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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