log poles is defined for Y \subset X over S} if for all y \in Y and local equation f \in \mathcal{O}_{X, y} of Y we have \begin{enumerate} \item \mathcal{O}_{X, x} \to \Omega_{X/S, x}, g \mapsto g \text{d}f is a split injection, and \item \Omega^p_{X/S, y} is f-torsion free for all p. \end{enumerate} \end{definition} \noindent An easy local calculation shows that it suffices to find one local equation f for which condition (1) holds. \begin{lemma} \label{lemma-log-complex} Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y \subset X over S. There is a canonical short exact sequence of complexes$$ 0 \to \Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}(\log Y) \to \Omega^\bullet_{Y/S}[-1] \to 0 $$\end{lemma} \begin{proof} Our assumption is that for every y \in Y and local equation f \in \mathcal{O}_{X, y} of Y we have$$ \Omega_{X/S, y} = \mathcal{O}_{X, y}\text{d}f \oplus M \quad\text{and}\quad \Omega^p_{X/S, y} = \wedge^{p - 1}(M)\text{d}f \oplus \wedge^p(M) $$for some module M with f-torsion free exterior powers \wedge^p(M). It follows that$$ \Omega^p_{Y/S, y} = \wedge^p(M/fM) = \wedge^p(M)/f\wedge^p(M) $$Below we will tacitly use these facts. In particular the sheaves \Omega^p_{X/S} have no nonzero torsion sections supported on Y and we have a canonical inclusion$$ \Omega^p_{X/S} \subset \Omega^p_{X/S}(Y) $$see More on Flatness, Section \ref{flat-section-eta}. Let U = \Spec(A) be an affine open subscheme such that Y \cap U = V(f) for some nonzerodivisor f \in A. Let us consider the \mathcal{O}_U-submodule of \Omega^p_{X/S}(Y)|_U generated by \Omega^p_{X/S}|_U and \text{d}\log(f) \wedge \Omega^{p - 1}_{X/S} where \text{d}\log(f) = f^{-1}\text{d}(f). This is independent of the choice of f as another generator of the ideal of Y on U is equal to uf for a unit u \in A and we get$$ \text{d}\log(uf) - \text{d}\log(f) = \text{d}\log(u) = u^{-1}\text{d}u $$which is a section of \Omega_{X/S} over U. Obviously, these local sheaves glue to give a quasi-coherent submodule$$ \Omega^p_{X/S} \subset \Omega^p_{X/S}(\log Y) \subset \Omega^p_{X/S}(Y) $$Let us agree to think of \Omega^p_{Y/S} as a quasi-coherent \mathcal{O}_X-module. There is a unique surjective \mathcal{O}_X-linear map$$ \Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_Y $$which over U sends a local section \text{d}\log(f) \wedge \eta to the image \eta in \Omega^{p - 1}_{Y/S} and annihilates the submodule \Omega^p_{X/S}. If a form \eta over U restricts to zero on \Omega_{Y/S}, then \eta = \text{d}f \wedge \eta' + f\eta'' for some forms \eta' and \eta'' over U. We conclude that we have a short exact sequence$$ 0 \to \Omega^p_{X/S} \to \Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_{Y/S} \to 0 $$for all p. \medskip\noindent We still have to define the differentials \Omega^p_{X/S}(\log Y) \to \Omega^{p + 1}_{X/S}(\log Y). On the subsheaf \Omega^p_{X/S} we use the differential of the de Rham complex of X over S. Finally, we define \text{d}(\text{d}\log(f) \wedge \eta) = -\text{d}\log(f) \wedge \text{d}\eta. It is a pleasant exercise to show that we obtain a short exact sequence of complexes as stated in the lemma. \end{proof} \begin{lemma} \label{lemma-log-complex-consequence} Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. Assume \begin{enumerate} \item the diagonal of X is affine, and \item the de Rham complex of log poles is defined for Y \subset X over S. \end{enumerate} Let b \in H^m_{dR}(X/S) be a de Rham cohomology class whose restriction to Y is zero. Then c_1^{dR}(\mathcal{O}_X(Y)) \cup b = 0 in H^{m + 2}_{dR}(X/S). \end{lemma} \noindent This lemma is true without the assumption on the diagonal