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An Introduction to Homological Algebra

Joseph J. Rotman

Chapter 2

Hom and Tensor - all with Video Answers

Educators

Chapter Questions

03:56

Problem 1

Let $R$ and $S$ be rings, and let $\varphi: R \rightarrow S$ be a ring homomorphism. If $M$ is a left $S$-module, prove that $M$ is also a left $R$-module if we define
$$
r m=\varphi(r) m,
$$
for all $r \in R$ and $m \in M$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:13

Problem 2

Give an example of a left $R$-module $M=S \oplus T$ having a submodule $N$ such that $N \neq(N \cap S) \oplus(N \cap T)$.

Fan Yang
Fan Yang
Numerade Educator
00:42

Problem 3

Let $f, g: M \rightarrow N$ be $R$-maps between left $R$-modules. If $M=\langle X\rangle$ and $f|X=g| X$, prove that $f=g$.

Linh Vu
Linh Vu
Numerade Educator
02:46

Problem 4

Let $\left(M_{i}\right)_{i \in I}$ be a (possibly infinite) family of left $R$-modules and, for each $i$, let $N_{i}$ be a submodule of $M_{i}$. Prove that

Michael Jacobsen
Michael Jacobsen
Numerade Educator
05:10

Problem 5

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of left $R$-modules. If $M$ is any left $R$-module, prove that there are exact sequences
$$
0 \rightarrow A \oplus M \rightarrow B \oplus M \rightarrow C \rightarrow 0
$$
and
$$
0 \rightarrow A \rightarrow B \oplus M \rightarrow C \oplus M \rightarrow 0
$$

Doruk Isik
Doruk Isik
Numerade Educator
01:05

Problem 6

(i) Let $\rightarrow A_{n+1} \stackrel{d_{n+1}}{\longrightarrow} A_{n} \stackrel{d_{n}}{\longrightarrow} A_{n-1} \rightarrow$ be an exact sequence, and let $\operatorname{im} d_{n+1}=K_{n}=\operatorname{ker} d_{n}$ for all $n$. Prove that
$$
0 \rightarrow K_{n} \stackrel{i_{n}}{\longrightarrow} A_{n} \stackrel{d_{n}^{\prime}}{\longrightarrow} K_{n-1} \rightarrow 0
$$
is an exact sequence for all $n$, where $i_{n}$ is the inclusion and $d_{n}^{\prime}$ is obtained from $d_{n}$ by changing its target. We say that the original sequence has been factored into these short exact sequences.
(ii) Let
$$
\rightarrow A_{1} \stackrel{f_{1}}{\longrightarrow} A_{0} \stackrel{f_{0}}{\longrightarrow} K \rightarrow 0
$$
and
$$
0 \rightarrow K \stackrel{g_{0}}{\longrightarrow} B_{0} \stackrel{g_{1}}{\longrightarrow} B_{1} \rightarrow
$$
be exact sequences. Prove that
$$
\rightarrow A_{1} \stackrel{f_{1}}{\longrightarrow} A_{0} \stackrel{g_{0} f_{0}}{\longrightarrow} B_{0} \stackrel{g_{1}}{\longrightarrow} B_{1} \rightarrow
$$
is an exact sequence. We say that the original two sequences have been spliced to form the new exact sequence.

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 7

Use left exactness of Hom to prove that if $G$ is an abelian group, then $\operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{I}_{n}, G\right) \cong G[n]$, where $G[n]=\{g \in G: n g=0\}$.

Nick Johnson
Nick Johnson
Numerade Educator
08:25

Problem 8

(i) Prove that a short exact sequence in ${ }_{R}$ Mod,
$$
0 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 0
$$
splits if and only if there exists $q: B \rightarrow A$ with $q i=1_{A}$.
(Note that $q$ is a retraction $B \rightarrow$ im $i$.)
(ii) A sequence $A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C$ in Groups is exact if im $i=$ ker $p$; an exact sequence
$$
1 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 1
$$
in Groups is split if there is a homomorphism $j: C \rightarrow B$ with $p j=1_{C}$. Prove that $1 \rightarrow A_{3} \rightarrow S_{3} \rightarrow \mathbb{I}_{2} \rightarrow 1$ is a split exact sequence. In contrast to part (i), show, in a split exact sequence in Groups, that there may not be a homomorphism $q: B \rightarrow A$ with $q i=1_{A}$.

