(i) Let →An+1 dn+1⟶ An dn⟶ An-1 →be an exact sequence, and let im dn+1=Kn=ker dn for all n. Prov

step 1 So for this given chapter, we started with the natural numbers and built up to the real numbers.

(i) Let →An+1 dn+1⟶ An dn⟶ An-1 →be an exact sequence, and...

(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.

(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.

Key Concepts

Factorization of Exact Sequences

Factorizing an exact sequence involves breaking a long exact sequence into a series of short exact sequences, where each piece captures the relationship between successive kernels and images. This technique is useful for analyzing the structure of complex sequences by isolating simpler, more manageable components, and understanding the internal architecture of the sequence via the interaction of kernels and images.

Short Exact Sequence

A short exact sequence is a particular kind of exact sequence of the form 0 ? A ? B ? C ? 0. It serves as a building block in homological algebra because it encapsulates an entire module or group B, showing how it is constructed from a subobject A and a quotient object C. The exactness condition here implies that A embeds into B and that C is the exact quotient of B by A.

Splicing of Exact Sequences

Splicing refers to the process of joining two exact sequences together along a common segment to form a new exact sequence. When the end of one exact sequence overlaps with the beginning of another, the splicing procedure creates a longer exact sequence, thereby synthesizing local information from one sequence with global structure from another. This method is often employed to piece together known data about different parts of an algebraic structure to deduce properties of the entire structure.

Exact Sequence

An exact sequence is a sequence of algebraic objects and homomorphisms between them such that the image of one homomorphism equals the kernel of the next. This property ensures that there is a precise relationship between the substructures of these objects, capturing the idea of how one structure embeds into another or maps onto another, and offering a concise way to track loss and gain of information across the sequence.

Kernel and Image

The kernel of a homomorphism is the set of elements that map to the zero element of the codomain, while the image is the set of all outputs under that homomorphism. In an exact sequence, the condition that the image of one map equals the kernel of the next is key to understanding how the structures relate to each other, enforcing a strict chain of dependencies that is common in homological algebra.

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