Structured chain complexes V (Ex)

Let $f \colon C \rightarrow D$$<wikitex>; Let $f \colon C \rightarrow D$ and $g \colon D \rightarrow E$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$ to~<script type="math/tex">f \colon C \rightarrow D$ and $g \colon D \rightarrow E$$g \colon D \rightarrow E$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$$j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$$gf$ to~$0$$0$. Then the formula

$\displaystyle \Phi_j (y,x) = g(y) + j(x)$ $$\displaystyle \Phi_j (y,x) = g(y) + j(x)$$

gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$$\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram commutative

$\displaystyle \xymatrix{C \ar[r]^{f} & D \ar[r]^{e} \ar[dr]_{g} & \mathcal{C}(f) \ar@{-->}[d]^{\Phi_j} \\ & & E }$

Find appropriate $j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$$j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$ for which we have

$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$ $$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$$

and

$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$ $$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$$

References

$. Then the formula $$ \Phi_j (y,x) = g(y) + j(x) $$ gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram commutative $$ \xymatrix{C \ar[r]^{f} & D \ar[r]^{e} \ar[dr]_{g} & \mathcal{C}(f) \ar@{-->}[d]^{\Phi_j} \ & & E } $$ Find appropriate $j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$ for which we have $$ \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D. $$ and $$ \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f). $$ == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]f \colon C \rightarrow D and $g \colon D \rightarrow E$$g \colon D \rightarrow E$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$$j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$$gf$ to~$0$$0$. Then the formula

$\displaystyle \Phi_j (y,x) = g(y) + j(x)$ $$\displaystyle \Phi_j (y,x) = g(y) + j(x)$$

gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$$\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram commutative

$\displaystyle \xymatrix{C \ar[r]^{f} & D \ar[r]^{e} \ar[dr]_{g} & \mathcal{C}(f) \ar@{-->}[d]^{\Phi_j} \\ & & E }$

Find appropriate $j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$$j \colon ef \simeq 0 \colon C \rightarrow \mathcal{C} (f)$ for which we have

$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$ $$\displaystyle \mathrm{ev}_l (\delta \varphi,\varphi) = \delta\varphi_0 + f \varphi_0 \colon \mathcal{C} (f)^{n+1-\ast} \rightarrow D.$$

and

$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$ $$\displaystyle \mathrm{ev}_r (\delta \varphi,\varphi) = \begin{pmatrix} \delta \varphi_0 \\ \varphi_0 f^\ast \end{pmatrix} \colon D^{n+1-\ast} \rightarrow \mathcal{C} (f).$$