Ely Crowder
Ely Crowder
Numerade Educator
01:21

Problem 9

(i) Let $v_{1}, \ldots, v_{n}$ be a basis of a vector space $V$ over a field $k$. Let $v_{i}^{*}: V \rightarrow k$ be the evaluation $V^{*} \rightarrow k$ defined by $v_{i}^{*}=\left(\square, v_{i}\right)$ (see Example 1.16). Prove that $v_{1}^{*}, \ldots, v_{n}^{*}$ is a basis of $V^{*}$ (it is called the dual basis of $v_{1}, \ldots, v_{n}$ ).
Hint. Use Corollary 2.22(ii) and Example 2.27.
(ii) Let $f: V \rightarrow V$ be a linear transformation, and let $A$ be the matrix of $f$ with respect to a basis $v_{1}, \ldots, v_{n}$ of $V$; that is, the $i$ th column of $A$ consists of the coordinates of $f\left(v_{i}\right)$ with respect to the given basis $v_{1}, \ldots, v_{n}$. Prove that the matrix of the induced map $f^{*}: V^{*} \rightarrow V^{*}$ with respect to the dual basis is the transpose $A^{T}$ of $A$.

Arun Bana
Arun Bana
Numerade Educator
03:13

Problem 10

If $X$ is a subset of a left $R$-module $M$, prove that $\langle X\rangle$, the submodule of $M$ generated by $X$, is equal to $\bigcap S$, where the intersection ranges over all those submodules $S$ of $M$ that contain $X$.

Gideon Idumah
Gideon Idumah
Numerade Educator
10:50

Problem 11

Prove that if $f: M \rightarrow N$ is an $R$-map and $K$ is a submodule of a left $R$-module $M$ with $K \subseteq \operatorname{ker} f$, then $f$ induces an $R$-map $\widehat{f}: M / K \rightarrow N$ by $\widehat{f}: m+K \mapsto f(m)$

Chris Trentman
Chris Trentman
Numerade Educator
04:07

Problem 12

(i) Let $R$ be a commutative ring and let $J$ be an ideal in $R$. Recall Example $2.8$ (iv): if $M$ is an $R$-module, then $J M$ is a submodule of $M$. Prove that $M / J M$ is an $R / J$-module if we define scalar multiplication:
$$
(r+J)(m+J M)=r m+J M .
$$
Conclude that if $J M=\{0\}$, then $M$ itself is an $R / J-$ module. In particular, if $J$ is a maximal ideal in $R$ and $J M=\{0\}$, then $M$ is a vector space over $R / J$.
(ii) Let $I$ be a maximal ideal in a commutative ring $R$. If $X$ is a basis of a free $R$-module $F$, prove that $F / I F$ is a vector space over $R / I$ and that $\{\operatorname{cosets} x+I F: x \in X\}$ is a basis.

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 13

Let $M$ be a left $R$-module.
(i) Prove that the map $\varphi_{M}: \operatorname{Hom}_{R}(R, M) \rightarrow M$, given by $\varphi_{M}: f \mapsto f(1)$, is an $R$-isomorphism.
Hint. Make the abelian group $\operatorname{Hom}_{R}(R, M)$ into a left $R$ module by defining $r f$ (for $f: R \rightarrow M$ and $r \in R$ ) by $r f: s \mapsto f(s r)$ for all $s \in R .$
(ii) If $g: M \rightarrow N$, prove that the following diagram commutes:
Conclude that $\varphi=\left(\varphi_{M}\right)_{M \in o b j(R M o d)}$ is a natural isomorphism from $\operatorname{Hom}_{R}(R, \square)$ to the identity functor on ${ }_{R} \mathbf{M o d}$.
[Compare with Example 1.16(ii).]

