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An Introduction to Homological Algebra

Joseph J. Rotman

Chapter 10

Spectral Sequences - all with Video Answers

Educators

Chapter Questions

06:51

Problem 1

If $\left(M, d^{\prime}, d^{\prime \prime}\right)$ is a bicomplex of left $R$-modules, and if $F: R$ Mod $\rightarrow$ ${ }_{S} \mathbf{M o d}$ is an additive functor, prove that $\left(F M, F d^{\prime}, F d^{\prime \prime}\right)$ is a bicomplex of left $S$-modules.

Linda Hand
Linda Hand
Numerade Educator
01:28

Problem 2

(i) For a ring $R$ and a fixed $k \geq 0$, prove that $\operatorname{Tor}_{k}^{R}(\square, \square)$ is a bifunctor.
(ii) For a ring $R$ and a fixed $k \geq 0$, prove that $\operatorname{Ext}_{R}^{k}(\square, \square$ ) is a bifunctor.

Harshita Goel
Harshita Goel
Numerade Educator
02:24

Problem 3

Let $\left(E^{r}, d^{r}\right)_{r \geq 0}$ be a spectral sequence. If there is an integer $s$ with $E_{p, q}^{s}=\{0\}$ for some $(p, q)$, prove that $E_{p, q}^{r}=\{0\}$ for all $r \geq s .$

Nick Johnson
Nick Johnson
Numerade Educator
02:24

Problem 4

Let $\left(E^{r}, d^{r}\right)_{r \geq 0}$ be a spectral sequence. If there is an integer $s$ with $E_{p, q}^{s}=\{0\}$ for some $(p, q)$, prove that $E_{p, q}^{r}=\{0\}$ for all $r \geq s .$

Nick Johnson
Nick Johnson
Numerade Educator
12:13

Problem 5

Consider the commutative diagram in which the top row is exact,
where $\operatorname{im}\left(A_{p+1} \rightarrow A_{p}\right)=K_{p}=\operatorname{ker}\left(A_{p} \rightarrow A_{p-1}\right)$. For each $p$ and module $C$, the exact sequence $0 \rightarrow K_{p} \rightarrow A_{p} \rightarrow K_{p-1} \rightarrow 0$ gives the long exact sequence
$\operatorname{Tor}_{q}\left(C, K_{p}\right) \stackrel{\beta}{\rightarrow} \operatorname{Tor}_{q}\left(C, A_{p}\right) \stackrel{y}{\rightarrow} \operatorname{Tor}_{q}\left(C, K_{p-1}\right) \stackrel{\alpha}{\rightarrow} \operatorname{Tor}_{q-1}\left(C, K_{p}\right) .$
If $D_{p, q}=\operatorname{Tor}_{q}\left(C, K_{p}\right)$ and $E_{p, q}=\operatorname{Tor}_{q}\left(C, A_{p}\right)$, prove that $\alpha, \beta$, and $\gamma$ are bigraded maps with respective bidegrees $(1,-1),(0,0)$, and $(-1,0)$ and that ( $D, E, \alpha, \beta, \gamma$ ) is an exact couple. (Notice that this exact couple does not arise from a filtration.)

Anthony Ramos
Anthony Ramos
Numerade Educator
14:32

Problem 6

(Mapping Theorem) Let $\left(E^{r}, d^{r}\right)$ and $\left(E^{\prime r}, d^{\prime r}\right)$ be spectral sequences. A map of spectral sequences is a family of bigraded maps $\left.f=\left(f^{r}: E^{r} \rightarrow E^{\prime r}\right)\right)$, each of bidegree $(0,0)$, such that, for all $r$, we have $d^{r} f^{r}=f^{r} d^{r}$ and $f^{r+1}$ is the map induced by $f^{r}$ in homology; that is, since $E^{r+1}=H\left(E^{r}, d^{r}\right)$, we have $f^{r+1}: \operatorname{cls}(z) \mapsto \operatorname{cls}\left(f^{r} z\right)$
(i) Prove that if $f=\left(f^{r}:\left(E^{r}, d^{r}\right) \rightarrow\left(E^{\prime r}, d^{\prime r}\right)\right)$ is a map of spectral sequences for which $f^{t}$ is an isomorphism for some $t$, then $f^{r}$ is an isomorphism for all $r \geq t .$
(ii) Let $E^{2}$ and $E^{\prime 2}$ be either first or third quadrant spectral sequences. If $f=\left(f^{r}:\left(E^{r}, d^{r}\right) \rightarrow\left(E^{\prime r}, d^{\prime r}\right)\right)$ is a map of spectral sequences for which $f^{t}$ is an isomorphism for some $t$, prove that $f$ induces an isomorphism $E^{\infty} \cong E^{\prime \infty} .$

