Chain duality I (Ex)

Revision as of 09:29, 16 May 2012 (view source) Tibor Macko (Talk | contribs) (Created page with "<wikitex>; Let $\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant fu...")		Revision as of 12:24, 1 June 2012 (view source) Spiros Adams-Florou (Talk | contribs) (The question was incorrect before the alteration) Newer edit →
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−	Let $\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor	+	(i) Let $\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor
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−	T \colon \Aa \rightarrow \Bb (\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).	+	T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).
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−	Observe that if $C \in \Bb(\Aa)$ is concentrated between dimensions $0$ and $n$ then $T(C)$ is concentrated between dimensions $0$ and $-n$.	+	(ii) Suppose instead that $T \colon \Aa \to \Aa$ and that $C \in \Bb(\Aa)$ is concentrated between dimensions $0$ and $n$. Observe that the extension $T: \Bb(\Aa) \to \Bb(\Aa)$ has $T(C)$ concentrated between dimensions $0$ and $-n$.
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	== References ==		== References ==

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Revision as of 12:24, 1 June 2012

(i) Let $<wikitex>; Let $\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor $$ T \colon \Aa \rightarrow \Bb (\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa). $$ Observe that if $C \in \Bb(\Aa)$ is concentrated between dimensions <script type="math/tex">\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor

(ii) Suppose instead that $T \colon \Aa \to \Aa$ and that $C \in \Bb(\Aa)$ is concentrated between dimensions $0$ and $n$. Observe that the extension $T: \Bb(\Aa) \to \Bb(\Aa)$ has $T(C)$ concentrated between dimensions $0$ and $-n$.

References

$ and $n$ then $T(C)$ is concentrated between dimensions $\Aa$ be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor

(ii) Suppose instead that $T \colon \Aa \to \Aa$ and that $C \in \Bb(\Aa)$ is concentrated between dimensions $0$ and $n$. Observe that the extension $T: \Bb(\Aa) \to \Bb(\Aa)$ has $T(C)$ concentrated between dimensions $0$ and $-n$.

References

$ and $-n$. == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]\Aa be an additive category and let $\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$. Using the double complex extend $T$ to a contravariant functor

(ii) Suppose instead that $T \colon \Aa \to \Aa$ and that $C \in \Bb(\Aa)$ is concentrated between dimensions $0$ and $n$. Observe that the extension $T: \Bb(\Aa) \to \Bb(\Aa)$ has $T(C)$ concentrated between dimensions $0$ and $-n$.

References