Chain duality I (Ex)

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

$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$ $$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$$

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

References

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

$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$ $$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$$

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

References

$ and $-n$. == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]\Aa be an additive category and let $\Bb (\Aa)$$\Bb (\Aa)$ be the category of bounded chain complexes in $\Aa$$\Aa$. Using the double complex extend $T$$T$ to a contravariant functor

$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$ $$\displaystyle T \colon \Aa \rightarrow \Bb(\Aa) \quad \textup{to} \quad T \co \Bb(\Aa) \rightarrow \Bb (\Aa).$$

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