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Fix some formatting internal references

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aisejohan committed Nov 8, 2019
1 parent 86638d2 commit 3117ef43fcf96d9a409db0b7826cc6a6059dc456
Showing with 9 additions and 6 deletions.
  1. +2 −1 cohomology.tex
  2. +3 −3 dga.tex
  3. +2 −1 sdga.tex
  4. +2 −1 sites-cohomology.tex
@@ -10588,7 +10588,8 @@ \section{Duals}
the category of complexes and of the derived category.
In particular, we will see that an object of $D(\mathcal{O}_X)$
has a dual if and only if it is perfect (this follows from
Example \ref{example-dual-derived} and Lemma \ref{lemma-left-dual-derived}).
Example \ref{example-dual-derived} and
Lemma \ref{lemma-left-dual-derived}).

\begin{lemma}
\label{lemma-symmetric-monoidal-cat-complexes}
@@ -1517,7 +1517,7 @@ \section{Hom complexes and differential graded modules}
compatible with differentials. We have seen this is true for the right
hand side in Section \ref{section-tensor-product}. Given an element
$g$ of the left hand side, the equality of
More Algebra, Lemma \ref{more-algebra-lemma-compose}
More on Algebra, Lemma \ref{more-algebra-lemma-compose}
determines a map of complexes of $R$-modules
$g' : \text{Tot}(M' \otimes_R M) \to N^\bullet$.
In other words, we obtain a graded $R$-bilinear
@@ -1641,7 +1641,7 @@ \section{Hom complexes and differential graded modules}
compatible with differentials. We have seen this is true for the right
hand side in Section \ref{section-tensor-product}. Given an element
$g$ of the left hand side, the equality of
More Algebra, Lemma \ref{more-algebra-lemma-compose}
More on Algebra, Lemma \ref{more-algebra-lemma-compose}
determines a map of complexes
$g' : \text{Tot}(M' \otimes_R M) \to N^\bullet$.
We precompose with the commutativity constraint to get
@@ -1746,7 +1746,7 @@ \section{Hom complexes and differential graded modules}
We omit the verification; we warn the reader that the
$A$-module structure on the shift of a left graded $A$-module
is defined using a sign, see
Definiton \ref{definition-shift-graded-module}.
Definition \ref{definition-shift-graded-module}.
\end{remark}


@@ -4837,7 +4837,8 @@ \section{Derived pullback}
is a quasi-isomorphism. Choose a quasi-isomorphism
$\mathcal{Q} \to \mathcal{N}'$ where $\mathcal{Q}$
is a good left differential graded $\mathcal{A}'$-module
(Lemma \ref{lemma-resolve}). By Lemma \ref{lemma-good-on-other-side} the map
(Lemma \ref{lemma-resolve}).
By Lemma \ref{lemma-good-on-other-side} the map
above as a map in the derived category of $\mathcal{O}$-modules is the map
$$
\mathcal{P} \otimes_\mathcal{A} \mathcal{N}
@@ -11433,7 +11433,8 @@ \section{Duals}
the category of complexes and of the derived category.
In particular, we will see that an object of $D(\mathcal{O})$
has a dual if and only if it is perfect (this follows from
Example \ref{example-dual-derived} and Lemma \ref{lemma-left-dual-derived}).
Example \ref{example-dual-derived} and
Lemma \ref{lemma-left-dual-derived}).

\begin{lemma}
\label{lemma-symmetric-monoidal-cat-complexes}

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