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Let $K$ be a field and consider $K[[x,y]]$, which is the usual ring of power series in two variables.

We have the prime ideal

$$(y)=yK[[x,y]]$$

and consider the localization $K[[x,y]]_{(y)}$ at $(y)$. Is there any explicit way to describe the elements in $K[[x,y]]_{(y)}$?

Example:

I'm asking some expression similar to the following one for $(y)^{-r}K[[x,y]]$

$(y)^{-r}K[[x,y]]=y^{-r}K[[x,y]]=\{\sum_{i\ge 0, j\ge -r}a_{ij} x^iy^j\}\subset K((x,y))$

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    I don't know if you can something more explicit that $y^r \frac{f}{g}$, with $f,g$ formal series which are not multiple of $y$ and $r \geq 0$. – 2017-02-07
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    It is the union of the rings you wrote down, so $\{f = \sum_{i \geq 0, j > -\infty} a_ij x^i y^j | \exists N< 0 a_{ij} = 0 \forall j < N \}$. Note that the negative powers of $y$ are bounded below for any individual member of the ring. – 2017-02-07
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    @hunter : I believe this is the _localization_ at the prime ideal $(y)$, so every element becomes inversible excepted multiple of $y$, which is precisely the opposite of the ring $k[[x,y]][y^{-1}]$. If I'm not mistaken, notation can be confusing ... – 2017-02-07
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    I suspect that it should be something like $K(x)[[y]]$, but I'm not sure (like my username says) – 2017-02-07
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    Surely not ! For example $\frac{1}{y+1}$ is in the localisation but not in your ring. – 2017-02-07
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    @N.H. you're quite right! oops – 2017-02-07

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Let me give you a geometric motivation. I am on purpose a bit vague here. For a given point $x \in X$, you can look at the functions $f$ defined "in a little neighborhood of $x$, where $f,g$ will be equivalent if there is a little neighborhood $U$ of $x$ where $f_{|U} = g_{|U}$. More formally we are looking at the stalk at $x$ of the sheaf of function on $X$.

For example, let's take the origin in $\mathbb A^1$. the function $\frac{1}{x +1}$ is not defined at $-1$, but it is certainly at $0$ and should be in this ring of "local functions". Our strategy will be to write our function $\frac{f}{g}$ and verify that $g$ is non-zero at $0$. But such functions are exactly the $x^r \frac{f}{g}$ where $f(0) \neq 0, g(0) \neq 0$ and $r \geq 0$.

The localization $K[[x,y]]_{(y)}$ has exactly the same interpretaion : we are localizing at the $x$-axis (defined by the equation $y = 0$), so we are looking in all the functions which are defined in a neighborhood of the $x$-axis, and as before this is the functions which can be written as $y^r \frac{f(x,y)}{g(x,y)}$ where $y$ does not divide $f$ and $g$, and $r \geq 0$.

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    If you want a description in Laurent series, I think this is the Laurent series with no term in $\frac{1}{y}$. – 2017-02-07