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An Introduction to Homological Algebra

Joseph J. Rotman

Chapter 5

Setting the Stage - all with Video Answers

Educators

Chapter Questions

02:34

Problem 1

(i) Prove that $\varnothing$ is an initial object in Sets.
(ii) Prove that any one-point set $\Omega=\left\{x_{0}\right\}$ is a terminal object in Sets. In particular, what is the function $\varnothing \rightarrow \Omega ?$

Chris Trentman
Chris Trentman
Numerade Educator
05:40

Problem 2

A zero object in a category $\mathcal{C}$ is an object that is both an initial object and a terminal object.
(i) Prove the uniqueness to isomorphism of initial, terminal, and zero objects, if they exist.
(ii) Prove that $\{0\}$ is a zero object in ${ }_{R}$ Mod and that $\{1\}$ is a zero object in Groups.
(iii) Prove that neither Sets nor Top has a zero object.
(iv) Prove that if $A=\{a\}$ is a set with one element, then $(A, a)$ is a zero object in Sets $_{*}$, the category of pointed sets. If $A$ is given the discrete topology, prove that $(A, a)$ is a zero object in Top $_{*}$, the category of pointed topological spaces.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:47

Problem 3

(i) Prove that the zero ring is not an initial object in ComRings.
(ii) If $k$ is a commutative ring, prove that $k$ is an initial object in ComAlg $_{k}$, the category of all commutative $k$-algebras.
(iii) In ComRings, prove that $\mathbb{Z}$ is an initial object and that the zero ring $\{0\}$ is a terminal object.

Willis James
Willis James
Numerade Educator
01:34

Problem 4

For every commutative ring $k$, prove that the direct product $R \times S$ is the categorical product in ComAlg $_{k}$ (in particular, direct product is the categorical product in ComAlg $_{Z}=$ ComRings).

Narayan Hari
Narayan Hari
Numerade Educator
03:13

Problem 5

Let $k$ be a commutative ring.
(i) Prove that $k[x, y]$ is a free commutative $k$-algebra with basis $\{x, y\}$.
(ii) Use Proposition $5.2$ to prove that $k[x] \otimes_{k} k[y]$ is a free $k$ algebra with basis $\{x, y\}$.
(iii) Use Proposition $5.4$ to prove that $k[x] \otimes_{k} k[y] \cong k[x, y]$ as $k$-algebras.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:14

Problem 6

(i) Let $Y$ be a set, and let $\mathcal{P}(Y)$ denote its power set; that is, $\mathcal{P}(Y)$ is the partially ordered set of all the subsets of $Y$. As in Example 1.3(iii), view $\mathcal{P}(Y)$ as a category. If $A, B \in$ $\mathcal{P}(Y)$, prove that the coproduct $A \sqcup B=A \cup B$ and that the product $A \sqcap B=A \cap B$.
(ii) Generalize part (i) as follows. If $X$ is a partially ordered set viewed as a category, and $a, b \in X$, prove that the coproduct $a \sqcup b$ is the least upper bound of $a$ and $b$, and that the product $a \sqcap b$ is the greatest lower bound.
(iii) Give an example of a category in which there are two objects whose coproduct does not exist.

Manisha Sarker
Manisha Sarker
Numerade Educator
09:31

Problem 7

Define the wedge of pointed spaces $\left(X, x_{0}\right),\left(Y, y_{0}\right) \in \operatorname{Top}_{*}$ to be $\left(X \vee Y, z_{0}\right)$, where $X \vee Y$ is the quotient space of the disjoint union $X \sqcup Y$ in which the basepoints are identified to $z_{0}$. Prove that wedge is the coproduct in $\mathbf{T o p}_{*}$.

Chris Trentman
Chris Trentman
Numerade Educator
02:25

Problem 8

Give an example of a covariant functor that does not preserve coproducts.

Mj Santos
Mj Santos
Numerade Educator
01:35

Problem 9

If $A$ and $B$ are (not necessarily abelian) groups, prove that $A \cap B=$ $A \times B$ (direct product) in Groups. For readers familiar with group theory, prove that $A \sqcup B=A * B$ (free product) in Groups.

