Exact sequence properties
If $0 stackrel{gamma}to C_n stackrel{d_n}to C_{n-1} to dots C_0to 0$ is exact, then it induces the short exact sequence
$$0 to ker d_n to C_n to text{img}d_{n} to 0?$$
This is part of the Euler Characteristic proof. I stared at this for a while, but it isn't too clear to me. Is this the correct justification?
Since $ker d_n hookrightarrow{} C_n$ in a one-to-one fashion, the boundary map $C_n stackrel{d_{n}}totext{img}d_{n} subset C_{n-1}$ maps back into $C_{n-1}$ (actually it is $0$ from $d^2 = 0$?). So $ker d_n subset text{img}d_{n}$. Also $text{img}d_{n} approx C_n/ker d_n = C_n/text{img}gamma_{n} = C_n/{0} approx C_n$. Actually I am not sure how to find the other inclusion.
algebraic-topology exact-sequence
asked Nov 25 at 1:23
Hawk
5,4531138103
add a comment |
If $0 stackrel{gamma}to C_n stackrel{d_n}to C_{n-1} to dots C_0to 0$ is exact, then it induces the short exact sequence
$$0 to ker d_n to C_n to text{img}d_{n} to 0?$$
This is part of the Euler Characteristic proof. I stared at this for a while, but it isn't too clear to me. Is this the correct justification?
Since $ker d_n hookrightarrow{} C_n$ in a one-to-one fashion, the boundary map $C_n stackrel{d_{n}}totext{img}d_{n} subset C_{n-1}$ maps back into $C_{n-1}$ (actually it is $0$ from $d^2 = 0$?). So $ker d_n subset text{img}d_{n}$. Also $text{img}d_{n} approx C_n/ker d_n = C_n/text{img}gamma_{n} = C_n/{0} approx C_n$. Actually I am not sure how to find the other inclusion.
algebraic-topology exact-sequence
asked Nov 25 at 1:23
Hawk
5,4531138103
Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50
add a comment |
If $0 stackrel{gamma}to C_n stackrel{d_n}to C_{n-1} to dots C_0to 0$ is exact, then it induces the short exact sequence
$$0 to ker d_n to C_n to text{img}d_{n} to 0?$$
This is part of the Euler Characteristic proof. I stared at this for a while, but it isn't too clear to me. Is this the correct justification?
Since $ker d_n hookrightarrow{} C_n$ in a one-to-one fashion, the boundary map $C_n stackrel{d_{n}}totext{img}d_{n} subset C_{n-1}$ maps back into $C_{n-1}$ (actually it is $0$ from $d^2 = 0$?). So $ker d_n subset text{img}d_{n}$. Also $text{img}d_{n} approx C_n/ker d_n = C_n/text{img}gamma_{n} = C_n/{0} approx C_n$. Actually I am not sure how to find the other inclusion.
algebraic-topology exact-sequence
asked Nov 25 at 1:23
Hawk
5,4531138103
If $0 stackrel{gamma}to C_n stackrel{d_n}to C_{n-1} to dots C_0to 0$ is exact, then it induces the short exact sequence
$$0 to ker d_n to C_n to text{img}d_{n} to 0?$$
This is part of the Euler Characteristic proof. I stared at this for a while, but it isn't too clear to me. Is this the correct justification?
Since $ker d_n hookrightarrow{} C_n$ in a one-to-one fashion, the boundary map $C_n stackrel{d_{n}}totext{img}d_{n} subset C_{n-1}$ maps back into $C_{n-1}$ (actually it is $0$ from $d^2 = 0$?). So $ker d_n subset text{img}d_{n}$. Also $text{img}d_{n} approx C_n/ker d_n = C_n/text{img}gamma_{n} = C_n/{0} approx C_n$. Actually I am not sure how to find the other inclusion.
algebraic-topology exact-sequence
algebraic-topology exact-sequence
asked Nov 25 at 1:23
Hawk
5,4531138103
asked Nov 25 at 1:23
Hawk
5,4531138103
asked Nov 25 at 1:23
Hawk
5,4531138103
asked Nov 25 at 1:23
Hawk
5,4531138103
asked Nov 25 at 1:23
Hawk
5,4531138103
5,4531138103
Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50
add a comment |
Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50
Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50
add a comment |
1 Answer
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If $Astackrel fto B$ is a homomorphism, then
$$0to ker fstackrel subseteqto A stackrel ftooperatorname{img}f to 0$$
is a short exact sequence. Clearly, the incusion is injective and $f$ is surjective (to $operatorname{img}f$). Finally the kernel of $f$ is ... $ker f$, as desired.
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
add a comment |
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1 Answer
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If $Astackrel fto B$ is a homomorphism, then
$$0to ker fstackrel subseteqto A stackrel ftooperatorname{img}f to 0$$
is a short exact sequence. Clearly, the incusion is injective and $f$ is surjective (to $operatorname{img}f$). Finally the kernel of $f$ is ... $ker f$, as desired.
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
add a comment |
If $Astackrel fto B$ is a homomorphism, then
$$0to ker fstackrel subseteqto A stackrel ftooperatorname{img}f to 0$$
is a short exact sequence. Clearly, the incusion is injective and $f$ is surjective (to $operatorname{img}f$). Finally the kernel of $f$ is ... $ker f$, as desired.
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
add a comment |
If $Astackrel fto B$ is a homomorphism, then
$$0to ker fstackrel subseteqto A stackrel ftooperatorname{img}f to 0$$
is a short exact sequence. Clearly, the incusion is injective and $f$ is surjective (to $operatorname{img}f$). Finally the kernel of $f$ is ... $ker f$, as desired.
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
If $Astackrel fto B$ is a homomorphism, then
$$0to ker fstackrel subseteqto A stackrel ftooperatorname{img}f to 0$$
is a short exact sequence. Clearly, the incusion is injective and $f$ is surjective (to $operatorname{img}f$). Finally the kernel of $f$ is ... $ker f$, as desired.
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
answered Nov 25 at 1:43
Hagen von Eitzen
276k21269496
276k21269496
add a comment |
add a comment |
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Your justification is not clear at all, and statements like "$ker d_n subset text{img}d_{n}$" make no sense (the two sides aren't even subsets of the same module). Try to be less handwavy (what does "maps back" mean?) and more formal, even at the cost of greater length.
– darij grinberg
Nov 25 at 1:38
@darijgrinberg $iota(ker d_n) subset C_n$, then $d_n(iota(ker d_n)) subset img d_n subset C_{n-1}$
– Hawk
Nov 25 at 1:50