It's My Turn

Kate Gunzinger:
Let me just show you how to *construct* the map S, which is the fun of the lemma anyhow, okay? So you assume you have an element in the kernel of gamma, that is, an element in C, such that gamma takes you to 0 in C-prime. You pull it back to B, via map g, which is surjective...

Cooperman:
Hold it, hold it, hold it. That's -- that's not unique.

Kate Gunzinger:
Yes, it is unique, Mr. Cooperman. Up to an element of the image of f, all right? So we've pulled it back to a fixed B here. Then you take beta of B, which takes you to 0 in C-prime, by the commutivity of the diagram. It's therefore in the kernel of the map g-prime, hence is in the image of the map f-prime, by the exactness of the lower sequence...

Cooperman:
No.

Kate Gunzinger:
...so we can pull it back...

Cooperman:
No.

Kate Gunzinger:
...to an element in A-prime...

Cooperman:
It's not well defined!

Kate Gunzinger:
...which it turns out is *well* defined *modulo* the image of alpha. And thus defines the element in the co-kernel of alpha...

Kate Gunzinger:
and that's the "snake"! And on Monday, we'll address ourselves to

Kate Gunzinger:
the co-homology of groups... and Mr. Cooperman's next objections.