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Let $k$ be a field.

Consider the inverse limit

$\varprojlim k[x,y]/(y\cdot x^n)$.

I wonder if there is a nice description of this ring?

Geometrically, we look at the union of the line $y=0$ along with an infinitesimal neighborhood of the line $x=0$. But what happens in the limit? I think we get $k[y][[x]]$? What is the geometric interpretation of this?

Any help will be appreciated.

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    Thank you for your comment. I now think you are right. I wonder what is the answer.2011-12-30

1 Answers 1

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Let $R=\varprojlim k[x]/(x^n) = k[[x]]$, the ring of power series in $x$. The natural map $k[x]\rightarrow R$ is injective.

Then your limit is the subring of $R[y]$ with constant terms in $k[x]$.

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    @anonymous: Geometrically, you give yourself a function defined on the $x$-axis, the $y$-axis, *and* on some neighbourhood of the $y$-axis minus origin. You identify a function with the one obtained by shrinking the neighbourhood of the punctured $y$-axis.The projective limit describes the equivalence classes of these functions.2011-12-30