# Talk:Chain duality III (Ex)

(Difference between revisions)

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We have to check the equality of | We have to check the equality of | ||

− | $T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$ | + | $$T'_{F(M),F(M)}(F (\varphi) \circ G(M))=e'_{F(M)}\circ T'G(M) \circ T'F(\varphi)$$ |

− | and $F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M)$. | + | and $$F(T_{M,M}\varphi)\circ G(M)=F(e_M)\circ FT\varphi\circ G(M).$$ |

+ | |||

+ | This follows from the commutative diagram | ||

+ | $$\xymatrix{ | ||

+ | T'F(M)\ar[r]^{T'F(\varphi)} \ar[d]_{G(M)} & | ||

+ | T'FT(M) \ar[r]^{T'G(M)} \ar[d]_{G(TM)} & | ||

+ | T'^2F(M) \ar[d]_{e'_{F(M)}}\\ | ||

+ | FT(M)\ar[r]^{FT\varphi} & | ||

+ | FT^2M\ar[r]^{Fe_M}& | ||

+ | F(M) | ||

+ | }.$$ | ||

</wikitex> | </wikitex> |

## Revision as of 11:32, 1 June 2012

We check this for objects .

Let be an element of .

We have to check the equality of

and

This follows from the commutative diagram