Nick Johnson
Nick Johnson
Numerade Educator
14:32

Problem 14

Let $A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C$ be a sequence of module maps. Prove that $g f=0$ if and only if im $f \subseteq \operatorname{ker} g$. Give an example of such a sequence that is not exact.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:18

Problem 15

(i) Prove that $f: M \rightarrow N$ is surjective if and only if coker $f=$ $\{0\}$.
(ii) If $f: M \rightarrow N$ is a map, prove that there is an exact sequence
$$
0 \rightarrow \operatorname{ker} f \rightarrow M \stackrel{f}{\rightarrow} N \rightarrow \operatorname{coker} f \rightarrow 0
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:18

Problem 16

(i) If $0 \rightarrow M \rightarrow 0$ is an exact sequence, prove that $M=\{0\}$.
(ii) If $A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \stackrel{h}{\rightarrow} D$ is an exact sequence, prove that $f$ is surjective if and only if $h$ is injective.
(iii) Let $A \stackrel{\alpha}{\longrightarrow} B \stackrel{\beta}{\longrightarrow} C \stackrel{\gamma}{\longrightarrow} D \stackrel{\delta}{\longrightarrow} E$ be exact. If $\alpha$ and $\delta$ are isomorphisms, prove that $C=\{0\}$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
05:49

Problem 17

If $A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} D \stackrel{k}{\longrightarrow} E$ is exact, prove that there is an exact sequence
$$
0 \rightarrow \operatorname{coker} f \stackrel{\alpha}{\longrightarrow} C \stackrel{\beta}{\longrightarrow} \operatorname{ker} k \rightarrow 0
$$
where $\alpha: b+\operatorname{im} f \mapsto g b$ and $\beta: c \mapsto h c$.

Anas Venkitta
Anas Venkitta
Numerade Educator
03:13

Problem 18

Let $0 \rightarrow A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C \rightarrow 0$ be a short exact sequence.
(i) Assume that $A=\langle X\rangle$ and $C=\langle Y\rangle$. For each $y \in Y$, choose $y^{\prime} \in B$ with $p\left(y^{\prime}\right)=y$. Prove that
$$
B=\left\langle i(X) \cup\left\{y^{\prime}: y \in Y\right\}\right) .
$$
(ii) Prove that if both $A$ and $C$ are finitely generated, then $B$ is finitely generated. More precisely, prove that if $A$ can be generated by $m$ elements and $C$ can be generated by $n$ elements, then $B$ can be generated by $m+n$ elements.

Gideon Idumah
Gideon Idumah
Numerade Educator
03:58

Problem 19

Let $R$ be a ring, let $A$ and $B$ be left $R$-modules, and let $r \in Z(R)$.
(i) If $\mu_{r}: B \rightarrow B$ is multiplication by $r$, prove that the induced map $\left(\mu_{r}\right)_{*}: \operatorname{Hom}_{R}(A, B) \rightarrow \operatorname{Hom}_{R}(A, B)$ is also multiplication by $r$.
(ii) If $m_{r}: A \rightarrow A$ is multiplication by $r$, prove that the induced map $\left(m_{r}\right)^{*}: \operatorname{Hom}_{R}(A, B) \rightarrow \operatorname{Hom}_{R}(A, B)$ is also multiplication by $r$.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:01

Problem 20

Suppose one assumes, in the hypothesis of Proposition 2.42, that the induced map $i^{*}: \operatorname{Hom}_{R}(B, M) \rightarrow \operatorname{Hom}_{R}\left(B^{\prime}, M\right)$ is surjective for every $M$. Prove that $0 \rightarrow B^{\prime} \stackrel{i}{\longrightarrow} B \stackrel{p}{\longrightarrow} B^{\prime \prime} \rightarrow 0$ is a split short exact sequence.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:59

Problem 21

If $T: \mathbf{A b} \rightarrow \mathbf{A b}$ is an additive functor, prove, for every abelian group $G$, that the function End $(G) \rightarrow \operatorname{End}(T G)$, given by $f \mapsto$ $T f$, is a ring homomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 22

(i) Prove that $\operatorname{Hom}_{\mathrm{Z}}(\mathbb{Q}, C)=\{0\}$ for every cyclic group $C$.
(ii) Let $R$ be a commutative ring. If $M$ is an $R$-module such that $\operatorname{Hom}_{R}(M, R / I)=\{0\}$ for every nonzero ideal $I$, prove that $\operatorname{im} f \subseteq \bigcap I$ for every $R$-map $f: M \rightarrow R$, where the intersection is over all nonzero ideals $I$ in $R$.
(iii) Let $R$ be a domain and suppose that $M$ is an $R$-module with $\operatorname{Hom}_{R}(M, R / I)=\{0\}$ for all nonzero ideals $I$ in $R$. Prove that $\operatorname{Hom}_{R}(M, R)=\{0\}$.
Hint. Every $r \in \bigcap_{l \neq 0} I$ is nilpotent.

