Kevin Wald: Commutative Algebra 101 S: I am Sam. Sam I am. N: That Sam-I-am! That Sam-I-am! I do not like that Sam-I-am! S: Do you like this diagram? N: I do not like it, Sam-I-am. I do not like this Diagram. S: Would you chase it here or there? N: I would not chase it here or there. I would not chase it anywhere. I do not like this diagram. I do not like it, Sam-I-am. S: Would you draw it with a p-group Would you draw it with a Lie group? N: I would not draw it with a p-group. I would not draw it with a Lie group. I would not chase it here or there. I would not chase it anywhere. I do not like this diagram. I do not like it, Sam-I-am. S: Would you draw it with a Tor group? How about the cyclic four-group? N: Not with a Tor group. Not with a four-group. Not with a p-group. Not with a Lie group. I would not chase it here or there. I would not chase it anywhere. I do not like this diagram. I do not like it, Sam-I-am. S: Would you? Could you? With a star? Chase it! Chase it! Into \R. N: I would not, could not, into \R S: You can chase it into \C. Or \Q adjoin root minus 3. N: I would not, could not into \C Nor into \R! You let me be. I will not draw it with a Tor group. I will not draw it with a four-group I will not draw it with a p-group. I will not draw it with a Lie-group. I will not chase it here or there. I will not chase it anywhere. I do not like this diagram. I do not like it, Sam-I-am. S: A chain! A chain! A chain! A chain! Could you, would you, With a chain? N: Not with a chain! Not into \C! Not into \R! Sam! Let me be! I would not, could not, with a Tor group. I could not, would not, with a four-group. I will not draw it with a p-group. I will not draw it with a Lie-group. I will not chase it here or there. I will not chase it anywhere. I will not chase this diagram. I do not like it, Sam-I-am. S: Could you draw it like McLane? N: I could not draw it like McLane. I would not draw it with a chain. Not into \R. Not into \C. Nor \Q adjoin root minus three. Not with a Tor group. Not with a p-group. Not with a four-group. Not with a Lie group. I will not chase it here or there. I will not chase it anywhere! S: You do not like this diagram? N: I do not like it, Sam-I-am. S: Would you tensor with Z-hat? N: I would not tensor with Z-hat. S: Could you show that it is flat? N: I cannot show that it is flat. I will not tensor with Z-hat. I could not draw it like McLane. I would not draw it with a chain. Not into \R! Not into \C! Not with a star! You let me be! I do not like it with a Tor group. I do not like it with a four-group. I will not draw it with a p-group. I will not draw it with a Lie group. I will not chase it here or there. I will not chase it ANYWHERE! I do not like this diagram! I do not like it, Sam-I-am. S: If you don't chase this diagram, You're gonna flunk your topic exam. N: Oh! Well a here maps to b. Then to c, here, then to d. Say! I like this diagram! I do! I like it, Sam-I-am. And I can tensor with Z-hat, And use the fact that it is flat, (As shown in Eilenburg-McLane) Which means it's an acyclic chain, And so, from \R (or even \C) By straight commutativity We get a mapping through the Tor group (Mod out by the cyclic four-group) Into SU_5, our Lie group. Consequently, from the p-group, We get mappings here and there, Which shows exactness EVERYWHERE! I do so like this diagram! Thank you! Thank you, Sam-I-am!