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 10

(i) Given a pushout diagram in ${ }_{R}$ Mod:
prove that $g$ injective implies $\alpha$ injective and that $g$ surjective implies $\alpha$ surjective. Thus, parallel arrows have the same properties.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:21

Problem 11

(i) Assuming that coproducts exist, prove commutativity:
$$
A \sqcup B \cong B \sqcup A .
$$
(ii) Assuming that coproducts exist, prove associativity:
$$
A \sqcup(B \sqcup C) \cong(A \sqcup B) \sqcup C .
$$

Urvashi Arora
Urvashi Arora
Numerade Educator
03:06

Problem 12

(i) Assuming that products exist, prove commutativity:
$$
A \sqcap B \cong B \sqcap A .
$$
(ii) Assuming that products exist, prove associativity:
$$
A \sqcap(B \sqcap C) \cong(A \sqcap B) \sqcap C .
$$

Chandra Jain
Chandra Jain
Numerade Educator
01:05

Problem 13

(i) If $\Omega$ is a terminal object in a category $\mathcal{C}$, prove, for any $G \in \operatorname{obj}(\mathcal{C})$, that the projections $\lambda: G \sqcap \Omega \rightarrow G$ and $\rho: \Omega \sqcap G \rightarrow G$ are isomorphisms.
(ii) If $A$ is an initial object in a category $\mathcal{C}$, prove, for any $G \in$ obj $(\mathcal{C})$, that the injections $\lambda: G \rightarrow G \sqcup \Omega$ and $\rho: G \rightarrow$ $\Omega \sqcup G$ are isomorphisms.

Anthony Ramos
Anthony Ramos
Numerade Educator
14:32

Problem 14

Let $C_{1}, C_{2}, D_{1}, D_{2}$ be objects in a category $\mathcal{C}$.
(i) If there are morphisms $f_{i}: C_{i} \rightarrow D_{i}$, for $i=1,2$, and if $C_{1} \sqcap C_{2}$ and $D_{1} \sqcap D_{2}$ exist, prove that there exists a unique morphism $f_{1} \sqcap f_{2}$ making the following diagram commute for $i=1,2$ :
where $p_{i}$ and $q_{i}$ are projections.
(ii) If there are morphisms $g_{i}: X \rightarrow C_{i}$, where $X$ is an object in $\mathcal{C}$ and $i=1,2$, prove that there is a unique morphism $\left(g_{1}, g_{2}\right)$ making the following diagram commute:
where the $p_{i}$ are projections.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:07

Problem 15

Let $\mathcal{C}$ be a category having finite products and a terminal object $\Omega$. A group object in $\mathcal{C}$ is a quadruple $(G, \mu, \eta, \epsilon)$, where $G$ is an object in $\mathcal{C}, \mu: G \prod G \rightarrow G, \eta: G \rightarrow G$, and $\epsilon: \Omega \rightarrow G$ are morphisms, so that the following diagrams commute:
Associativity:
where $\omega: G \rightarrow \Omega$ is the unique morphism to the terminal object.
(i) Prove that a group object in Sets is a group.
(ii) Prove that a group object in Groups is an abelian group.
(iii) Define a morphism between group objects in a category $\mathcal{C}$, and prove that all the group objects form a subcategory of $\mathcal{C}$.
(iv) Define the dual notion cogroup object, and prove the dual of (iii).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:12

Problem 16

Prove that every left exact covariant functor $T: R$ Mod $\rightarrow$ Ab preserves pullbacks. Conclude that if $B$ and $C$ are submodules of a module $A$, then for every module $M$, we have
$$
\operatorname{Hom}_{R}(M, B \cap C)=\operatorname{Hom}_{R}(M, B) \cap \operatorname{Hom}_{R}(M, C) .
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
08:25

Problem 17

(i) Let $\left(A_{n}\right)_{n \in \mathbb{N}}$ be a family of isomorphic abelian groups; say, $A_{n} \cong A$ for all $n$. Consider inverse systems $\left\{A_{n}, f_{n}^{m}\right\}$ and $\left\{A_{n}, g_{n}^{m}\right\}$, where each $f_{n}^{m}=0$ and each $g_{n}^{m}$ is an isomorphism. Prove that the inverse limit of the first inverse system is $\{0\}$ while the inverse limit of the second inverse system is $A$. Conclude that inverse limits depend on the morphisms in the inverse systems.
(ii) Give an example of two direct systems having the same abelian groups and whose direct limits are not isomorphic.

Ely Crowder
Ely Crowder
Numerade Educator
01:23

Problem 18

Let $\left\{M_{i}, \varphi_{j}^{i}\right\}$ be a direct system of $R$-modules over an index set $I$, and let $F:{ }_{R} \operatorname{Mod} \rightarrow \mathcal{C}$ be a functor to some category $\mathcal{C}$. Prove that $\left\{F M_{i}, F \varphi_{j}^{i}\right\}$ is a direct system in $\mathcal{C}$ if $F$ is covariant, while it is an inverse system if $F$ is contravariant.

