Chapter 7, Tor and Ext Video Solutions, An Introduction to Homological Algebra

Problem 1

If $R$ is right hereditary, prove that $\operatorname{Tor}_{j}^{R}(A, B)=\{0\}$ for all $j \geq 2$ and for all right $R$-modules $A$ and $B$.
Hint. Every submodule of a projective module is projective.

Lucía Guerrero

Numerade Educator

05:21

Problem 2

If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence of right $R$-modules with both $A$ and $C$ flat, prove that $B$ is flat.

Sirat Shah

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01:23

Problem 3

If $F$ is flat and $\pi: P \rightarrow F$ is a surjection with $P$ flat, prove that ker $\pi$ is flat.

Joseph Liao

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Problem 4

If $A, B$ are finite abelian groups, prove that $\operatorname{Tor}_{1}^{\mathbb{Z}}(A, B) \cong A \otimes \mathbb{Z} B$.

Nick Johnson

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03:13

Problem 5

Let $R$ be a domain with $\operatorname{Frac}(R)=Q$ and $K=Q / R$. Prove that the right derived functors of $t$ (the torsion submodule functor) are
$$
R^{0} t=t, \quad R^{1} t=K \otimes_{R} \square, \quad R^{n} t=0 \quad \text { for all } n \geq 2 .
$$

Gideon Idumah

Numerade Educator

03:53

Problem 6

Let $k$ be a field, let $R=k[x, y]$, and let $I$ be the ideal $(x, y)$.
(i) Prove that $x \otimes y-y \otimes x \in I \otimes_{R} I$ is nonzero.
Hint. Consider $\left(I / I^{2}\right) \otimes\left(I / I^{2}\right)$.
(ii) Prove that $x(x \otimes y-y \otimes x)=0$, and conclude that $I \otimes_{R} I$ is not torsion-free.

Joy Carpio

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Problem 7

Prove that the functor $T=\operatorname{Tor}_{1}^{Z}(G, \square)$ is left exact for every abelian group $G$, and compute its right derived functors $L_{n} T$.

Nick Johnson

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06:47

Problem 8

(i) Let $G$ be a $p$-primary abelian group, where $p$ is prime. If $(m, p)=1$, prove that $x \mapsto m x$ is an automorphism of $G$.
(ii) If $p$ is an odd prime and $G=\langle g\rangle$ is a cyclic group of order $p^{2}$, prove that $\varphi: x \mapsto 2 x$ is the unique automorphism with $\varphi(p g)=2 p g .$

Brandon Collins

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05:17

Problem 9

Prove that any two split extensions of modules $A$ by $C$ are equivalent.

Anthony Ramos

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00:59

Problem 10

Prove that if $A$ is an abelian group with $n A=A$ for some positive integer $n$, then every extension $0 \rightarrow A \rightarrow E \rightarrow \mathbb{I}_{n} \rightarrow 0$ splits.

Varsha Aggarwal

Numerade Educator

01:35

Problem 11

(i) Find an abelian group $B$ for which $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$.
(ii) Prove that $\mathbb{Q} \otimes_{\mathbb{Z}} \operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q}, B) \neq\{0\}$ for the group $B$ in (i).
(iii) Prove that Proposition $7.39$ may be false when $A$ is not finitely generated, even when $R=\mathbb{Z}$.

Nick Johnson

Numerade Educator

01:58

Problem 12

Let $E$ be a left $R$-module. Prove that $E$ is injective if and only if $\operatorname{Ext}_{R}^{1}(A, E)=\{0\}$ for every left $R$-module $A$.

Mohamed Mohamed

Numerade Educator

00:59

Problem 13

(i) Prove that the covariant functor $E=\operatorname{Ext}_{\mathbb{Z}}^{1}(G, \square)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} E$.
(ii) Prove that the contravariant functor $F=\operatorname{Ext}_{\mathbb{Z}}^{1}(\square, G)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} F$. (See the footnote on page 370 .)

