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Fix ampersand replacement
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Signed-off-by: zeramorphic <[email protected]>
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zeramorphic committed Dec 1, 2023
1 parent 71ed9a4 commit 2c53c3b
Showing 1 changed file with 5 additions and 5 deletions.
10 changes: 5 additions & 5 deletions iii/cat/07_additive_and_abelian_categories.tex
Original file line number Diff line number Diff line change
Expand Up @@ -959,10 +959,10 @@ \subsection{Derived functors}
We choose \( P_\bullet, R_\bullet \) arbitrarily, and take \( Q_n = P_n \oplus R_n \); this is projective as the coproduct of projective objects is projective.
Consider the diagram
% https://q.uiver.app/#q=WzAsMTIsWzQsMCwiQSJdLFs0LDEsIkIiXSxbNCwyLCJDIl0sWzMsMCwiUF8wIl0sWzIsMCwiS18wIl0sWzEsMCwiUF8xIl0sWzAsMCwiXFxjZG90cyJdLFswLDIsIlxcY2RvdHMiXSxbMSwyLCJSXzEiXSxbMiwyLCJNXzAiXSxbMywyLCJSXzAiXSxbMywxLCJQXzAgXFxvcGx1cyBSXzAiXSxbMCwxLCJmIl0sWzEsMiwiZyJdLFszLDAsImVfMSJdLFs0LDNdLFs1LDRdLFs2LDVdLFs3LDhdLFs4LDldLFs5LDEwXSxbMTAsMiwiZV8zIl0sWzMsMTEsIlxcYmVnaW57cG1hdHJpeH0xXFxcXDBcXGVuZHtwbWF0cml4fSIsMl0sWzExLDEwLCJcXGJlZ2lue3BtYXRyaXh9MCYxXFxlbmR7cG1hdHJpeH0iLDJdXQ==
\[\begin{tikzcd}
\cdots & {P_1} & {K_0} & {P_0} & A \\
&&& {P_0 \oplus R_0} & B \\
\cdots & {R_1} & {M_0} & {R_0} & C
\[\begin{tikzcd}[ampersand replacement=\&]
\cdots \& {P_1} \& {K_0} \& {P_0} \& A \\
\&\&\& {P_0 \oplus R_0} \& B \\
\cdots \& {R_1} \& {M_0} \& {R_0} \& C
\arrow["f", from=1-5, to=2-5]
\arrow["g", from=2-5, to=3-5]
\arrow["{e_1}", from=1-4, to=1-5]
Expand Down Expand Up @@ -1181,4 +1181,4 @@ \subsection{Derived functors}
in \( \mathcal B \).
Then the result follows from the Mayer--Vietoris sequence.
\end{proof}
In particular, \( L^0F \) is always right exact, so \( L^0F \cong F \) if and only if \( F \) is right exact.
In particular, \( L^0F \) is always right exact, so \( L^0F \cong F \) if and only if \( F \) is right exact.

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