Raj Bala
Raj Bala
Numerade Educator
01:59

Problem 19

Give an example of a direct system of modules, $\left\{A_{i}, \alpha_{j}^{i}\right\}$, over some directed index set $I$, for which $A_{i} \neq\{0\}$ for all $i$ and $\lim _{\rightarrow} A_{i}=\{0\}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:59

Problem 20

(i) Prove that End $\left(\mathbb{Z}\left(p^{\infty}\right)\right) \cong \mathbb{Z}_{p}$ as rings, where $\mathbb{Z}_{p}$ is the ring of $p$-adic integers.
$$
\left(a_{0}, a_{1}, a_{2} \ldots, \mid p a_{0}=0, p a_{n}=a_{n-1} \text { for } n \geq 1\right)
$$
(ii) Prove that the additive group of $\mathbb{Z}_{p}$ is torsion-free.

Aayush Gupta
Aayush Gupta
Numerade Educator
09:31

Problem 21

Let $0 \rightarrow U \rightarrow V \rightarrow V / U \rightarrow 0$ be an exact sequence of left $R$-modules.
(i) Let $\left\{U_{i}, \alpha_{j}^{i}\right\}$ be a direct system of submodules of $U$, where $\left(\alpha_{j}^{i}: U_{i} \rightarrow U_{j}\right)_{i \leq j}$ are inclusions. Prove that $\left\{V / U_{i}, e_{j}^{i}\right\}$ is a direct system, where each $e_{j}^{i}: V / U_{i} \rightarrow V / U_{j}$ is enlargement of coset.
(ii) If $\underline{\lim } U_{i}=U$, prove that $\lim _{\rightarrow}\left(V / U_{i}\right) \cong V / U$.

Chris Trentman
Chris Trentman
Numerade Educator
02:16

Problem 22

(i) Let $K$ be a cofinal subset of a directed index set $I$ (that is, for each $i \in I$, there is $k \in K$ with $i \preceq k$ ). Let $\left\{M_{i}, \varphi_{j}^{i}\right\}$ be a direct system over $I$, and let $\left\{M_{i}, \varphi_{j}^{i}\right\}$ be the subdirect system whose indices lie in $K$. Prove that the direct limit over $I$ is isomorphic to the direct limit over $K$.
(ii) Let $K$ be a cofinal subset of a directed index set $I$, let $\left\{M_{i}, \varphi_{j}^{i}\right\}$ be an inverse system over $I$, and let $\left\{M_{i}, \varphi_{j}^{i}\right\}$ be the subinverse system whose indices lie in $K$. Prove that the inverse limit over $I$ is isomorphic to the inverse limit over $K$.
(iii) A partially ordered set $I$ has a top element if there exists $\infty \in I$ with $i \preceq \infty$ for all $i \in I$. If $\left\{M_{i}, \varphi_{j}^{i}\right\}$ is a direct system over $I$, prove that
$$
\lim _{\longrightarrow} M_{i} \cong M_{\infty}
$$
(iv) Show that part (i) may not be true if the index set is not directed.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:19

Problem 23

Prove that a ring $R$ is left noetherian if and only if every direct limit (with directed index set) of injective left $R$-modules is itself injective.

James Chok
James Chok
Numerade Educator
01:49

Problem 24

Let
be a pullback diagram in $\mathbf{A b}$. If there are $c \in C$ and $b \in B$ with $g c=f b$, prove that there exists $d \in D$ with $c \alpha(d)$ and $b=\beta(d)$.

Nishant Tyagi
Nishant Tyagi
Numerade Educator
01:03

Problem 25

Consider the ideal $J=(x)$ in $k[x]$, where $k$ is a commutative ring. Prove that the completion $\lim \left(k[x] / J^{n}\right)$ of the polynomial ring $k[x]$ is $k[[x]]$, the ring of formai power series.

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 26

In $_{R}$ Mod, let $r:\left\{A_{i}, \alpha_{j}^{i}\right\} \rightarrow\left\{B_{i}, \beta_{j}^{i}\right\}$ and $s:\left\{B_{i}, \beta_{j}^{i}\right\} \rightarrow\left\{C_{i}, \gamma_{j}^{i}\right\}$ be morphisms of inverse systems over any (not necessarily directed) index set $I$. If
$$
0 \rightarrow A_{i} \stackrel{r_{i}}{\rightarrow} B_{i} \stackrel{s i}{\rightarrow} C_{i}
$$
is exact for each $i \in I$, prove that there are homomorphisms $\overleftarrow{r}, \overleftarrow{s}$ given by the universal property of inverse limits, and an exact sequence
$$
0 \rightarrow \lim _{\longleftarrow} A_{i} \stackrel{\overleftarrow}{r} \lim _{\longleftarrow} B_{i} \stackrel{\leftarrow}{\leftrightarrow} \lim C_{i}
$$

Carson Merrill
Carson Merrill
Numerade Educator
02:39

Problem 27

A category $\mathcal{C}$ is complete if $\lim _{\longleftarrow} A_{i}$ exists in $\mathcal{C}$ for every inverse system $\left\{A_{i}, \psi_{i}^{j}\right\}$ in $\mathcal{C}$; a category $\mathcal{C}$ is cocomplete if $\underline{\lim } A_{i}$ exists in $\mathcal{C}$ for every direct system $\left\{A_{i}, \varphi_{j}^{i}\right\}$ in $\mathcal{C}$.

