Example 15.90.13 (Non glueable module). Let $R$ be the ring of germs at $0$ of $C^\infty $ functions on $\mathbf{R}$. Let $f \in R$ be the function $f(x) = x$. Then $f$ is a nonzerodivisor in $R$, so $(R, f)$ is a glueing pair and $R^\wedge \cong \mathbf{R}[[x]]$. Let $\varphi \in R$ be the function $\varphi (x) = \text{exp}(-1/x^2)$. Then $\varphi $ has zero Taylor series, so $\varphi \in \mathop{\mathrm{Ker}}(R \to R^\wedge )$. Since $\varphi (x) \neq 0$ for $x \neq 0$, we see that $\varphi $ is a nonzerodivisor in $R$. The function $\varphi /f$ also has zero Taylor series, so its image in $M = R/\varphi R$ is a nonzero element of $M[f]$ which maps to zero in $M \otimes _ R R^\wedge = R^\wedge /\varphi R^\wedge = R^\wedge $. Hence $M$ is not glueable.

[ยง4, Remarques, Beauville-Laszlo]

## 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Your email address will not be published.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)

There are also: