View Single Post
John Baez
#7
Nov4-06, 03:20 PM
P: n/a
In article <1132065631.635708.191820@g14g2000cwa.googlegroups.com>,
<thomas_larsson_01@hotmail.com> wrote:

>John Baez wrote:


>> And, extensions
>>
>> 1 -> F -> E -> B -> 1
>>
>> ....
>>
>> People have also thought about "abelian extensions". That's an even
>> more special case where all three groups are abelian. The set of
>> isomorphism classes of such extensions is called Ext(B,F).


>I am pretty sure that is not standard terminology. In an abelian
>extension, only F is required to be abelian, not E and B.


Hmm. Maybe you're right. But then I'm not sure what the standard
terminology *is* for the kind of extension where all three groups
are abelian.

There should be *some* name for them, since they're wildly popular
in homological algebra, which is a kind of algebra used by topologists.
When kids take an introductory course in homology and cohomology, they
typically learn a bit of this - mainly stuff about "Tor" and "Ext".
Ext(B,F) is the set - actually an abelian group - of isomorphism
classes of extensions

1 -> F -> E -> B -> 1

where F, E and B are all abelian. But, when I pulled down Rotman's book
"An Introduction to Homological Algebra" and tried to see what he
called these extensions, I found he just called them "extensions" -
because in that chapter, he's not considering any other kind!

Well, actually he's considering short exact sequences of R-modules
for any ring R. When R = Z such R-modules are abelian groups - so
nonabelian groups don't even get in the door, in this chapter. There's
lots of nice machinery that works only in this "completely abelian"
case.

My excitement with Schreier theory came from seeing more clearly
than before the nice machinery that works when you're trying to
classify extensions without assuming ANY abelianness. It's
less familiar, because to see how nice it is, you need to understand
2-groups.

Btw, I put a more detailed explanation of this stuff in the "addendum"
to week223 here:

http://math.ucr.edu/home/baez/week223.html