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There are many times in mathematics that one needs to make verifications that are annoying and distract from the main point of the argument. Often, there are lemmas that can make this much easier, at least in many important cases.

For instance, in topology, it can be quite annoying to verify directly from the definition that a particular quotient space is what you think it is, and not something else with the same underlying set. (In fact, I suspect that many topologists habitually skip this verification.) However, the following lemma can in many cases make this verification much simpler:

Lemma: If $X$ is compact, $Y$ is Hausdorff, and $f \colon X \to Y$ is surjective, then $f$ is a quotient map.

This lemma can be made more powerful using the fact that it suffices to show a map is locally a quotient map.

Another such difficulty is to verify that a category is abelian; if you go directly from the definition, there is an annoyingly long list of things to verify. However, unless I am mistaken, once you have an abelian category $\mathcal{A}$, there are a number of other categories that are guaranteed to give other abelian categories. These (I think) include the category of functors into $\mathcal{A}$ from a fixed other category, the category of sheaves in $\mathcal{A}$ on a (topological space? other category?) (assuming $\mathcal{A}$ is nice enough for this to make sense), and any full subcategory of $\mathcal{A}$ that is closed under 0, $\oplus$, kernels, and cokernels. Using these in combination, together with the fact that $R$-mod is an abelian category for every ring $R$, I believe one can get to every abelian category used in Hartshorne. (Note: I am not too confident in this example, so if someone wanted to elaborate this in an answer, it would be appreciated.)

EDIT: As is pointed out in the comments below, the category of $\mathcal{O}_X$-modules is not of this form. (I came up with this example while writing the question, and did not think it through too carefully.) Thus, I would doubly appreciate a good answer specifically addressing, "How do you show a category is abelian?"

Question: What are some more of these useful lemmas / collections of lemmas, and how are they used?

# Slick ways to make annoying verifications

There are many times in mathematics that one needs to make verifications that are annoying and distract from the main point of the argument. Often, there are lemmas that can make this much easier, at least in many important cases.

For instance, in topology, it can be quite annoying to verify directly from the definition that a particular quotient space is what you think it is, and not something else with the same underlying set. (In fact, I suspect that many topologists habitually skip this verification.) However, the following lemma can in many cases make this verification much simpler:

Lemma: If $X$ is compact, $Y$ is Hausdorff, and $f \colon X \to Y$ is surjective, then $f$ is a quotient map.

This lemma can be made more powerful using the fact that it suffices to show a map is locally a quotient map.

Another such difficulty is to verify that a category is abelian; if you go directly from the definition, there is an annoyingly long list of things to verify. However, unless I am mistaken, once you have an abelian category $\mathcal{A}$, there are a number of other categories that are guaranteed to give other abelian categories. These (I think) include the category of functors into $\mathcal{A}$ from a fixed other category, the category of sheaves in $\mathcal{A}$ on a (topological space? other category?) (assuming $\mathcal{A}$ is nice enough for this to make sense), and any full subcategory of $\mathcal{A}$ that is closed under 0, $\oplus$, kernels, and cokernels. Using these in combination, together with the fact that $R$-mod is an abelian category for every ring $R$, I believe one can get to every abelian category used in Hartshorne. (Note: I am not too confident in this example, so if someone wanted to elaborate this in an answer, it would be appreciated.)

Question: What are some more of these useful lemmas / collections of lemmas, and how are they used?