3.4 Five lemma and nine lemma

The five lemma and nine lemma are powerful tools for proving isomorphisms in commutative diagrams with exact rows. They extend the concept of exact sequences, allowing us to analyze more complex relationships between objects and morphisms.

These lemmas build on our understanding of exact sequences, providing a framework for tackling intricate diagrams. By applying these tools, we can simplify proofs and gain deeper insights into the structure of algebraic objects and their connections.

Five and Nine Lemmas

The Five Lemma and its Variations

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• Five lemma states that given a commutative diagram with exact rows, if four out of five vertical arrows are isomorphisms, then the fifth one is also an isomorphism
• Weak five lemma relaxes the conditions of the five lemma by requiring only that the four vertical arrows are either epimorphisms or monomorphisms, concluding that the fifth arrow has the same property
• These lemmas are powerful tools for proving the existence of isomorphisms between objects in a commutative diagram with exact rows ($A \to B \to C \to D \to E$)
• Applying the five lemma or weak five lemma can simplify the process of proving isomorphisms in complex diagrams

The Nine Lemma and 3x3 Lemma

• Nine lemma extends the five lemma to a 3x3 grid of commutative diagrams with exact rows and columns
• It states that if the three objects in the top left corner, the three objects in the bottom right corner, and the object in the center are zero, and the remaining two objects in the middle row and column are isomorphic, then the objects in the four corners are also isomorphic
• 3x3 lemma is a special case of the nine lemma where all the objects in the diagram are zero, except for the four corner objects
• These lemmas are useful for proving isomorphisms in more intricate commutative diagrams ($A \to B \to C, D \to E \to F, G \to H \to I$) where the five lemma alone may not be sufficient

Exact Sequences and Isomorphism Theorems

Exact Sequences and their Properties

• Exact sequences are sequences of objects and morphisms ($A \to B \to C$) where the image of each morphism is equal to the kernel of the next morphism
• Short exact sequences ($0 \to A \to B \to C \to 0$) are exact sequences where the first object is the zero object, the morphism $A \to B$ is injective, and the morphism $B \to C$ is surjective
• Splitting lemma states that a short exact sequence splits (i.e., $B \cong A \oplus C$) if and only if there exists a morphism $C \to B$ such that the composition $C \to B \to C$ is the identity morphism on $C$
• Exact sequences provide a way to study the relationships between objects and morphisms in a category, and they are fundamental in homological algebra

Isomorphism Theorems and Long Exact Sequences

• Isomorphism theorems relate quotients, subobjects, and homomorphisms in a category
  • First isomorphism theorem (or fundamental theorem of homomorphisms) states that for a homomorphism $f: A \to B$, there is an isomorphism between the quotient $A / \ker(f)$ and the image $\operatorname{im}(f)$
  • Second isomorphism theorem states that for a subobject $A$ and a normal subobject $B$ of an object $C$, there is an isomorphism between $(A + B) / B$ and $A / (A \cap B)$
  • Third isomorphism theorem states that for normal subobjects $A$ and $B$ of an object $C$ with $A \subseteq B$, there is an isomorphism between $(C / A) / (B / A)$ and $C / B$
• Exact sequences of complexes are sequences of chain complexes and chain maps between them, where the composition of any two consecutive chain maps is zero
• Long exact sequence in homology is an exact sequence that relates the homology groups of a short exact sequence of chain complexes ($0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$)
  • Connects homology groups of the three complexes: $\cdots \to H_n(A_\bullet) \to H_n(B_\bullet) \to H_n(C_\bullet) \to H_{n-1}(A_\bullet) \to \cdots$
  • Useful for computing homology groups of complexes by relating them to the homology groups of simpler complexes