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John Baez
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 220

Posted: Mon May 29, 2006 1:30 am    Post subject: Nine Lemma

What's a famous, catchy, exciting or illuminating application
of the Nine Lemma in homological algebra? The Wikipedia article
on this subject needs some help, and someone asked me this...

Just to give information as I ask for it: this lemma says that
if you have a 3x3 grid of objects in an abelian category, like this:

0 0 0
| | |
v v v
0 -> * -> * -> * -> 0
| | |
v v v
0 -> * -> * -> * -> 0
| | |
v v v
0 -> * -> * -> * -> 0
| | |
v v v
0 0 0

and the columns are exact, and the first two or last two
rows are exact, then the remaining row is exact.

(What if the top and bottom row are exact?)
victor_meldrew_666@yahoo.
science forum beginner

Joined: 19 May 2006
Posts: 17

Posted: Mon May 29, 2006 5:00 pm    Post subject: Re: Nine Lemma

John Baez wrote:
 Quote: What's a famous, catchy, exciting or illuminating application of the Nine Lemma in homological algebra? The Wikipedia article on this subject needs some help, and someone asked me this... Just to give information as I ask for it: this lemma says that if you have a 3x3 grid of objects in an abelian category, like this: 0 0 0 | | | v v v 0 -> * -> * -> * -> 0 | | | v v v 0 -> * -> * -> * -> 0 | | | v v v 0 -> * -> * -> * -> 0 | | | v v v 0 0 0 and the columns are exact, and the first two or last two rows are exact, then the remaining row is exact. (What if the top and bottom row are exact?)

Then the middle row need not be exact; it need not even be a complex.
Call the nine objects A_1, ..., C_3 in an obvious fashion.
Work inside the category of Abelian groups and let A_1 = C_3 = 0,
A_2 = A_3 = B_1 = B_3 = C_1 = C_2 = Z and B_2 = Z^2.
Make any arrow between two Zs be the identity map.
To make the middle column exact let A_2 -> B_2 take x to
(x,0) and let B_2 -> C_2 take (x, y) to y.
Now the diagram will commute if the map B_1 -> B_2 takes
x to (ax, x) for some a and B_2 -> B_3 takes (x, y) to x + by
for some b. Although these maps must be injective and
surjective respectively, their composite takes x to (a+b)x
which need not be zero.

Victor Meldrew
John Baez
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 220

Posted: Tue Jun 27, 2006 4:50 am    Post subject: Re: Nine Lemma

In article <e5diqt\$r21\$1@news.ks.uiuc.edu>,
John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

 Quote: What's a famous, catchy, exciting or illuminating application of the Nine Lemma in homological algebra? The Wikipedia article on this subject needs some help, and someone asked me this...

So nobody knows any use for the Nine Lemma? Interesting....
Dimitri Ara
science forum beginner

Joined: 27 Jun 2006
Posts: 1

Posted: Tue Jun 27, 2006 11:58 am    Post subject: Re: Nine Lemma

baez@math.removethis.ucr.andthis.edu (John Baez) a écrit :

 Quote: So nobody knows any use for the Nine Lemma? Interesting....

One can prove the exactness of the relative Mayer-Vietoris sequence
(for singular homology) using the nine lemma.

Let (A_1, A_2) -> (X_1, X_2) be an inclusion of excisive couples. We
have a commutative diagram :

0 0 0
| | |
V V V
0 -> C(A_12) -> C(A_1) (+) C(A_2) -> C(A_1) + C(A_2) -> 0
| | |
V V V
0 -> C(X_12) -> C(X_1) (+) C(X_2) -> C(X_1) + C(X_2) -> 0
| | |
V V V
0 -> C(X_12, A_12) -> C(X_1, A_1) + C(X_2, A_2) -> (C(A_1) + C(A_2))/(C(X_1) + C(X_2)) -> 0
| | |
V V V
0 0 0

where Y_12 = Y_1 \cap Y_2, Y = Y_1 \cup Y_2 and C(Y,B) is the relative
singular chain complex of (Y, B).

The columns and the top rows are exact, so the bottom row is
exact by the nine lemma. I claim that the long exact sequence
associated to this row is the relative Mayer-Vietoris sequence.

To see that, let's consider this morphism :

0 -> C(A_1) + C(A_2) -> C(X_1) + C(X_2) -> (C(A_1) + C(A_2))/(C(X_1) + C(X_2)) -> 0
| | |
V V V
0 -> C(A) -> C(X) -> C(A,X) -> 0

The first arrows are quasi-isomorphisms by hypothesis. So is the third
by the five lemma. QED.

This proof is essentially a reformulation of Spanier's proof.

--
Dimitri Ara
Agustí Roig
science forum beginner

Joined: 18 Jul 2005
Posts: 11

Posted: Tue Jun 27, 2006 1:05 pm    Post subject: Re: Nine Lemma

John Baez ha escrit:

 Quote: In article , John Baez wrote: What's a famous, catchy, exciting or illuminating application of the Nine Lemma in homological algebra? The Wikipedia article on this subject needs some help, and someone asked me this... So nobody knows any use for the Nine Lemma? Interesting....

It can be used in the proof of the excision theorem for singular
homology (although there are proofs without it).

If you apply it to the commutative diagram with exact rows and columns:

S(A - U) ---> S(A - U) ---> 0
| | |
| | |
V V V
S(A) + S(A - U) ---> S(A) + S(X - U) ---> S(X - U) / S(A - U)
| | |
| | |
V V V
S({A, A - U}) ---> S({A, X - U}) ---> S({A, X - U}) / S({A,
A - U})

you get an isomorphism of complexes

S(X - U) / S(A - U) ---> S({A, X - U}) / S({A, A - U})

which composed with the isomorphism

H( S({A, X - U}) / S({A, A - U})) ---> H(X,A)

gives the excision isomorphism

H(X - U, A - U) ---> H(X,A)

Notations and conventions:

(1) U is a subspace of A , A a subspace of X , with the closure of
U contained in the interior of A .
(2) S(X), S(X,A) stand for the singular and relative chain complexes,
respectively.
(3) S({A, X - U}) is the subcomplex of S(X) generated by those
simplices with their images contained in A or in X - U .
(4) + means direct sum.
(5) H(X), H(X,A) mean singular homology and relative singular
homology, respectively.

Agustí Roig
Mike1161

Joined: 14 Oct 2005
Posts: 50

Posted: Wed Jun 28, 2006 6:24 am    Post subject: Re: Nine Lemma

John Baez wrote:
 Quote: In article , John Baez wrote: What's a famous, catchy, exciting or illuminating application of the Nine Lemma in homological algebra? The Wikipedia article on this subject needs some help, and someone asked me this... So nobody knows any use for the Nine Lemma? Interesting....

An overstatement to say that nobody knows any use for it. To my
knowledge the 9 lemma is never strictly NECESSARY. But it is handy to
have around. It conceptually illuminates homological algebra (assuming
that one has a brain as strange as mine that feels illuminated by the
result). One one has the9 lemma one can give tidy proofs of relative
Mayer-Vietoris and excision as others in the thread have pointed out.

Actually, I have always thought that the hexagonal lemma is
overrated as to usefulness. Opinions everybody?

Mike

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