The Stacks project

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

  1. 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.

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

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View Exercise 111.11.2 as pdf