From: dtd@world.std.com (Donald T. Davis, Jr.)
Subject: Re: What is an exact sequence?
Date: Sat, 06 Feb 1999 13:48:24 -0500
Newsgroups: [missing]
To: rusin@vesuvius.math.niu.edu (Dave Rusin)
In article <79gqla$bo3$1@gannett.math.niu.edu>,
rusin@vesuvius.math.niu.edu (Dave Rusin) wrote:
>no, you're close, but a little a bit off. every
>element of G1 gets mapped to G3's identity, G2
>gets mapped to G4, etc. thus
>
> phi2( phi1) = 0 = phi3( phi2) .
Actually, that should be
phi2( phi1) = 0 and phi3( phi2) =0 .
If you think that looks the same as what Davis posted, then
you're not ready for exact sequences :-) .
>the value of exactness is as a means of calculating
>unknown groups' structure. if you have a short exact
>sequence in which you know two of the three groups,
>then the corresponding long-exact sequence enables
>you to calculate the 3rd, unknown group.
Change "enables" to "assists", or "calculate" to "estimate",
or "group" to "vector space" or something. For example, if
1 -> Z/2Z -> G -> Z/2Z -> 1
is exact then you only know G is isomorphic to Z/4Z or (Z/2Z)^2.
dave
-----------------------------------------------------------
thanks for the correction (i think), but i didn't
understand your first point. could you explain
how "a = 0 = b" is different from "a = 0 & b = 0"?
i take it for granted that you're on to something i
missed, but i'm damned if i can see it.
also, i don't think you took my second point in the
spirit with which i offered it. salzman was asking
for "how do i think about this" guidance, not for a
repeat of his textbook's definition. i was a little
loose for that reason. i suppose i should have said,
"if you have a short exact sequence of chain groups
H_*, G_*,... then the corresponding long exact seq
enables..." the whole point of the nudge i was
trying to give, was to get quickly to the sense, and
let the guy's textbooks take care of the rigor.
somehow, my alg-top prof never really made it explicit
that that's what exactness was for; he was too busy
conveying the relevant definitions, theorems and
calculations. i didn't get this bit about the purpose,
value, and central role of exactness as a device for
calculation, until a couple years later. i had hoped
in my post to save salzman the trouble of figuring it
out for himself, as i'd had to do.
- don davis, boston
==============================================================================
[Reply not saved; the answer to Davis's query is that in the statement
phi2( phi1) = 0 and phi3( phi2) =0
the two symbols "0" stand for maps in different groups, and so cannot be
equal. -- djr]