# stacks/stacks-project

Improve section on alternative Weil cohomology

Added some lemmas on zero dimensional schemes.
aisejohan committed Dec 2, 2019
1 parent 8a49bd8 commit 194d6c9acd9c7dc222d191a81765050ea885ae1a
Showing with 285 additions and 39 deletions.
1. +18 −0 chow.tex
2. +55 −0 morphisms.tex
3. +16 −0 varieties.tex
4. +196 −39 weil.tex
 @@ -7843,6 +7843,24 @@ \section{The splitting principle} $\pi_i \circ \pi : P \to X$ does the job. Some details omitted. \end{proof} \begin{remark} \label{remark-the-proof-shows-more} The proof of Lemma \ref{lemma-splitting-principle} shows that the morphism $\pi : P \to X$ has the following additional properties: \begin{enumerate} \item $\pi$ is a finite composition of projective space bundles associated to locally free modules of finite constant rank, and \item for every $\alpha \in \CH_k(X)$ we have $\alpha = \pi_*(\xi_1 \cap \ldots \cap \xi_d \cap \pi^*\alpha)$ where $\xi_i$ is the first chern class of some invertible $\mathcal{O}_P$-module. \end{enumerate} The second observation follows from the first and Lemma \ref{lemma-cap-projective-bundle}. We will add more observations here as needed. \end{remark} \noindent Let $(S, \delta)$, $X$, and $\mathcal{E}_i$ be as in Lemma \ref{lemma-splitting-principle}.
 @@ -9030,6 +9030,61 @@ \section{Ample and very ample sheaves relative to finite type morphisms} Lemma \ref{lemma-characterize-relatively-ample}. \end{proof} \begin{lemma} \label{lemma-invertible-add-enough-ample-very-ample} Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{N}$, $\mathcal{L}$ be invertible $\mathcal{O}_X$-modules. Assume $S$ is quasi-compact, $f$ is of finite type, and $\mathcal{L}$ is $f$-ample. Then $\mathcal{N} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$. \end{lemma} \begin{proof} By Lemma \ref{lemma-characterize-very-ample-on-finite-type} we reduce to the case $S$ is affine. Combining Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} and Properties, Proposition \ref{properties-proposition-characterize-ample} we can find an integer $d_0$ such that $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ is globally generated. Choose global sections $s_0, \ldots, s_n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ which generate it. This determines a morphism $j : X \to \mathbf{P}^n_S$ over $S$. By Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} we can also pick an integer $d_1$ such that for all $d \geq d_1$ there exist sections $t_{d, 0}, \ldots, t_{d, n(d)}$ of $\mathcal{L}^{\otimes d}$ which generate it and define an immersion $$j_d = \varphi_{\mathcal{L}^{\otimes d}, t_{d, 0}, \ldots, t_{d, n(d)}} : X \longrightarrow \mathbf{P}^{n(d)}_S$$ over $S$. Then for $d \geq d_0 + d_1$ we can consider the morphism $$\varphi_{\mathcal{N} \otimes \mathcal{L}^{\otimes d}, s_j \otimes t_{d - d_0, i}} : X \longrightarrow \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S$$ This morphism is an immersion as it is the composition $$S \to \mathbf{P}^n_S \times_S \mathbf{P}^{n(d - d_0)}_S \to \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S$$ where the first morphism is $(j, j_{d - d_0})$ and the second is the Segre embedding (Constructions, Lemma \ref{constructions-lemma-segre-embedding}). Since $j$ is an immersion, so is $(j, j_{d - d_0})$ (apply Lemma \ref{lemma-immersion-permanence}). We have a composition of immersions and hence an immersion (Schemes, Lemma \ref{schemes-lemma-composition-immersion}). \end{proof}
 @@ -10644,6 +10644,22 @@ \section{Bertini theorems} $r - \dim \mathfrak m_x/\mathfrak m_x^2 - 1$ as desired. \end{proof} \section{Enriques-Severi-Zariski} \label{section-vanishing-negative} \noindent In this section we prove some results of the form: twisting by a very negative'' invertible module kills low degree cohomology. We also deduce the connectedness of a hypersurface section of a normal proper scheme of dimension $\geq 2$. \begin{lemma} \label{lemma-vanishin-h0-negative} Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$