Suppose I have a local morphism of local noetherian rings $\phi:A\to B$, $m_A, m_B$ the maximal ideals. Then the $m_B$-adic completion of $B/m_AB$ is: $$(B/m_AB)\hat{}= \varprojlim(B/m_AB/m_B^n) $$ I wanted to know if there is a better way to write that "double quotient" better, as a single quotient, in particular relating that to $m_A^nB$. My guess by now is that $(B/m_AB)/m_B^n=B/(m_AB,m^n_B)$, but still I wanted to find a better relation.
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$\mathfrak m_B^n$ is not a submodule of $B/\mathfrak m_AB$. – Bernard Mar 15 '18 at 10:17
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you are right. So the induced maximal ideal in $B/m_AB$ is $m_BB/m_AB $, so I should look at $ m_B^nB/m_AB $ I guess. Still, does it have some nice description? – Serser Mar 15 '18 at 11:45
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Rather $(\mathfrak m_B^n+\mathfrak m_AB)/\mathfrak m_AB$. – Bernard Mar 15 '18 at 11:50
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Whatever you have to prove with with, I guess it's linked with Artin-Rees' lemma. – Bernard Mar 15 '18 at 11:56