Give an example of a commutative diagram with exact rows and...

Give an example of a commutative diagram with exact rows and vertical maps $h_{1}, h_{2}, h_{4}, h_{5}$ isomorphisms
for which there does not exist a map $h_{3}: A_{3} \rightarrow B_{3}$ making the diagram commute.

Key Concepts

Exact Sequences

An exact sequence in algebra is a collection of objects and morphisms between them in which the image of one morphism equals the kernel of the next. This condition expresses a precise balance between the algebraic structures, ensuring that no information is lost or extraneously added between subsequent mappings. Exact sequences are central in homological algebra and are used to capture the structure and relationships within algebraic systems.

Commutative Diagrams

A commutative diagram is a graphical display of objects and morphisms arranged in a shape (often a square, triangle, or more complex structure) such that for any two paths in the diagram with the same start and end object, the composed morphisms are equal. This notion is critical in many areas of mathematics because it assures that different sequences of operations lead to the same result, reflecting structural consistency across the algebraic or categorical system.

Isomorphisms in Diagrams

An isomorphism is a morphism that has an inverse, meaning it establishes a perfect 'structural equivalence' between objects. When isomorphisms appear as vertical maps in commutative diagrams, they indicate that the corresponding slices of the diagram are essentially the same from an algebraic perspective. However, even when many vertical maps are isomorphisms, it does not automatically guarantee that every segment of the diagram will admit a compatible map, highlighting subtle issues in diagram lifting problems.

Lifting Problems and Counterexamples

Lifting problems involve finding a map that makes an incomplete commutative diagram fully commute, aligning with the given structure. In many cases, powerful tools like the Five Lemma can ensure such a map exists under certain conditions, but counterexamples demonstrate that these conditions are necessary and that isomorphisms on nearby objects do not always force the existence of a compatible map in the middle. Constructing a counterexample shows where the logical extension of maps fails, emphasizing the delicate interplay between exactness, commutativity, and the existence of lifting maps.

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