Consider the following commutative diagram in R Mod having...

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.

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.

Key Concepts

Exact Sequences

Exact sequences are sequences of module homomorphisms where the image of each homomorphism is equal to the kernel of the next. This concept is fundamental in homological algebra because it provides a precise framework to capture concepts like kernel, image, and cokernel, and allows one to understand how different algebraic structures relate and fit together.

Short Exact Sequences

Short exact sequences, which are exact sequences of the form 0 ? A ? B ? C ? 0, not only capture the idea of one module being embedded in another but also the idea of a quotient module. They are essential in studying extensions of modules and in many areas of algebra, providing a concise way to represent exactness when dealing with submodules and factor modules.

Injective Maps (Monomorphisms)

An injective map, or monomorphism, is a homomorphism that preserves distinctiveness, meaning if the map sends two elements to the same result, then those elements must have been identical. In the context of exact sequences, injectivity ensures that certain parts of the sequence capture the full information of the module being embedded, and plays a critical role in determining the exactness of rows or columns in a commutative diagram.

Commutative Diagrams

Commutative diagrams are diagrams of modules and module homomorphisms that illustrate how compositions of maps yield the same result regardless of the path taken. They are a central visualization tool in homological algebra, enabling the tracking of how various components of modules and maps interrelate in proofs and constructions, particularly when dealing with multiple interconnected exact sequences.

Diagram Chasing

Diagram chasing is a technique used in homological algebra to prove properties such as exactness or injectivity by carefully analyzing the elements in a commutative diagram. This method involves following elements through the diagram to derive necessary conditions or contradictions, and is especially useful in proving results about exact sequences and when demonstrating that certain maps are injective or surjective.

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