The Stacks project

Example 15.90.12 (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.