= Power-series presentation of an adic completion
{title2=$R[[X_1,\ldots,X_s]]\twoheadrightarrow\widehat R$}
If $I=(a_1,\ldots,a_s)$, evaluation $X_i\mapsto a_i$ gives a ring map to the $I$-adic completion, with each series evaluated modulo $I^r$ using only terms of total degree below $r$. It is surjective: successively represent the residual class in $I^r/I^{r+1}$ by a homogeneous degree-$r$ polynomial in the $a_i$. If $R$ is a commutative <Noetherian ring>, <Noetherianity of formal power series rings> therefore makes its completion Noetherian.
Back to article page