Prove that a category is complete if and only if it has equalizers and products (over any index set). Dually, prove that a category is cocomplete if and only if it has coequalizers and coproducts (over any index set).

James Chok
James Chok
Numerade Educator
01:13

Problem 28

Prove that if $T:{ }_{R} \mathbf{M o d} \rightarrow \mathbf{A} \mathbf{b}$ is an additive left exact functor preserving direct products, then $T$ preserves inverse limits.

Arun Bana
Arun Bana
Numerade Educator
02:41

Problem 29

Give an example of an additive functor $H: \mathbf{A b} \rightarrow \mathbf{A b}$ that has neither a left nor a right adjoint.

Monica Miller
Monica Miller
Numerade Educator
01:48

Problem 30

Let $(F, G)$ be an adjoint pair, where $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{D} \rightarrow \mathcal{C}$, and let $\tau_{C, D}: \operatorname{Hom}(F C, D) \rightarrow \operatorname{Hom}(C, G C)$ be the natural bijection.
(i) If $D=F C$, there is a natural bijection
$$
\tau_{C, F C}: \operatorname{Hom}(F C, F C) \rightarrow \operatorname{Hom}(C, G F C)
$$
with $\tau\left(1_{F C}\right)=\eta_{C}: C \rightarrow G F C$. Prove that $\eta: 1_{C} \rightarrow G F$ is a natural transformation.
(ii) If $C=G D$, there is a natural bijection
$$
\tau_{G D, D}^{-1}: \operatorname{Hom}(G D, G D) \rightarrow \operatorname{Hom}(F G D, D)
$$
with $\tau^{-1}\left(1_{D}\right)=\varepsilon_{D}: F G D \rightarrow D .$ Prove that $\varepsilon: F G \rightarrow$ $1_{\mathcal{D}}$ is a natural transformation. (We call $\varepsilon$ the counit of the adjoint pair.)

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
04:07

Problem 31

Let $(F, G)$ be an adjoint pair of functors between module categories. Prove that if $G$ is exact, then $F$ preserves projectives; that is, if $P$ is a projective module, then $F P$ is projective. Dually, prove that if $F$ is exact, then $G$ preserves injectives.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:02

Problem 32

(i) Let $F:$ Groups $\rightarrow$ Ab be the functor with $F(G)=G / G^{\prime}$, where $G^{\prime}$ is the commutator subgroup of a group $G$, and let $U:$ Ab $\rightarrow$ Groups be the functor taking every abelian group $A$ into itself (that is, UA regards $A$ as a not necessarily abelian group). Prove that $(F, U)$ is an adjoint pair of functors.
(ii) Prove that the unit of the adjoint pair $(F, U)$ is the natural $\operatorname{map} G \rightarrow G / G^{\prime}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:17

Problem 33

(i) If $I$ is a partially ordered set, let $\operatorname{Dir}(I, R$ Mod) denote all direct systems of left $R$-modules over $I$. Prove that $\operatorname{Dir}(I, R$ Mod) is a category and that $\underline{\lim }: \operatorname{Dir}(I, R \operatorname{Mod}) \rightarrow$ $R$ Mod is a functor.
(ii) In Example $1.19$ (ii), we saw that constant functors define a functor $|\square|: \mathcal{C} \rightarrow \mathcal{C}^{\mathcal{D}}$; to each object $C$ in $\mathcal{C}$ assign the constant functor $|C|$, and to each morphism $\varphi: C \rightarrow C^{\prime}$ in $\mathcal{C}$, assign the natural transformation $|\varphi|:|C| \rightarrow\left|C^{\prime}\right|$ defined by $|\varphi|_{D}=\varphi .$ If $\mathcal{C}$ is cocomplete, prove that $\left(\lim _{\longrightarrow}|\square|\right)$ is an adjoint pair, and conclude that $\underline{\lim }$ preserves direct limits.
(iii) Let $I$ be a partially ordered set and let $\operatorname{Inv}(I, R$ Mod) denote the class of all inverse systems, together with their morphisms, of left $R$-modules over $I$. Prove that Inv( $I, R$ Mod) is a category and that $\lim : \operatorname{Inv}\left(I,{ }_{R} \mathbf{M o d}\right) \rightarrow{ }_{R} \mathbf{M o d}$ is a functor.
(iv) Prove that if $\mathcal{C}$ is complete, then $(|\square|, \lim )$ is an adjoint pair and lim preserves inverse limits.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
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Problem 34