Anthony Ramos
Anthony Ramos
Numerade Educator
05:06

Problem 7

(i) If $\mathbf{P}$ is a projective resolution of a right $R$-module $A$ and $\mathbf{Q}$ is a projective resolution of a left $R$-module $B$, prove that $\mathbf{P} \otimes_{R} \mathbf{Q}$ is a projective resolution of $A \otimes_{R} B$.
(ii) Use part (i) to redo Exercise $9.34$ on page 594 : if $G$ is a free abelian group of rank $r$, then $\operatorname{cd}(G)<r$.

WZ
Wen Zheng
Numerade Educator
06:42

Problem 8

Prove that there is an analog of the adjoint isomorphism for complexes: if $R$ and $S$ are rings, $A$ is a complex of right $R$-modules, $\mathbf{B}$ is a complex of $(R, S)$-bimodules, and $\mathbf{C}$ is a complex of right $S$-modules, then
$$
\operatorname{Hom}_{S}\left(\mathbf{A} \otimes_{R} \mathbf{B}, \mathbf{C}\right) \cong \mathbf{H o m}_{R}\left(\mathbf{A}, \mathbf{H o m}_{S}(\mathbf{B}, \mathbf{C})\right)
$$

Donald Albin
Donald Albin
Numerade Educator
01:46

Problem 9

(i) For fixed $k \geq 0$ and $(\mathbf{A}, d) \in \operatorname{Comp}\left(\operatorname{Mod}_{R}\right),(\mathbf{C}, \delta) \in$ $\operatorname{Comp}\left({ }_{R}\right.$ Mod), prove that there is a bicomplex $\left(M, d^{\prime}, d^{\prime \prime}\right)$ with $M_{p, q}=\operatorname{Tor}_{k}^{R}\left(A_{p}, B_{q}\right), d_{p, q}^{\prime}=\left(d_{p} \otimes 1_{C_{q}}\right)_{*}$, and $d_{p, q}^{\prime \prime}=(-1)^{p}\left(1_{A_{p}} \otimes \delta_{q}\right)_{*}$. We $\operatorname{denote} \operatorname{Tot}(M)$ by
$$
\operatorname{Tor}_{k}^{R}(\mathbf{A}, \mathbf{C})
$$
[See Example 10.6(i).]
(ii) Prove, as in part (i), that there is a bicomplex with ( p.q) term $\operatorname{Ext}_{R}^{k}\left(A_{-p}, C_{q}\right) ;$ its total complex is denoted by
$$
\mathbf{E x t}_{R}^{k}(\mathbf{A}, \mathbf{C})
$$

ha
Hunny Aggarwal
Numerade Educator
02:45

Problem 10

i) If $R$ is a ring and $0 \rightarrow \mathbf{A}^{\prime} \rightarrow \mathbf{A} \rightarrow \mathbf{A}^{\prime \prime} \rightarrow 0$ is an exact sequence in Comp(Mod $_{R}$ ), prove, for any complex $\mathbf{C}$ of left $R$-modules, that there is an exact sequence in $\operatorname{Comp}(\mathbb{Z})$ :
$$
\begin{aligned}
\rightarrow & \operatorname{Tor}_{1}^{R}\left(\mathbf{A}^{\prime}, \mathbf{C}\right) \rightarrow \operatorname{Tor}_{1}^{R}(\mathbf{A}, \mathbf{C}) \rightarrow \operatorname{Tor}_{1}^{R}\left(\mathbf{A}^{\prime \prime}, \mathbf{C}\right) \\
& \rightarrow \mathbf{A}^{\prime} \otimes_{R} \mathbf{C} \rightarrow \mathbf{A} \otimes_{R} \mathbf{C} \rightarrow \mathbf{A}^{\prime \prime} \otimes_{R} \mathbf{C} \rightarrow 0
\end{aligned}
$$
ii) Definition. A complex A flat if $0 \rightarrow \mathbf{C}^{\prime} \stackrel{i}{\longrightarrow} \mathbf{C}$ is exact in $\operatorname{Comp}\left(\operatorname{Mod}_{R}\right)$, then $0 \rightarrow \mathbf{A} \otimes_{R} \mathbf{C}^{\prime} \stackrel{1 \otimes i}{\longrightarrow} \mathbf{A} \otimes_{R} \mathbf{C}$ is exact in Comp (Ab).
Prove that the following statements are equivalent.
(i) $\mathbf{A}$ is a complex having all terms flat.
(ii) $\operatorname{Tor}_{1}^{R}(\mathbf{A}, \mathbf{C})=0$ for every $\mathbf{C}$; i.e., every term of the complex Tor ${ }_{1}^{R}(\mathbf{A}, \mathbf{C})$ is $\{0\}$.
(iii) $\mathbf{A}$ is flat. ${ }^{3}$

