Exercise 111.64.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

Artinian rings,

flatness and prime ideals,

lengths of $A/\mathfrak m^ n$ for $(A, \mathfrak m)$ Noetherian local,

the dimension formula for universally catenary Noetherian rings,

completion of a Noetherian local ring, and

Matlis duality for Artinian local rings.

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