(i) If $A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \cdots$ is an ascending sequence of submodules of a module $A$, prove that $A / \bigcup A_{i} \cong \bigcup A / A_{i} ;$ that is, coker $\left(\lim _{\rightarrow} A_{i} \subseteq A\right) \cong \underline{\lim } \operatorname{coker}\left(A_{i} \rightarrow A\right)$.
(ii) Generalize part (i): prove that any two direct limits (perhaps with distinct index sets) commute.
(iii) Prove that any two inverse limits (perhaps with distinct index sets) commute.
(iv) Give an example in which direct limit and inverse limit do not commute.

Nick Johnson
Nick Johnson
Numerade Educator
03:58

Problem 35

(i) Define $\mathrm{ACC}$ in $_{R}$ Mod, and prove that if $S$ Mod $\cong{R}$ Mod, then ${ }_{S}$ Mod has ACC. Conclude that if $R$ is left noetherian, then $S$ is left noetherian.
(ii) Give an example showing that ${ }_{R}$ Mod and Mod $_{R}$ are not isomorphic.

Anthony Ramos
Anthony Ramos
Numerade Educator
14:32

Problem 36

(i) Recall that a cogenerator of a category $\mathcal{C}$ is an object $C$ such that $\operatorname{Hom}(\square, C): \mathcal{C} \rightarrow$ Sets is a faithful functor; that is, if $f, g: A \rightarrow B$ are distinct morphisms in $\mathcal{C}$, then there exists a morphism $h: B \rightarrow C$ with $h f \neq h g$. Prove, when $\mathcal{C}={ }_{R}$ Mod, that this definition coincides with the definition of cogenerator on page 264 .
(ii) A generator of a category $\mathcal{C}$ is an object $G$ such that $\operatorname{Hom}(G, \square): \mathcal{C} \rightarrow$ Sets is a faithful functor; that is, if $f, g: A \rightarrow B$ are distinct morphisms in $\mathcal{C}$, then there exists a morphism $h: G \rightarrow A$ with $f h \neq g h$. Prove, when $\mathcal{C}={ }_{R}$ Mod, that this definition coincides with the definition of cogenerator on page 269 .

Anthony Ramos
Anthony Ramos
Numerade Educator
01:05

Problem 37

We call a functor $F: \mathcal{A} \rightarrow \mathcal{B}$ a strong isomorphism if there exists a functor $G: \mathcal{B} \rightarrow \mathcal{A}$ with $G F=1_{\mathcal{A}}$ and $F G=1_{\mathcal{B}}$. If $R$ is a ring, show that $\operatorname{Hom}_{R}(R, \square):{ }_{R} \operatorname{Mod} \rightarrow{ }_{R}$ Mod (which is naturally isomorphic to $1_{R \text { Mod }}$, by Exercise $2.13$ on page 66 ) is not a strong isomorphism. Conclude that strong isomorphism is not an interesting idea.

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 38

(i) Prove that the zero sheaf is a zero object in $\mathbf{S h}(X, \mathbf{A b})$ and in $\mathbf{p S h}(X, \mathbf{A b})$.
(ii) Prove that $\operatorname{Hom}\left(\mathcal{P}, \mathcal{P}^{\prime}\right)$ is an additive abelian group when $\mathcal{P}, \mathcal{P}^{\prime}$ are presheaves or when $\mathcal{P}, \mathcal{P}^{\prime}$ are sheaves.
(iii) The distributive laws hold: given presheaf maps
$$
\mathcal{X} \stackrel{\alpha}{\longrightarrow} \mathcal{P} \underset{\psi}{\stackrel{\varphi}{=}} \mathcal{Q} \stackrel{\beta}{\longrightarrow} \mathcal{Y},
$$
where $\mathcal{X}$ and $\mathcal{Y}$ are presheaves over a space $X$, prove that
$$
\beta(\varphi+\psi)=\beta \varphi+\beta \psi \text { and }(\varphi+\psi) \alpha=\varphi \alpha+\psi \alpha \text {. }
$$

