I have a question about the proof of the Five Lemma. For the sake of simplicity, I will talk about the proof provided on Wikipedia. In the proof, there are two things I am confused about: this line "Then $t(n(c)) = p(h(c)) = t(c′)$. Since t is a homomorphism, it follows that $t(c′ − n(c)) = 0$" where $t$ is exact, and this line "Since n is a homomorphism, $n(g(b) + c) = n(g(b)) + n(c) = c′ − n(c) + n(c) = c′$." I have no background knowledge in algebra or topology, and have had struggles finding the answer to my question elsewhere. So, my question is this: are exact sequences and homomorphisms linear over addition (i.e. f(a+b) = f(a)+f(b)) as used in the proof? And if they are not generally, what would make this work in this case? If possible, a more intuitive answer would be appreciated as I am very new to this. Thank you to anyone for helping.
1 Answers
Generally speaking, if you want to understand things like the $5$-Lemma, you will want to learn some amount of Algebra, as this is the basis for Homological Algebra. Despite the time investment, it's well worth it; it's a great topic area!
But as to your question, a homomorphism is a function from one "object" to another, e.g. one vector space to another, one group to another, etc. This function must preserve the "structure" from one object to the other. Because linear algebra in some form is familiar to most, I'll use this as an example.
Suppose $A$ and $B$ are vector spaces. So each comes with an addition structure, i.e. you can add vectors (whatever they are or that means) in each vector space. A homomorphism from $A$ to $B$ would be a function $f: A \to B$ that preserves this addition, i.e. if $a_1, a_2 \in A$, then $f(a_1 +_A a_2) = f(a_1) +_B f(a_2)$.
Notice that on the left, addition is happening in $A$, i.e. $a_1 +_A a_2$ (hence the $A$ symbol I put on the addition). Whereas on the right, addition is happening in $B$, i.e. $f(a_1) +_B f(a_2)$ (note that $f(a_1), f(a_2) \in B$).
The general idea of a homomorphism is that if you "do something" and then apply the morphism that this should be the same as applying the morphism individually and then applying the operation. What this "looks like" depends on the operation. For instance, if an operation in a group is written multiplicatively, i.e. $x \cdot y$ instead of $x + y$, then the function should have the property that $f(x \cdot y)= f(x) \cdot f(y)$ in order to be a homomorphism. Note that on the left the $\cdot$ is "multiplication" in the domain whereas on the right the multiplication is happening in the codomain.
The $5$-Lemma you are looking at most likely is happening over modules, so that the operation is addition; hence, a homomorphism $f: A \to B$ should have the property that $f(a_1 + a_2)= f(a_1) + f(a_2)$. It makes no sense to talk about "linearity" with exact sequences - as they are something else entirely. To define exact sequences, you really need to know some algebra as their very definition is abstract nonsense involving kernels and images - although in the topological context can have a nice interpretation.
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