differential on total chain complex
$begingroup$
There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(partial b)_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.
homological-algebra
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
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add a comment |
$begingroup$
There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(partial b)_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.
homological-algebra
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
$endgroup$
add a comment |
$begingroup$
There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(partial b)_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.
homological-algebra
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
$endgroup$
There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(partial b)_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.
homological-algebra
homological-algebra
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
asked Jan 5 at 6:19
Daniel XuDaniel Xu
457
457
add a comment |
add a comment |
1 Answer
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$begingroup$
An element of the total complex consists of a choice of an element
$b_{p,q}in B_{p,q}$ for each pair of integers $p$ and $q$. We need
to define what its boundary is. The definition is that is consists of the
elements $c_{p,q}$ determined by the formula
$$c_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}.$$
As $b_{p+1,q}in B_{p+1,q}$ then $partial'b_{p+1,q}in B_{p,q}$ etc., so that
$c_{p,q}in B_{p,q}$.
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
An element of the total complex consists of a choice of an element
$b_{p,q}in B_{p,q}$ for each pair of integers $p$ and $q$. We need
to define what its boundary is. The definition is that is consists of the
elements $c_{p,q}$ determined by the formula
$$c_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}.$$
As $b_{p+1,q}in B_{p+1,q}$ then $partial'b_{p+1,q}in B_{p,q}$ etc., so that
$c_{p,q}in B_{p,q}$.
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
$endgroup$
add a comment |
$begingroup$
An element of the total complex consists of a choice of an element
$b_{p,q}in B_{p,q}$ for each pair of integers $p$ and $q$. We need
to define what its boundary is. The definition is that is consists of the
elements $c_{p,q}$ determined by the formula
$$c_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}.$$
As $b_{p+1,q}in B_{p+1,q}$ then $partial'b_{p+1,q}in B_{p,q}$ etc., so that
$c_{p,q}in B_{p,q}$.
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
$endgroup$
add a comment |
$begingroup$
An element of the total complex consists of a choice of an element
$b_{p,q}in B_{p,q}$ for each pair of integers $p$ and $q$. We need
to define what its boundary is. The definition is that is consists of the
elements $c_{p,q}$ determined by the formula
$$c_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}.$$
As $b_{p+1,q}in B_{p+1,q}$ then $partial'b_{p+1,q}in B_{p,q}$ etc., so that
$c_{p,q}in B_{p,q}$.
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
$endgroup$
An element of the total complex consists of a choice of an element
$b_{p,q}in B_{p,q}$ for each pair of integers $p$ and $q$. We need
to define what its boundary is. The definition is that is consists of the
elements $c_{p,q}$ determined by the formula
$$c_{p,q}=partial'b_{p+1,q}+partial''b_{p,q+1}.$$
As $b_{p+1,q}in B_{p+1,q}$ then $partial'b_{p+1,q}in B_{p,q}$ etc., so that
$c_{p,q}in B_{p,q}$.
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
answered Jan 5 at 6:28
Lord Shark the UnknownLord Shark the Unknown
102k959132
102k959132
add a comment |
add a comment |
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