Nick Johnson
Nick Johnson
Numerade Educator
03:13

Problem 39

Let $(E, p, X)$ be an etale-sheaf, and let $\mathcal{F}$ be its sheaf of sections.
(i) Prove that a subset $G \subseteq E$ is a sheet if and only if $G=$ $\sigma(U)$ for some open $U \subseteq X$ and $\sigma \in \mathcal{F}(U)$.
(ii) Prove that $G \subseteq E$ is a sheet if and only if $G$ is an open subset of $E$ and $p \mid G$ is a homeomorphism.
(iii) If $G=\sigma(U)$ and $H=\tau(V)$ are sheets, where $\sigma \in \mathcal{F}(U)$ and $\tau \in \mathcal{F}(V)$, prove that $G \cap H$ is a sheet.
(iv) If $\sigma \in \mathcal{F}(U)$, prove that
$$
\operatorname{supp}(\sigma)=\left\{x \in X: \sigma(x) \neq 0_{x} \in E_{x}\right\}
$$
is a closed subset of $X$.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:40

Problem 40

Prove that an etale-map $\varphi:(E, p, X) \rightarrow\left(E^{\prime}, p^{\prime}, X\right)$ is an isomorphism in $\mathbf{S h}_{\mathrm{et}}(X, \mathbf{A b})$ if and only if $\varphi: E \rightarrow E^{\prime}$ is a homeomorphism.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:09

Problem 41

Let $\varphi: \mathcal{P} \rightarrow \mathcal{P}^{\prime}$ be a presheaf map. Prove that the following statements are equivalent:
(i) $\varphi$ is an isomorphism;
(ii) $\varphi \mid \mathcal{P}(U): \mathcal{P}(U) \rightarrow \mathcal{P}^{\prime}(U)$ is an isomorphism for every open setU;
(iii) $\varphi \mid \mathcal{P}(U): \mathcal{P}(U) \rightarrow \mathcal{P}^{\prime}(U)$ is a bijection for every open set $U$.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
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Problem 42

Prove that every presheaf of abelian groups $\mathcal{P}$ on a discrete space $X$ is a sheaf.

Nick Johnson
Nick Johnson
Numerade Educator
03:27

Problem 43

If $X=\{x\}$ is a space with only one point, prove that
$$
\mathbf{p S h}(X, \mathbf{A} \mathbf{b})=\mathbf{S h}(X, \mathbf{A} \mathbf{b}) \cong \mathbf{A} \mathbf{b} .
$$

James Chok
James Chok
Numerade Educator
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Problem 44

Let $x_{*} A$ be a skyscraper sheaf, as in Example $5.72$.
(i) Prove, for every sheaf $\mathcal{G}$, that there is an isomorphism
$$
\operatorname{Hom}_{\mathbb{Z}}\left(\mathcal{G}_{x}, A\right) \cong \operatorname{Hom}_{S h}\left(\mathcal{G}, x_{*} A\right)
$$
that is natural in $\mathcal{G}$.
(ii) Every sheaf map $\varphi: \mathcal{F} \rightarrow \mathcal{G}$ induces homomorphisms of stalks $\varphi_{y}: \mathcal{F}_{y} \rightarrow \mathcal{G}_{y}$ for all $y \in X .$ Choose $x \in X .$ If $\mathcal{F}$ is a sheaf over $X$ with stalk $\mathcal{F}_{x}=A$, prove that there is a sheaf map $\varphi: \mathcal{F} \rightarrow x_{*} A$ with $\varphi_{x}=1_{A}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:02

Problem 45

Let $v: \mathcal{P} \rightarrow \Gamma\left(\square, \mathcal{P}^{\text {et }}\right)$ be the natural map in Theorem 5.68: in the notation of this proposition, if $U$ is an open set in $X$, then $v_{U}: \mathcal{P}(U) \rightarrow \Gamma\left(U, \mathcal{P}^{\mathrm{et}}\right)$ is given by $\sigma \mapsto \sigma^{\mathrm{et}}$. If $x \in X$, prove that $v_{x}: \sigma(x) \mapsto \sigma^{\text {et }}(x)=\sigma(x)$.

Raj Bala
Raj Bala
Numerade Educator
03:58

Problem 46

Let $X$ be a topological space and let $\mathcal{B}$ be a base for the topology $\mathcal{U}$ on $X$. Viewing $\mathcal{B}$ as a partially ordered set, we may define a presheaf on $\mathcal{B}$ to be a contravariant functor $\mathcal{Q}: \mathcal{B} \rightarrow \mathbf{A b}$. Prove that $\mathcal{Q}$ can be extended to a presheaf $\widetilde{\mathcal{Q}}: \mathcal{U} \rightarrow \mathbf{A} \mathbf{b}$ by defining
$$
\widetilde{\mathcal{Q}}(U)=\lim _{V \in \mathcal{B} \atop V \subseteq U} \mathcal{Q}(V) .
$$
If $U \in \mathcal{B}$, prove that $\widetilde{\mathcal{Q}}(U)$ is canonically isomorphic to $\mathcal{Q}(U)$.

