Exercise 111.9.4. Let $A = k[x, y]_{(x, y)}$ be the local ring of the affine plane at the origin. Make any assumption you like about the field $k$. Suppose that $f = x^3 + x^2y^2 + y^{100}$ and $g = y^3 - x^{999}$. What is the length of $A/(f, g)$ as an $A$-module? (Possible way to proceed: think about the ideal that $f$ and $g$ generate in quotients of the form $A/{\mathfrak m}_ A^ n= k[x, y]/(x, y)^ n$ for varying $n$. Try to find $n$ such that $A/(f, g)+{\mathfrak m}_ A^ n \cong A/(f, g)+{\mathfrak m}_ A^{n + 1}$ and use NAK.)

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)