Structured chain complexes II (Ex)

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Latest revision as of 12:46, 30 July 2013

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$<wikitex>; Let $f,g \co C \ra D$ be two chain maps between bounded chain complexes over~$R$. Show that the following hold: 1.) $$((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_0 = (1+T) (f \otimes g) \varphi_0, \ ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_s = \; 0 \; \textup{for} \; s \geq 1,$$ 2.) $$((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_0 = (f \otimes g) (1+T) \psi_0, \ ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_s = 0 \; \textup{for} \; s \geq 1,$$ 3.) $$ ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s = 0 \; \textup{for} \; s \geq 0.$$ </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]f,g \co C \ra D$ be two chain maps between bounded chain complexes over~ $R$ $R$ . Show that the following hold:

1.)

$\displaystyle ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_0 = (1+T) (f \otimes g) \varphi_0, \\ ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_s = \; 0 \; \textup{for} \; s \geq 1,$ $\displaystyle ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_0 = (1+T) (f \otimes g) \varphi_0, \\ ((f + g)^{\%} (\varphi) - f^{\%} (\varphi) - g^{\%} (\varphi))_s = \; 0 \; \textup{for} \; s \geq 1,$

2.)

$\displaystyle ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_0 = (f \otimes g) (1+T) \psi_0, \\ ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_s = 0 \; \textup{for} \; s \geq 1,$ $\displaystyle ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_0 = (f \otimes g) (1+T) \psi_0, \\ ((f + g)_{\%} (\psi) - f_{\%} (\psi) - g_{\%} (\psi))_s = 0 \; \textup{for} \; s \geq 1,$

3.)

$\displaystyle ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s = 0 \; \textup{for} \; s \geq 0.$ $\displaystyle ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s = 0 \; \textup{for} \; s \geq 0.$

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Structured chain complexes II (Ex)

Latest revision as of 12:46, 30 July 2013

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@@ Line 11: / Line 11: @@
   ((\widehat{f + g})^{\%} (\varphi) - \widehat{f}^{\%} (\varphi) - \widehat{g}^{\%} (\varphi))_s  = 0 \; \textup{for} \; s \geq 0.$$
 </wikitex>
-== References ==
+== References==
 {{#RefList:}}
 [[Category:Exercises]]
 [[Category:Exercises without solution]]