Exact sequence properties
up vote
1
down vote
favorite
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
5,4381138102
add a comment |
up vote
1
down vote
favorite
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
5,4381138102
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 |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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
5,4381138102
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
5,4381138102
asked Nov 25 at 1:23
Hawk
5,4381138102
asked Nov 25 at 1:23
Hawk
5,4381138102
asked Nov 25 at 1:23
Hawk
5,4381138102
asked Nov 25 at 1:23
Hawk
5,4381138102
5,4381138102
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
1
active
oldest
votes
up vote
2
down vote
accepted
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
275k21268494
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
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
275k21268494
add a comment |
up vote
2
down vote
accepted
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
275k21268494
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
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
275k21268494
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
275k21268494
answered Nov 25 at 1:43
Hagen von Eitzen
275k21268494
answered Nov 25 at 1:43
Hagen von Eitzen
275k21268494
answered Nov 25 at 1:43
Hagen von Eitzen
275k21268494
275k21268494
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012314%2fexact-sequence-properties%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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