Varsha Aggarwal

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Problem 14

(i) If $A$ is an abelian group with $m A=A$ for some nonzero $m \in \mathbb{Z}$, prove that every exact sequence $0 \rightarrow A \rightarrow G \rightarrow$ $\mathbb{I}_{m} \rightarrow 0$ splits. Conclude that $m \operatorname{Ext}_{\mathbb{Z}}^{1}(A, B)=\{0\}=$ $m \operatorname{Ext}_{\mathbb{Z}}^{1}(B, A)$.
(ii) If $A$ and $C$ are abelian groups with $m A=\{0\}=n C$, where $(m, n)=1$, prove that every extension of $A$ by $C$ splits.

Nick Johnson

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Problem 15

(i) For any ring $R$, prove that a left $R$-module $B$ is injective if and only if $\operatorname{Ext}_{R}^{1}(R / I, B)=\{0\}$ for every left ideal $I$.
Hint. Use the Baer criterion.
(ii) If $D$ is an abelian group and $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q} / \mathbb{Z}, D)=\{0\}$, prove that $D$ is divisible. The converse is true because divisible abelian groups are injective. Does this hold if we replace $\mathbb{Z}$ by a domain $R$ and $\mathbb{Q} / \mathbb{Z}$ by $\operatorname{Frac}(R) / R ?$

Nick Johnson

Numerade Educator

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Problem 16

Let $G$ be an abelian group $G$. Prove that $G$ is free abelian if and only if $\operatorname{Ext}_{\mathbb{Z}}^{1}(G, F)=\{0\}$ for every free abelian group $F$.

Nick Johnson

Numerade Educator

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Problem 17

Let $A$ be a torsion abelian group and let $S^{1}$ be the circle group. Prove that $\operatorname{Ext}_{\mathbb{Z}}^{1}(A, \mathbb{Z}) \cong \operatorname{Hom}_{\mathbb{Z}}\left(A, S^{1}\right)$.

Nick Johnson

Numerade Educator

08:25

Problem 18

An abelian group $W$ is a Whitehead group if $\operatorname{Ext}_{\mathbb{Z}}^{1}(W, \mathbb{Z})=\{0\} \cdot^{3}$
(i) Prove that every subgroup of a Whitehead group is a Whitehead group.
(ii) Prove that $\operatorname{Ext}_{\mathbb{Z}}^{1}(A, \mathbb{Z}) \cong \operatorname{Hom}_{\mathbb{Z}}\left(A, S^{1}\right)$ if $A$ is a torsion group and $S^{1}$ is the circle group. Prove that if $A \neq\{0\}$ is torsion, then $A$ is not a Whitehead group; conclude further that every Whitehead group is torsion-free.
Hint. Use Exercise $7.17 .$
(iii) Let $A$ be a torsion-free abelian group of rank 1; i.e., $A$ is a subgroup of $\mathbb{Q}$. Prove that $A \cong \mathbb{Z}$ if and only if $\operatorname{Hom}_{\mathbb{Z}}(A, \mathbb{Z}) \neq\{0\} .$
(iv) Let $A$ be a torsion-free abelian group of rank 1. Prove that if $A$ is a Whitehead group, then $A \cong \mathbb{Z}$.
Hint. Use an exact sequence $0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow T \rightarrow 0$, where $T$ is a torsion group whose $p$-primary component is either cyclic or isomorphic to Prüfer's group of type $p^{\infty}$.
(v) (K. Stein). Prove that every countable $^{4}$ Whitehead group is free abelian.
Hint. Use Exercise $3.4$ on page 114 , Pontrjagin's Lemma: if $A$ is a countable torsion-free group and every subgroup of $A$ having finite rank is free abelian, then $A$ is free abelian.

Ely Crowder

Numerade Educator

02:10

Problem 19

We have constructed the bijection $\psi: e(C, A) \rightarrow \operatorname{Ext}^{1}(C, A)$ using a projective resolution of $C$. Define a function $\psi^{\prime}: e(C, A) \rightarrow$ $\operatorname{Ext}^{1}(C, A)$ using an injective resolution of $A$, and prove that $\psi^{\prime}$ is a bijection.

