# Completion of a local noetherian $\Gamma$-algbera

I am in trouble with a result . Let $\Gamma$ is a complete local noetherian $k$-algbera with residue field $k$ and $(R,m)$ is a local noetherian $\Gamma$-algebra with residue field $k$. Also let $\hat{R}$ is the completion of $R$ then we know that $$\hat{R} \cong \Gamma[[ X_1,...,X_d]] / J$$ where $\Gamma[[ X_1,...,X_d]]$ is the formal power series on $d$ variables for some $d$ and $J$ is some ideal .

Question:

1. Why $J$ is supposed to be in the ideal $(X_1,...,X_d)^2$ ?

2 . What is a presentation of the $m$-adic completion $\hat{R}$ ?

Edit: This is actually Proposition $2.1.1 (ii)$ page $38$ - Book : Deformation of algebraic schemes - Sernesi.

## migrated from mathoverflow.netOct 1 '18 at 18:27

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• It will be helpful if you can indicate from which paper/book is your assertions/questions. – xarles Sep 13 '18 at 13:16
• Hi Xarles, thank you. I have just edited in my post. – An Khuong Doan Sep 13 '18 at 14:01
• The point is that you can choose your presentation so that $J$ is contained in $(X_1,\ldots ,X_d)^2$: choose elements $x_1,\ldots ,x_d$ of $\mathfrak{m}$ whose images in $\mathfrak{m}/\mathfrak{m}^2$ form a basis, and define your homomorphism by $X_i\mapsto x_i$. Then any $F\in \Gamma [[X_1,\ldots ,X_d]]$ such that $F(x_1,\ldots ,x_d)=0$ cannot contain terms of degree $\leq 1$, hence belongs to $(X_1,\ldots ,X_d)^2$. – abx Sep 13 '18 at 15:10
• Dear abx,Is your defined homomorphism is from $\Gamma[[X_1,...,X_d]]$ to $m/m^2$ ? – An Khuong Doan Sep 13 '18 at 17:16
• This is standard — one proves by induction on $n$ that the induced map to $\hat{R}/\mathfrak{m}^n$ is surjective for all $n$. – abx Sep 13 '18 at 19:46