Write a few:
- as ever in discussing PIDs, bear in mind Z and K[x]
- statement & interpretation of result
+ equivalence via Chinese Remainder Theorem
+ naturalness & lack thereof
+ comment on the free part being a bit weird
(prime ideals of different heights/lengths??)
algebraic geometry POV: slice horizontally vs. vertically
(module is a (coherent) sheaf over affine scheme;
over point, it's not 2 points, but a 2D vector space)
- free resolution proof: presentation, Smith Normal Form
--> homological algebra
- intrinsic proof via filtration
--> Wederburn-Artin
- primary resolution is a Hasse principle:
if know module at every prime
(and modules over a DVR are very easy;
next simpler than vector space)
...then know whole module
- subtle homological interpretation:
torsion is weird! (Tor = Tor^1 !!)
tensoring is easier (from this POV)