Motivation for cochains, cocycles and coboundaries
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.
The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence
$$0 to P to M to N to 0,$$
we only have the following partial sequence for the $0$-th cohomology groups:
$$0 to P^G to M^G to N^G$$
and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?
abstract-algebra reference-request homology-cohomology intuition group-cohomology
edited Sep 4 at 10:53
Mark S.
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
add a comment |Â
up vote
2
down vote
favorite
I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.
The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence
$$0 to P to M to N to 0,$$
we only have the following partial sequence for the $0$-th cohomology groups:
$$0 to P^G to M^G to N^G$$
and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?
abstract-algebra reference-request homology-cohomology intuition group-cohomology
edited Sep 4 at 10:53
Mark S.
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
1
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
1
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.
The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence
$$0 to P to M to N to 0,$$
we only have the following partial sequence for the $0$-th cohomology groups:
$$0 to P^G to M^G to N^G$$
and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?
abstract-algebra reference-request homology-cohomology intuition group-cohomology
edited Sep 4 at 10:53
Mark S.
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.
The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence
$$0 to P to M to N to 0,$$
we only have the following partial sequence for the $0$-th cohomology groups:
$$0 to P^G to M^G to N^G$$
and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?
abstract-algebra reference-request homology-cohomology intuition group-cohomology
abstract-algebra reference-request homology-cohomology intuition group-cohomology
edited Sep 4 at 10:53
Mark S.
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
edited Sep 4 at 10:53
Mark S.
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
edited Sep 4 at 10:53
Mark S.
10.8k22365
edited Sep 4 at 10:53
Mark S.
10.8k22365
edited Sep 4 at 10:53
Mark S.
10.8k22365
10.8k22365
asked Sep 3 at 7:59
TheStudent
836
asked Sep 3 at 7:59
TheStudent
836
asked Sep 3 at 7:59
TheStudent
836
836
1
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
1
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57
add a comment |Â
1
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
1
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57
1
1
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
1
1
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2903632%2fmotivation-for-cochains-cocycles-and-coboundaries%23new-answer', 'question_page');
);
Post as a guest
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
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
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
1
How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation...
– Mark S.
Sep 3 at 15:59
@MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome!
– TheStudent
Sep 4 at 2:35
1
I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice.
– Mark S.
Sep 4 at 10:57