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Make script happier

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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

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