## 111.11 Ext groups

Ext groups are defined in Algebra, Section 10.71.

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).

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

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

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

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

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

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.

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.)

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