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Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m< n)$.

Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ?

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    Looks like $k[x_1,x_2,\ldots]$ to me...2012-09-30
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    I think you are mixing up projective limit and direct limit: $k[x_1,...]$ is the direct limit of $k[x_1,...,x_n] \hookrightarrow k[x_1,...,x_m]\;(n < m)$.2012-10-01
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    I also thought about this power series ring. But have have no idea on how to define the projections $k[[x_1,x_2,...]] \to k[x_1,...,x_n]$ ?2012-10-01
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    Ups, the previous comment concerns a comment of John Stalfos who suggested the projective limit in question is $k[[x_1,x_2,...]]$ (and was deleted while I was typing my comment).2012-10-01
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    What was the problem with $k[x_1,x_2,...],$ again? There *are* projections onto each element of your inverse system which commute with the maps in the system, though I haven't tried to check universality.2012-10-01
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    No element of $k[x_1, x_2, \cdots]$ (with the canonical projection) is capable of mapping to $\sum_{i=1}^n x_i \in k[x_1, \cdots, x_n]$ for every $n$... but there is a diagram of maps from $k[y]$ that does. (I assume the "canonical projection" mentioned in the question sends $x_n \to 0$, although that really isn't canonical)2012-10-01
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    @Kevin: The sequence $x=(x_1+...+x_n)_{n\ge 1}$ is an element in the projective limit (considered as subring of $\prod_n k[x_1,...,x_n]$) which I think has no counterpart in $k[x_1,x_2,...]$. In analogy with p-adic numbers I would interpret $x$ as $x_1 + x_2 + ...$ which is an element in $k[[x_1,x_2,...]]$. But I don't know if this a correct.2012-10-01
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    Ah, yes, ignore what I said, $k[x_1,\ldots]$ is a direct limit.2012-10-01

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