Goutam Chand

Numerade Educator

Problem 20

Consider the diagram
$$
\begin{aligned}
&\xi_{1}=\quad 0 \longrightarrow A_{1} \longrightarrow B_{1} \longrightarrow C_{1} \longrightarrow 0 \\
&\xi_{2}=\quad 0 \rightarrow A_{2} \longrightarrow B_{2} \longrightarrow C_{2} \longrightarrow 0
\end{aligned}
$$
Prove that there is a map $\beta: B_{1} \rightarrow B_{2}$ making the diagram commute if and only if $\left[h \xi_{1}\right]=\left[\xi_{2} k\right]$.

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01:02

Problem 21

(i) Prove, in $e(C, A)$, that $-[\xi]=\left[\left(-1_{A}\right) \xi\right]=\left[\xi\left(-1_{C}\right)\right]$.
(ii) Generalize (i) by replacing $\left(-1_{A}\right)$ and $\left(-1_{C}\right)$ by $\mu_{r}$ for any $r$ in the center of $R$.

Hoan Nguyen

Numerade Educator

01:37

Problem 22

Prove that $[\xi]=[0 \rightarrow A \stackrel{i}{\longrightarrow} B \rightarrow C \rightarrow 0] \in e(C, A)$ has finite order if and only if there are a nonzero $m \in \mathbb{Z}$ and a map $s: B \rightarrow A$ with $s i=m \cdot 1_{A}$.

James Chok

Numerade Educator

01:05

Problem 23

(i) Prove that $e(C, \square): R$ Mod $\rightarrow$ Ab is a covariant functor if, for $h: A \rightarrow A^{\prime}$, we define $h_{*}: e(C, A) \rightarrow e\left(C, A^{\prime}\right)$ by $[\xi] \mapsto[h \xi] .$
(ii) Prove that $e(C, \square)$ is naturally isomorphic to Ext $_{R}^{1}(C, \square)$.

Anthony Ramos

Numerade Educator

01:05

Problem 24

Consider the extension $\chi=0 \rightarrow A^{\prime} \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} A^{\prime \prime} \rightarrow 0$.
(i) Define $D: \operatorname{Hom}_{R}\left(C, A^{\prime \prime}\right) \rightarrow e\left(C, A^{\prime}\right)$ by $k \mapsto[\chi k]$, and prove exactness of
$\operatorname{Hom}(C, A) \stackrel{p_{*}}{\longrightarrow} \operatorname{Hom}\left(C, A^{\prime \prime}\right) \stackrel{D}{\longrightarrow} e\left(C, A^{\prime}\right)$
$$
\stackrel{i_{*}}{\longrightarrow} e(C, A) \stackrel{p_{*}}{\longrightarrow} e\left(C, A^{\prime \prime}\right)
$$
(ii) Prove commutativity of
where $\partial$ is the connecting homomorphism.

Anthony Ramos

Numerade Educator

01:05

Problem 25

(i) Prove that $e(\square, A):{ }_{R}$ Mod $\rightarrow \mathbf{A b}$ is a contravariant functor if, for $k: C^{\prime} \rightarrow C$, we define $k^{*}: e(C, A) \rightarrow e\left(C^{\prime}, A\right)$ by $[\xi] \mapsto[\xi k]$.
(ii) Prove that $e(\square, A)$ is naturally isomorphic to $\operatorname{Ext}_{R}^{1}(\square, A)$.

Anthony Ramos

Numerade Educator

01:05

Problem 26

Consider the extension $X=0 \rightarrow C^{\prime} \stackrel{i}{\longrightarrow} C \stackrel{p}{\longrightarrow} C^{\prime \prime} \rightarrow 0$.
(i) Define $D^{\prime}: \operatorname{Hom}_{R}\left(C^{\prime}, A\right) \rightarrow e\left(C^{\prime \prime}, A\right)$ by $h \mapsto\lfloor h X]$, and prove exactness of
$\operatorname{Hom}(C, A) \stackrel{i^{*}}{\longrightarrow} \operatorname{Hom}\left(C^{\prime}, A\right) \stackrel{D^{\prime}}{\longrightarrow} e\left(C^{\prime \prime}, A\right)$
$\stackrel{p^{*}}{\longrightarrow} e(C, A) \stackrel{i^{*}}{\longrightarrow} e\left(C^{\prime}, A\right) .$
(ii) Prove commutativity of
where $\partial^{\prime}$ is the connecting homomorphism.

Anthony Ramos

Numerade Educator