of X. It can be proven by replacing open coverings in the proof below by hypercoverings. \begin{proof} The short exact sequence of complexes of Lemma \ref{lemma-log-complex} gives a boundary map$$ H^m_{dR}(Y/S) = H^{m + 1}(X, \Omega^\bullet_{Y/S}[-1]) \to H^{m + 2}_{dR}(X, \Omega^\bullet_{X/S}) = H^{m + 2}_{dR}(X/S) $$We claim that b \cup c_1^{dR}(\mathcal{O}_X(-Y)) is equal to the boundary of the restriction of b to Y. The claim proves the lemma. \medskip\noindent To prove this, choose an affine open covering \mathcal{U} : X = \bigcup_{i \in I} U_i such that Y \cap U_i is the vanishing scheme of a nonzerodivisor f_i \in \mathcal{O}_X(U_i). Then U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p} is affine for all i_0, \ldots, i_p \in I by our assumption on the diagonal of X. Now the short exact sequence of complexes of Lemma \ref{lemma-log-complex} is termwise given by a short exact sequence of quasi-coherent modules. Hence taking sections over any affine open produces a short exact sequence. Thus we obtain a short exact sequence$$ 0 \to \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/S}^\bullet)) \to \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/S}^\bullet(\log Y)) \to \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1])) \to 0 $$of {\v C}ech complexes. By Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes} the associated long exact sequence of cohomology groups is the long exact sequence of cohomology associated to the short exact sequence of complexes in Lemma \ref{lemma-log-complex} (small detail omitted). \medskip\noindent Using notation as in Cohomology, Section \ref{cohomology-section-cech-cohomology-of-complexes}. let \beta = \{\beta_{i_0 \ldots i_p}\} be a cocycle representing b which has degree m. Let us compute the de Rham chern class of the invertible \mathcal{O}_X-module \mathcal{O}_X(-Y). On U_i the section f_i is a trivialization. On U_{i_0i_1} the regular functions f_{i_0} and f_{i_1} differ by a unit f_{i_0i_1}, like so$$ f_{1_1}|_{U_{i_0i_1}} = f_{i_0i_1} \cdot f_{i_0}|_{U_{i_0i_1}} $$By the discussion in Section \ref{section-first-chern-class} the de Rham cohomology class c_1^{dR}(\mathcal{O}_X(-Y)) is the class of the cocycle \alpha = \{\alpha_{i_0 \ldots i_p}\} with \alpha_{i_0i_1} = f_{i_0i_1}^{-1} \text{d}f_{i_0i_1} and zero in other degrees. The cup product is given by$$ (\alpha \cup \beta)_{i_0 \ldots i_p} = \sum\nolimits_{r = 0}^p \epsilon(2, m, p, r) \alpha_{i_0 \ldots i_r} \wedge \beta_{i_r \ldots i_p} = (-1)^{p + 1} \alpha_{i_0i_1} \wedge \beta_{i_1 \ldots i_p} $$because \epsilon(n, m, p, r) is -1 to the power (p + r)n + rp + r. \medskip\noindent On the other hand, the restriction of b to Y is represented by the cocycle \beta|_Y = \{\beta_{i_0 \ldots i_p}|_{U_{i_0 \ldots i_p} \cap Y}\} of the complex \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet)). This means that the cocycle in$$ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1])) $$corresponding to \beta|_Y has is given by \{(-1)^p\beta_{i_0 \ldots i_p}|_{U_{i_0 \ldots i_p} \cap Y}\} by the commutative diagram in Cohomology, Remark \ref{cohomology-remark-shift-complex-cech-complex} and the construction of the map$$ \gamma : \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet))[-1] \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1])) $$in Homology, Remark \ref{homology-remark-shift-double-complex}. By our construction of the complex with log poles in the proof of Lemma \ref{lemma-log-complex} this cocycle lifts to the cycle$$ \gamma = \{(-1)^p\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots i_p}\} in the complex \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/S}^\bullet(\log Y)). We have \begin{align*} \text{d}(\gamma)_{i_0 \ldots i_p} = & (-1)^{p - 1} \text{d}\log(f_{i_1}) \wedge \beta_{i_1 \ldots i_p} \\ & + \sum\nolimits_{j = 1, \ldots, p} (-1)^{j + p - 1} \text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots \hat i_j \ldots i_p} \\ & + (-1)^{p + p} \text{d}(\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots i_p}) \\ = & (-1)^{p - 1} (\text{d}\log(f_{i_1} - \text{d}\log(f_{i_0})) \wedge \beta_{i_1 \ldots i_p} \\ & + \sum\nolimits_{j = 0, \ldots, p} (-1)^{j + p - 1} \text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots \hat i_j \ldots i_p} \\ & - \text{d}\log(f_{i_0}) \wedge \text{d}(\beta_{i_0 \ldots i_p}) \\ = & (-1)^{p - 1} f_{i_0i_1}^{-1} \text{d}f_{i_0i_1} \wedge \beta_{i_1 \ldots i_p} \end{align*} The last equality because the remaining terms sum up to (-1)^{p - 1}\text{d}\log(f_{i_0}) \wedge d(\beta)_{i_0 \ldots i_p} which is zero as \beta is a cocycle. This is the same result we got above and the proof is complete. \end{proof} \begin{lemma} \label{lemma-check-log-smooth} Let X \to S be a morphism of schemes. Let Y \subset X be an effective Cartier divisor. If both X \to S and Y \to S are smooth, then the de Rham complex of log poles is defined for Y \subset X over S. \end{lemma} \begin{proof} In this case the modules \Omega^p_{X/S} are locally free, see Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}, and hence the second condition of Definition \ref{definition-local-product} holds. On the other hand, for s \in S the fibre Y_s \subset X_s is an effective Cartier divisor (Divisors, Lemma \ref{divisors-lemma-relative-Cartier}). Hence if y \in Y maps to s \in S we have \dim \mathcal{O}_{Y_s, y} = \dim \mathcal{O}_{X_s, y} - 1 for example by Algebra, Lemma \ref{algebra-lemma-one-equation}. Thus \dim_y(Y_s) = \dim_y(X_s) - 1 by Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}. Now \Omega_{X/S, x} is free of rank \dim_y(X_s) and \Omega_{Y/S, y} is free of rank \dim_y(Y_s) = \dim_y(X_s) - 1 by the already used Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}. Since \Omega_{Y/S, y} is the quotient of \Omega_{X/S, x} by the submodule f\Omega_{X/S, x} and \mathcal{O}_{X, x}\text{d}f we conclude that \text{d}f must map to a nonzero element of \Omega_{X/S, x} \otimes \kappa(y). Hence \text{d}f generates a direct summand and the proof is complete. \end{proof} \begin{remark} \label{remark-check-log-completion-1} Let S be a locally Noetherian scheme. Let X be locally of finite type over S. Let Y \subset X be an effective Cartier divisor. If the map \mathcal{O}_{X, y}^\wedge \longrightarrow \mathcal{O}_{Y, y}^\wedge $$has a section for all y \in Y, then the de Rham complex of log poles is defined for Y \subset X over S. If we ever need this result we will formulate a precise statement and add a proof here. \end{remark} \begin{remark} \label{remark-check-log-completion-2} Let S be a locally Noetherian scheme. Let X be locally of finite type over S. Let Y \subset X be an effective Cartier divisor. If for every y \in Y we can find a diagram of schemes over S$$ X \xleftarrow{\varphi} U \xrightarrow{\psi} V  with $\varphi$ \'etale and $\psi|_{\varphi^{-1}(Y)} : \varphi^{-1}(Y) \to V$ \'etale, then the de Rham complex of log poles is defined for $Y \subset X$ over $S$. A special case is when the pair $(X, Y)$ \'etale locally looks like $(V \times \mathbf{A}^1, V \times \{0\})$. If we ever need this result we will formulate a precise statement and add a proof here. \end{remark} \section{Comparing sheaves of differential forms} \label{section-quasi-finite-syntomic}