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!