Ely Crowder
Ely Crowder
Numerade Educator
02:13

Problem 23

Generalize Proposition 2.26. Let $\left(S_{i}\right)_{i \in I}$ be a family of submodules of a left $R$-module $M$. If $M=\left\langle\bigcup_{i \in I} S_{i}\right\rangle$, then the following conditions are equivalent.
(i) $M=\bigoplus_{i \in I} S_{i}$.
(ii) Every $a \in M$ has a unique expression of the form $a=s_{i_{1}}+$ $\cdots+s_{i_{n}}$, where $s_{i j} \in S_{i j} .$
(iii) $S_{i} \cap\left\langle\bigcup_{j \neq i} S_{j}\right\rangle=\{0\}$ for each $i \in I$.

Fan Yang
Fan Yang
Numerade Educator
05:10

Problem 24

(i) Prove that any family of $R$-maps $\left(f_{j}: U_{j} \rightarrow V_{j}\right)_{j \in J}$ can be assembled into an $R$-map $\varphi: \bigoplus_{j} U_{j} \rightarrow \bigoplus_{j} V_{j}$, namely, $\varphi:\left(u_{j}\right) \mapsto\left(f_{j}\left(u_{j}\right)\right)$.
(ii) Prove that $\varphi$ is an injection if and only if each $f_{j}$ is an injection.

Doruk Isik
Doruk Isik
Numerade Educator
01:40

Problem 25

(i) If $Z_{i} \cong \mathbb{Z}$ for all $i$, prove that
$$
\operatorname{Hom}_{\mathbb{Z}}\left(\prod_{i=1}^{\infty} Z_{i}, \mathbb{Z}\right) ¥ \prod_{i=1}^{\infty} \operatorname{Hom}_{\mathbb{Z}}\left(Z_{i}, \mathbb{Z}\right)
$$
Hint. A theorem of J. Los and, independently, of E. C. Zeeman (see Fuchs, Infinite Abelian Groups II, Section 94) says that
(ii) Let $p$ be a prime and let $B_{n}$ be a cyclic group of order $p^{n}$, where $n$ is a positive integer. If $A=\bigoplus_{n=1}^{\infty} B_{n}$, prove that
$$
\operatorname{Hom}_{k}\left(A, \bigoplus_{n=1}^{\infty} B_{n}\right) \neq \bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right)
$$
Hint. Prove that $\operatorname{Hom}(A, A)$ has an element of infinite order, while every element in $\bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right)$ has finite order.
(iii) Prove that Hom $_{2}\left(\prod_{n \geq 2} \mathbb{I}_{n}, \mathbb{Q}\right) \neq \prod_{n \geq 2} \operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{I}_{n}, \mathbb{Q}\right)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
10:42

Problem 26

Let $R$ be a ring with IBN.
(i) If $R^{\infty}$ is a free left $R$-module having an infinite basis, prove that $R \oplus R^{\infty} \cong R^{\infty}$.
(ii) Prove that $R^{\infty} \not R^{n}$ for any $n \in \mathbb{N}$.
(iii) If $X$ is a set, denote the free left $R$-module $\bigoplus_{x \in X} R x$ by $R^{(X)}$. Let $X$ and $Y$ be sets, and let $R^{(X)} \cong R^{(Y)}$. If $X$ is infinite, prove that $Y$ is infinite and that $|X|=|Y|$; that is, $X$ and $Y$ have the same cardinal.
Hint. Since $X$ is a basis of $R^{(X)}$, each $u \in R^{(X)}$ has a unique expression $u=\sum_{x \in X} r_{X} x ;$ define
$$
\operatorname{Supp}(u)=\left\{x \in X: r_{x} \neq 0\right\}
$$
Given a basis $B$ of $R^{(X)}$ and a finite subset $W \subseteq X$, prove that there are only finitely many elements $b \in B$ with $\operatorname{Supp}(b) \subseteq W .$ Conclude that $|B|=\operatorname{Fin}(X)$, where Fin $(X)$ is the family of all the finite subsets of $X$. Finally, using the fact that $|\operatorname{Fin}(X)|=|X|$ when $X$ is infinite, conclude that $R^{(X)} \cong R^{(Y)}$ implies $|X|=|Y| .$