Uma Kumari
Uma Kumari
Numerade Educator
02:33

Problem 11

Let $\mathbf{A}$ be a complex of right $R$-modules, and let $\mathbf{C}$ be a complex of left $R$-modules. If $\mathcal{A}$ has zero differentials and all its terms are flat, and if $H_{\bullet}(\mathbf{C})$ is viewed as a complex having zero differentials, prove that
$$
H_{n}\left(\mathbf{A} \otimes_{R} \mathbf{C}\right)=\left(\mathbf{A} \otimes_{R} H_{\bullet}(\mathbf{C})\right)_{n} .
$$

Foster Wisusik
Foster Wisusik
Numerade Educator
01:05

Problem 12

Let $C=\left(C_{p, q}\right)$ and $D=\left(D_{p, q}\right)$ be first quadrant bicomplexes, and let $f=\left(f_{p, q}: C_{p, q} \rightarrow D_{p, q}\right)$ be a map of bicomplexes. If, for all $p \geq 0$, we have $f_{p, *}: C_{p, *} \rightarrow D_{p, *}$ a quasi-isomorphism, prove that the map Tot $(C) \rightarrow \operatorname{Tot}(D)$ induced by $f$ is a quasiisomorphism.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:13

Problem 13

Let $M$ be a first quadrant or third quadrant bicomplex all of whose rows (or all of whose columns) are exact. Prove that $\operatorname{Tot}(M)$ is acyclic.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
02:55

Problem 14

Let $\mathbf{P}=\varrho^{0}(\mathbb{Z})$, the complex of abelian groups having $\mathbb{Z}$ concentrated in degree 0 . Prove, without using Proposition $10.42$, that $\mathbf{P}$ is not projective in $\operatorname{Comp}(\mathbf{A b})$.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:52

Problem 15

If $0 \rightarrow A^{\prime} \stackrel{\delta}{\longrightarrow} A \rightarrow A^{\prime \prime} \rightarrow 0$ is a split exact sequence in an abelian category $\mathcal{A}$, prove that $\Sigma^{k}(\delta)$ is a direct summand of $\Sigma^{k}\left(1_{A}\right)$, where $\Sigma^{k}(\delta)$ is the complex with $\delta$ concentrated in degrees $(k, k-1)$.

Wendi Zhao
Wendi Zhao
Numerade Educator
06:42

Problem 16

Let $\mathbf{C}=\mathbf{C}^{\prime} \oplus \mathbf{C}^{\prime \prime}$ be a direct sum of complexes. If $\mathbf{C}$ is split, prove that $\mathbf{C}^{\prime}$ is also split.

Donald Albin
Donald Albin
Numerade Educator
06:42

Problem 17

Let $\mathbf{C}=\mathbf{C}^{\prime} \oplus \mathbf{C}^{\prime \prime}$ be a direct sum of complexes. If $\mathbf{C}$ is split, prove that $\mathbf{C}^{\prime}$ is also split.

Donald Albin
Donald Albin
Numerade Educator
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Problem 18

(i) Prove that every contractible complex $\mathbf{C}$ is split exact.
(ii) Prove, using induction, that a positive or a negative complex $\mathbf{C}$ which is split exact is contractible.

Victor Salazar
Victor Salazar
Numerade Educator