# stacks/stacks-project

Make script happier

Please ignore this
aisejohan committed Jul 22, 2019
1 parent f4540e2 commit 272a396cc9028047bcedf18c8ce5ee0d7a7beefb
Showing with 4 additions and 9 deletions.
1. +1 −1 divisors.tex
2. +1 −6 intersection.tex
3. +1 −1 perfect.tex
4. +1 −1 tags/tags
 @@ -3332,7 +3332,7 @@ \section{Complements of affine opens} \end{proof} \begin{lemma} \label{lemma-generalization-of-complement-open-affine-effective-cartier-divisor} \label{lemma-complement-open-affine-effective-cartier-divisor-bis} Let $X$ be a Noetherian scheme with affine diagonal. Let $U \subset X$ be a dense affine open. If $\mathcal{O}_{X, x}$ is a UFD for all $x \in X \setminus U$, then there exists an effective Cartier
 @@ -2080,12 +2080,7 @@ \section{Projection formula for flat proper morphisms} Thus $V \to Y$ is finite in an open neighbourhood of $\xi$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood}. Using a very general projection formula\footnote{This can be avoided by working in an affine neighbourhood of $\xi$ as above, choosing an affine open of $X$ containing the generic points of the $Z_i$, and translating the question into algebra. Doing this will produce a relatively elementary proof of (\ref{equation-stalks}).} for derived tensor products, we get Using a very general projection formula for derived tensor products, we get  Rf_*(\mathcal{O}_V \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{O}_W) = Rf_*\mathcal{O}_V \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{O}_W
 @@ -7354,7 +7354,7 @@ \section{The resolution property} We may and do assume $U_j$ nonempty for all $j$. By More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD} the local rings of $X$ are UFDs and hence by Divisors, Lemma \ref{divisors-lemma-generalization-of-complement-open-affine-effective-cartier-divisor} \ref{divisors-lemma-complement-open-affine-effective-cartier-divisor-bis} we can find an effective Cartier divisors $D_j \subset X$ whose complement is $U_j$. Then the ideal sheaf of $D_j$ is invertible, hence a finite locally free module and we conclude
 0EGG,algebra-example-locally-nilpotent-not-nilpotent 0EGH,algebra-lemma-uple-generated-degree-1 0EGI,algebra-lemma-quotient-module-ML 0EGJ,divisors-lemma-generalization-of-complement-open-affine-effective-cartier-divisor 0EGJ,divisors-lemma-complement-open-affine-effective-cartier-divisor-bis 0EGK,varieties-remark-exact-sequence-induction-cohomology 0EGL,dualizing-remark-matlis 0EGM,obsolete-section-sites