The Stacks project

Exercise 111.11.1. Compute all the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i(M, N)$ of the given modules in the category of $\mathbf{Z}$-modules (also known as the category of abelian groups).

1. $M = \mathbf{Z}$ and $N = \mathbf{Z}$,

2. $M = \mathbf{Z}/4\mathbf{Z}$ and $N = \mathbf{Z}/8\mathbf{Z}$,

3. $M = \mathbf{Q}$ and $N = \mathbf{Z}/2\mathbf{Z}$, and

4. $M = \mathbf{Z}/2\mathbf{Z}$ and $N = \mathbf{Q}/\mathbf{Z}$.

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.

Exercise 111.11.3. Give an example of a Noetherian ring $R$ and finite modules $M$, $N$ such that $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ is nonzero for all $i \geq 0$.

Exercise 111.11.4. Give an example of a ring $R$ and ideal $I$ such that $\mathop{\mathrm{Ext}}\nolimits ^1_ R(R/I, R/I)$ is not a finite $R$-module. (We know this cannot happen if $R$ is Noetherian by Algebra, Lemma 10.71.9.)