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More clarify on target category
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aisejohan committed May 6, 2024
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Expand Up @@ -1135,12 +1135,15 @@ \section{Deriving torsion}
Let $A$ be a ring and $I \subset A$ a finitely generated ideal.
By More on Algebra, Lemma \ref{more-algebra-lemma-I-power-torsion}
the category of $I$-power torsion modules is a Serre subcategory
of the category of all $A$-modules, hence there is a functor
of the category of all $A$-modules. Hence there is a functor
\begin{equation}
\label{equation-compare-torsion}
D(I^\infty\text{-torsion}) \to D_{I^\infty\text{-torsion}}(A)
D(I^\infty\text{-torsion}) \longrightarrow D_{I^\infty\text{-torsion}}(A)
\end{equation}
see Derived Categories, Section \ref{derived-section-triangulated-sub}.
where the right hand side is the full subcategory of $D(A)$
consising of objects whose cohomologies are $I$-power torsion modules.
Both the functor and the category are discussed in
Derived Categories, Section \ref{derived-section-triangulated-sub}.

\begin{lemma}
\label{lemma-not-equal}
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