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When trying to understand the construction of the unipotent completion of a finitely generated group $\Gamma$, the following very basic question came to me.

Denote, as usual, $\mathbb{Q}[\Gamma]$ the group ring and $I \subset \mathbb{Q}[\Gamma]$ the augmentation ideal. One looks at $$ \varinjlim (\mathbb{Q}[\Gamma]/I^{n+1})^\vee $$ but... how are the transition maps defined? What seems natural to me is to project the quotient by $I^n$ into the quotient by $I^{n+1}$ but then, when passing to the dual, one gets maps in the opposite direction. So, how are the maps $$ \mathbb{Q}[\Gamma]/I^{n+1} \to \mathbb{Q}[\Gamma]/I^n $$ defined?

Thanks for your help!

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This is a really late answer. But I stumbled upon this question when seeking an introduction to the unipotent completion myself. Information can probably be found more easily when searching for "Malcev completion". A concise introduction can be found in Section 2 of this paper of Knudsen.

To answer your question. It seems like you just got the direction mixed up. For the completion $$ \mathbb Q[\Gamma]^{\wedge} = \varprojlim \mathbb Q[\Gamma]/I^n$$ one uses the inverse system $$ \cdots \twoheadrightarrow \mathbb Q[\Gamma]/I^2 \twoheadrightarrow\mathbb Q[\Gamma]/I \twoheadrightarrow\mathbb Q[\Gamma]$$ and for the regular functions on the unipotent completion of $\Gamma$ $$ \varinjlim (\mathbb Q[\Gamma]/I^n)^\vee$$ one uses the direct system $$ (\mathbb Q[\Gamma])^\vee \hookrightarrow(\mathbb Q[\Gamma]/I)^\vee \hookrightarrow(\mathbb Q[\Gamma]/I^2)^\vee \hookrightarrow \cdots .$$