Exercise 111.11.2. Let $R = k[x, y]$ where $k$ is a field.

Show by hand that the Koszul complex

\[ 0 \to R \xrightarrow { \left( \begin{matrix} y \\ -x \end{matrix} \right) } R^{\oplus 2} \xrightarrow {(x, y)} R \xrightarrow {f \mapsto f(0, 0)} k \to 0 \]is exact.

Compute $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(k, k)$ where $k = R/(x, y)$ as an $R$-module.

## Comments (0)