Mike Gaerlan
Mike Gaerlan
Numerade Educator
27:09

Problem 27

Let $V$ and $W$ be finite-dimensional vector spaces over a field $F$, say, and let $v_{1}, \ldots, v_{m}$ and $w_{1}, \ldots, w_{n}$ be bases of $V$ and $W$, respectively. Let $S: V \rightarrow V$ be a linear transformation having matrix $A=\left[a_{i j}\right]$, and let $T: W \rightarrow W$ be a linear transformation having matrix $B=\left[b_{k \ell}\right]$. Show that the matrix of $S \otimes T: V \otimes_{k} W \rightarrow$ $V \otimes_{k} W$, with respect to a suitable listing of the vectors $v_{i} \otimes w_{j}$, is the $n m \times n m$ matrix $K$, which we write in block form:
$$
A \otimes B=\left[\begin{array}{cccc}
a_{11} B & a_{12} B & \cdots & a_{1 m} B \\
a_{21} B & a_{22} B & \cdots & a_{2 m} B \\
\vdots & \vdots & \vdots & \vdots \\
a_{m 1} B & a_{m 2} B & \cdots & a_{m m} B
\end{array}\right]
$$
Remark. The matrix $A \otimes B$ is called the Kronecker product of the matrices $A$ and $B$.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
06:12

Problem 28

Let $R$ be a domain with $Q=\operatorname{Frac}(R)$, its field of fractions. If $A$ is an $R$-module, prove that every element in $Q \otimes_{R} A$ has the form $q \otimes a$ for $q \in Q$ and $a \in A$ (instead of $\sum_{i} q_{i} \otimes a_{i}$ ). (Compare this result with Example 2.67.)

Linda Hand
Linda Hand
Numerade Educator
01:40

Problem 29

(i) Let $p$ be a prime, and let $p, q$ be relatively prime. Prove that if $A$ is a $p$-primary group and $a \in A$, then there exists $x \in A$ with $q x=a$.
(ii) If $D$ is a finite cyclic group of order $m$, prove that $D / n D$ is a cyclic group of order $d=(m, n)$.
(iii) Let $m$ and $n$ be positive integers, and let $d=(m, n)$. Prove that there is an isomorphism of abelian groups
$$
\mathbb{I}_{m} \otimes \mathbb{I}_{n} \cong \mathbb{I}_{d} .
$$
(iv) Let $G$ and $H$ be finitely generated abelian groups, so that
$$
G=A_{1} \oplus \cdots \oplus A_{n} \quad \text { and } \quad H=B_{1} \oplus \cdots \oplus B_{m}
$$
where $A_{i}$ and $B_{j}$ are cyclic groups. Compute $G \otimes_{2} H$ explicitly.
Hint. $G \otimes_{\mathbb{Z}} H \cong \sum_{i, j} A_{i} \otimes_{\mathbb{Z}} B_{j}$. If $A_{i}$ or $B_{j}$ is infinite cyclic, use Proposition $2.58$; if both are finite, use part (ii).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
38:29

