The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.

The finite presentation trick

 Our goal here is:

Theorem 1  Let {A} be a Noetherian ring, and {I} an ideal. If we take all completions with respect to the {I}-adic topology,      

 

\displaystyle \hat{M} = \hat{A} \otimes_A M

for any f.g. {A}-module {M}.   (more…)

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