Anthony Ramos
Anthony Ramos
Numerade Educator
08:25

Problem 47

(i) If $f: A \rightarrow B$ is a homomorphism in $\mathbf{A b}$ and $K=\operatorname{ker} f$ is the usual kernel (which is a subgroup!), prove that the inclusion $i: K \rightarrow A$ is a categorical kernel of $f$.
(ii) If $f: A \rightarrow B$ is a homomorphism in $\mathbf{A b}$ and $C=B / \operatorname{im} f$ is the usual cokernel (which is a quotient group), then the natural map $p: B \rightarrow C$ is a categorical cokernel of $f$. Note that $\operatorname{im} f=\operatorname{ker} p$.

Ely Crowder
Ely Crowder
Numerade Educator
01:58

Problem 48

Let $\mathcal{S}=(E, p, X)$ be an etale-sheaf and let $\mathcal{G}=(G, p \mid G, X)$, where $G \subseteq E$. Prove that $\Gamma(\square, \mathcal{G})$ is a subsheaf of $\Gamma(\square, \mathcal{S})$ if and only if $G$ is open in $E$ and $G_{x}=G \cap E_{x}$ is a subgroup for all $x \in X$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:13

Problem 49

If $\mathcal{F}$ is a subsheaf of a sheaf $\mathcal{G}$, define the quotient sheaf as $(\mathcal{G} / \mathcal{F})^{*}$, the sheafification of the presheaf $\mathcal{G} / \mathcal{F}$. Define the natural map to be the composite $\pi: \mathcal{G} \rightarrow \mathcal{G} / \mathcal{F} \rightarrow(\mathcal{G} / \mathcal{F})^{*}$. Prove that if $\iota: \mathcal{F} \rightarrow \mathcal{G}$ is the inclusion, then the natural map is coker $\iota .$

Gideon Idumah
Gideon Idumah
Numerade Educator
02:00

Problem 50

Denote the sheafification functor $\mathrm{pSh}(X, \mathbf{A b}) \rightarrow \operatorname{Sh}(X, \mathbf{A b})$ by $\mathcal{P} \mapsto \mathcal{P}^{*}$. Prove that $^{*}$ is left adjoint to the inclusion functor $\operatorname{Sh}(X, \mathbf{A b}) \rightarrow \mathbf{p S h}(X, \mathbf{A} \mathbf{b})$. [Either prove this directly or use the fact that $f_{*} f^{*}$ is the unit of the adjoint pair $\left.\left(f_{*}, f^{*}\right) .\right]$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
06:02

Problem 51

If $\mathcal{C}$ is an additive category with zero object 0 , prove that the unique morphism $A \rightarrow 0$ [where $A \in \operatorname{obj}(\mathcal{C})$ ] and the unique morphism $0 \rightarrow A$ are the identity elements of the abelian groups $\operatorname{Hom}_{C}(A, 0)$ and $\operatorname{Hom}_{\mathcal{C}}(0, A)$.

Ely Crowder
Ely Crowder
Numerade Educator
01:07

Problem 52

If $\mathcal{C}$ is an additive category and $C \in \mathrm{obj}(\mathcal{C})$, prove that $\operatorname{Hom}(C, C)$ is a ring with composition as product.

Carson Merrill
Carson Merrill
Numerade Educator
08:25

Problem 53

In any category having a zero object, prove that every kernel is a monomorphism and, dually, every cokernel is an epimorphism.

Ely Crowder
Ely Crowder
Numerade Educator
02:36

Problem 54

Let $\mathcal{C}$ be an additive category and let $\mathcal{S}$ be a subcategory. Prove that $\mathcal{S}$ is an additive category if $\mathcal{S}$ is full, contains a zero object of $\mathcal{C}$, and contains the direct sum $A \oplus B$ (in $\mathcal{C}$ ) of all $A, B \in \operatorname{obj}(\mathcal{S})$.

James Chok
James Chok
Numerade Educator
01:40

Problem 55

(i) Prove that a function is epic in Sets if and only if it is surjective and that a function is monic in Sets if and only if it is injective.
(ii) Prove that an $R$-map is epic in ${ }_{R}$ Mod if and only if it is surjective and that an $R$-map is monic in ${ }_{R}$ Mod if and only if it is injective.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:50

Problem 56

Let $\mathcal{C}$ be the category of all divisible abelian groups.
(i) Prove that the natural map $\mathbb{Q} \rightarrow \mathbb{Q} / \mathbb{Z}$ is monic in $\mathcal{C}$.
(ii) Conclude that $\mathcal{C}$ is a concrete category in which monomorphisms and injections do not coincide.

