(i) If 0 →M →0 is an exact sequence, prove that M={0}. (ii) If A f→ B g→ C h→ D is an exact sequ

step 1 So we need to show that this is subjective, right? So we need to show that f of x is equal to c.

(i) If 0 →M →0 is an exact sequence, prove that M={0}. (ii)...

(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 $C=\{0\}$.

(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 $C=\{0\}$.

Key Concepts

Exact Sequence

An exact sequence is a sequence of modules (or more generally, objects in an abelian category) and homomorphisms between them in which the image of each map is exactly the kernel of the following map. This condition ensures a precise measure of how well the structure is 'transferred' from one module to the next, and forms the basis for many important results in homological algebra.

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 the homomorphism can produce. In the context of an exact sequence, the equality of the image of one map to the kernel of the next is a crucial mechanism that links the structures and dictates the outcomes of various properties such as triviality of a module when surrounded by zero modules.

Injective and Surjective Homomorphisms

An injective homomorphism (or monomorphism) is one in which different elements have distinct images, meaning no nonzero element maps to zero, ensuring a one-to-one correspondence on its domain. A surjective homomorphism (or epimorphism), on the other hand, means that every element in the codomain is an image of at least one element in the domain. These properties are pivotal in analyzing and deducing structural information in exact sequences, as they control the flow of information between modules.

Isomorphism

An isomorphism is a bijective homomorphism between two algebraic structures that preserves the structure, meaning both the operations and the identities are maintained. Isomorphisms indicate that two modules (or objects) are essentially the same in structure. In the context of exact sequences, having isomorphisms at specific points can yield critical implications, such as forcing an intermediate module to be trivial.

Zero Module

A zero module (or trivial module) is a module that contains only the zero element. In an exact sequence, when a module is positioned between two zero modules, or when the images and kernels force every element to be zero, this signifies that the module in question is trivial. The zero module plays a fundamental role in understanding the boundary conditions of exact sequences and serves as a benchmark for comparing other modules.

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