Problem 30

(i) Given $A_{R},{ }_{R} B_{S}$, and ${ }_{S} C$, define $T(A, B, C)=F / N$, where $F$ is the free abelian group on all ordered triples $(a, b, c) \in$ $A \times B \times C$, and $N$ is the subgroup generated by all
$$
(a r, b, c)-(a, r b, c),
$$
$$
\begin{gathered}
(a, b s, c)-(a, b, s c), \\
\left(a+a^{\prime}, b, c\right)-(a, b, c)-\left(a^{\prime}, b, c\right), \\
\left(a, b+b^{\prime}, c\right)-(a, b, c)-\left(a, b^{\prime}, c\right), \\
\left(a, b, c+c^{\prime}\right)-(a, b, c)-\left(a, b, c^{\prime}\right) .
\end{gathered}
$$
Define $h: A \times B \times C \rightarrow T(A, B, C)$ by $h:(a, b, c) \mapsto$ $a \otimes b \otimes c$, where $a \otimes b \otimes c=(a, b, c)+N$. Prove that this construction gives a solution to the universal mapping problem for triadditive functions.
(ii) Let $R$ be a commutative ring and let $A_{1}, \ldots, A_{n}, M$ be $R$-modules, where $n \geq 2$. An $R$-multilinear function is a function $h: A_{1} \times \cdots \times A_{n} \rightarrow M$ if $h$ is additive in each variable (when we fix the other $n-1$ variables), and $f\left(a_{1}, \ldots, r a_{i}, \ldots, a_{n}\right)=r f\left(a_{1}, \ldots, a_{i}, \ldots, a_{n}\right)$ for all $i$ and all $r \in R$. Let $F$ be the free $R$-module with basis $A_{1} \times \cdots \times A_{n}$, and define $N \subseteq F$ to be the submodule generated by all the elements of the form
$$
\left(a_{1}, \ldots, r a_{i}, \ldots, a_{n}\right)-r\left(a_{1}, \ldots, a_{i}, \ldots, a_{n}\right)
$$
and
$$
\left(\ldots, a_{i}+a_{i}^{\prime}, \ldots\right)-\left(\ldots, a_{i}, \ldots\right)-\left(\ldots, a_{i}^{\prime}, \ldots\right)
$$
Define $T\left(A_{1}, \ldots, A_{n}\right)=F / N$ and $h: A_{1} \times \cdots \times A_{n} \rightarrow$ $T\left(A_{1}, \ldots, A_{n}\right)$ by $\left(a_{1}, \ldots, a_{n}\right) \mapsto\left(a_{1}, \ldots, a_{n}\right)+N$. Prove that $h$ is $R$-multilinear, and that $h$ and $T\left(A_{1}, \ldots, A_{n}\right)$ solve the univeral mapping problem for $R$-multilinear functions.
(iii) Let $R$ be a commutative ring and prove generalized associativity for tensor products of $R$-modules.
Hint. Prove that any association of $A_{1} \otimes \cdots \otimes A_{n}$ is also a solution to the universal mapping problem.

Donald Albin
Donald Albin
Numerade Educator
01:05

Problem 31

Assume that the following diagram commutes, and that the vertical arrows are isomorphisms.
Prove that the bottom row is exact if and only if the top row is exact.

Anthony Ramos
Anthony Ramos
Numerade Educator
05:26

Problem 32

$(3 \times 3$ Lemma) Consider the following commutative diagram in ${ }_{R}$ Mod having exact columns.

If the bottom two rows are exact, prove that the top row is exact; if the top two rows are exact, prove that the bottom row is exact.

Wasim Sher
Wasim Sher
Numerade Educator
01:07

Problem 33

Consider the following commutative diagram in ${ }_{R}$ Mod having exact rows and columns.
If $A^{\prime \prime} \rightarrow B^{\prime \prime}$ and $B^{\prime} \rightarrow B$ are injections, prove that $C^{\prime} \rightarrow C$ is an injection. Similarly, if $C^{\prime} \rightarrow C$ and $A \rightarrow B$ are injections, then $A^{\prime \prime} \rightarrow B^{\prime \prime}$ is an injection. Conclude that if the last column and the second row are short exact sequences, then the third row is a short exact sequence and, similarly, if the bottom row and the second column are short exact sequences, then the third column is a short exact sequence.