Ely Crowder
Ely Crowder
Numerade Educator
03:33

Problem 57

Prove, in every category, that the injections of a coproduct are monic and the projections of a product are epic.

Donald Albin
Donald Albin
Numerade Educator
01:05

Problem 58

(i) Prove that every isomorphism in an additive category is both monic and epic.
(ii) Prove that a morphism in an abelian category is an isomorphism if and only if it is both monic and epic.
(iii) Prove, in ComRings, that $\varphi: R \rightarrow \operatorname{Frac}(R)$ is both monic and epic, but that $\varphi$ is not an isomorphism.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:02

Problem 59

(Eilenberg-Moore) Let $G$ be a (possibly nonabelian) group.
(i) If $H$ is a proper subgroup of a group $G$, prove that there exist a group $L$ and distinct homomorphisms $f, g: G \rightarrow L$ with $f|H=g| H$.
Hint. Define $L=S_{X}$, where $X$ denotes the family of all the left cosets of $H$ in $G$ together with an additional element, denoted $\infty$. If $a \in G$, define $f(a)=f_{a} \in S_{X}$ by $f_{a}(\infty)=$ $\infty$ and $f_{a}(b H)=a b H$. Define $g: G \rightarrow S_{X}$ by $g=\gamma f$, where $\gamma \in S_{X}$ is conjugation by the transposition $(H, \infty)$.
(ii) Prove that a homomorphism $\varphi: A \rightarrow G$, where $A$ and $G$ are groups, is surjective if and only if it is an epimorphism in Groups.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:46

Problem 60

We call $\lim _{I} F$ or $\lim _{I} F$ finite if the index set $I$ is finite. Prove that if $\mathcal{A}$ is an additive category having kernels and cokernels, then $\mathcal{A}$ has all finite inverse limits and direct limits. Conclude that $\mathcal{A}$ has pullbacks, pushouts, equalizers, and coequalizers.

R M
R M
Numerade Educator
01:05

Problem 61

State and prove the First Isomorphism Theorem in an abelian category $A$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:16

Problem 62

Prove that every object in Sets is projective and injective.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:22

Problem 63

(i) Let $X$ be a set and, for each subset $Y \subseteq X$, let $i_{Y}: Y \rightarrow$ $X$ be the inclusion. If $2^{X}$ is the family of all subsets of $X$, prove that the function $2^{X} \rightarrow\left\{\left[i_{y}\right]: Y \subseteq X\right\}$, given
by $Y \mapsto\left[i_{Y}\right]$, is a bijection, where $\left[i_{Y}\right]$ is the categorical subobject of $X$.
(ii) Prove that the analog of (i) is true for Groups, Rings, and ${ }_{R}$ Mod, but it is false for Top.

Adriano Chikande
Adriano Chikande
Numerade Educator
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Problem 64

(i) Prove that the category $\mathcal{T}$ of all torsion abelian groups is an abelian category having no nonzero projective objects.
(ii) Prove, for every index set $I$, that $\mathcal{T}$ has a product $\prod_{i \in I} G_{i}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:35

Problem 65

(i) Prove that the full subcategory $\mathcal{T}$ of $\mathbf{A b}$ consisting of all torsion abelian groups is an abelian category that is closed under (infinite) coproducts.
(ii) Prove that $\mathcal{T}$ has enough injectives.
(iii) Prove that $\mathcal{T}$ has no nonzero projective objects. Conclude that $\mathcal{T}$ is not isomorphic to a category of modules.

Nick Johnson
Nick Johnson
Numerade Educator
06:51

Problem 66

If $\mathcal{A}$ is an abelian category, prove that a morphism $f=\left(f_{n}\right)$ in $\operatorname{Comp}(\mathcal{A})$ [i.e., a chain map] is monic (or epic) if and only if each $f_{n}$ is monic (or epic) in $\mathcal{A}$.

Linda Hand
Linda Hand
Numerade Educator
03:26

Problem 67

Let $\mathcal{A}$ be an abelian category with enough projectives, and let $\mathcal{C} \subseteq$ obj $(\mathcal{A})$ satisfy
(i) for every object $A$ in $\mathcal{A}$, there exists $C \in \mathbb{C}$ and an epimorphism $C \rightarrow A$;
(ii) if $C \in C{C}$, then every direct summand of $C$ also lies in $C$.
Prove that every projective lies in $\mathfrak{C}$. The dual result also holds.

Anthony Ramos
Anthony Ramos
Numerade Educator