Oct. 26th, 2008

synchcola: (heyoka "ok")
Hey, how's it going? I'm trying to lower my stress levels. ^_^

The MATH 400 midterm is on Monday, so we don't have any homework to do this week, which is nice cause otherwise I would be doing it right now.

The statement "A ≅ C/B" can be expressed as an exact sequence like this.
0 \to B \to C \stackrel{\pi}{\to} A \to 0
ie, G is a subgroup of F (or more precisely, Im G is a subgroup of F) and A ≅ F/ker π = F/G.

Abelian groups all have free resolutions like this:
0 \to G \to F \stackrel{\pi}\to A \to 0

That's because every abelian group can be written as the quotient of a free abelian group with something (A ≅ F/G with G ⊂ F), and every subgroup of a free group is free (I don't think that I ever learned why that is, but I have just now thought of an argument via transfinite induction, so there).

The slow explanation of Tor continues!

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