Chandra Jain
Chandra Jain
Numerade Educator
00:38

Problem 34

Give an example of a commutative diagram with exact rows and vertical maps $h_{1}, h_{2}, h_{4}, h_{5}$ isomorphisms
for which there does not exist a map $h_{3}: A_{3} \rightarrow B_{3}$ making the diagram commute.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
14:32

Problem 35

If $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$ are categories, then a bifunctor $T: \mathcal{A} \times \mathcal{B} \rightarrow \mathcal{C}$ assigns, to each ordered pair of objects $(A, B)$, where $A \in \mathrm{ob}(\mathcal{A})$ and $B \in o b(\mathcal{B})$, an object $T(A, B) \in o b(\mathcal{C})$, and to each ordered pair of morphisms $f: A \rightarrow A^{\prime}$ in $\mathcal{A}$ and $g: B \rightarrow B^{\prime}$ in $\mathcal{B}$, a morphism $T(f, g): T(A, B) \rightarrow T\left(A^{\prime}, B^{\prime}\right)$, such that
(a) fixing either variable is a functor; for example, if $A \in \mathrm{ob}(\mathcal{A})$, then $T_{A}=T(A, \square): \mathcal{B} \rightarrow \mathcal{C}$ is a functor, where $T_{A}(B)=T(A, B)$ and $T_{A}(g)=T\left(1_{A}, g\right)$,
(b) the following diagram commutes:
$$
\begin{gathered}
T(A, B) \stackrel{T\left(1_{A}, g\right)}{\longrightarrow} T\left(A, B^{\prime}\right) \\
T\left(f, 1_{B}\right) \mid \\
T\left(A^{\prime}, B\right) \underset{T\left(1_{A^{\prime}}, g\right)}{\longrightarrow} T\left(A^{\prime}, B^{\prime}\right) .
\end{gathered}
$$
(i) Prove that $\otimes: \operatorname{Mod}_{R} \times{ }_{R} \operatorname{Mod} \rightarrow \mathbf{A b}$ is a bifunctor.
(ii) Prove that Hom: ${ }_{R}$ Mod $\times{ }_{R}$ Mod $\rightarrow \mathbf{A b}$ is a bifunctor if we modify the definition of bifunctor to allow contravariance in one variable.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:58

Problem 36

Let $R$ be a commutative ring, and let $F$ be a free $R$-module.
(i) If $\mathrm{m}$ is a maximal ideal in $R$, prove that $(R / \mathrm{m}) \otimes_{R} F$ and $F / \mathrm{m} F$ are isomorphic as vector spaces over $R / \mathrm{m}$.
(ii) Prove that $\operatorname{rank}(F)=\operatorname{dim}\left((R / \mathrm{m}) \otimes_{R} F\right)$.
(iii) If $R$ is a domain with fraction field $Q$, prove that $\operatorname{rank}(F)=$ $\operatorname{dim}\left(Q \otimes_{R} F\right)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:13

Problem 37

Assume that a ring $R$ has IBN; that is, if $R^{m} \cong R^{n}$ as left $R$ modules, then $m=n$. Prove that if $R^{m} \cong R^{n}$ as right $R$-modules, then $m=n$.
Hint. If $R^{m} \cong R^{n}$ as right $R$-modules, apply $\operatorname{Hom}_{R}(\square, R)$, using Proposition 2.54(iii).

Gideon Idumah
Gideon Idumah
Numerade Educator
03:58

Problem 38

Let $R$ be a domain and let $A$ be an $R$-module.
(i) Prove that if the multiplication $\mu_{r}: A \rightarrow A$ is an injection for all $r \neq 0$, then $A$ is torsion-free; that is, there are no nonzero $a \in A$ and $r \in R$ with $r a=0$.
(ii) Prove that if the multiplication $\mu_{r}: A \rightarrow A$ is a surjection for all $r \neq 0$, then $A$ is divisible.
(iii) Prove that if the multiplication $\mu_{r}: A \rightarrow A$ is an isomorphism for all $r \neq 0$, then $A$ is a vector space over $Q$, where $Q=\operatorname{Frac}(R)$.
Hint. A module $A$ is a vector space over $Q$ if and only if it is torsion-free and divisible.
(iv) If either $C$ or $A$ is a vector space over $Q$, prove that both $C \otimes_{R} A$ and $\operatorname{Hom}_{R}(C, A)$ are also vector spaces over $Q .$

Anthony Ramos
Anthony Ramos
Numerade Educator