a repository of mathematical know-how

Clarify your problem by making it more abstract

Stub iconThis article is a stub.This means that it cannot be considered to contain or lead to any mathematically interesting information.

Quick description

If you are asked to prove that a mathematical object X has a property P, you will often obtain a clearer and easier problem if you identify certain properties P_1,\dots,P_k of X and then prove the statement "Every object that has properties P_1,\dots,P_k has property P" instead of the original statement "X has property P."


Exact functors

As an example, how about proving that a functor that preserves exactness in sequences of the form 0 \to A \to B \to C \to 0 necessarily preserves exactness in sequences of the form D \to E \to F? One could then apply this to tensor products with a flat module.

Similar articles?

Based on the quick descriptions of this article, and the following article:


It seems to me that they are getting toward a similar point. Of course both articles are still stubs; if the intention is to explain different tricks, perhaps someone could explain in the quick descriptions how they are different. Just my two cents.

I had noticed that too, and

I had noticed that too, and considered merging them. I may yet do so, but I want to wait until I have thought of good examples for each. Somehow, with this article the focus is on generalization via abstraction, and thereby obtaining a question that gets more to the heart of the problem, whereas with the other one you may not be generalizing at all, because you may be using an abstract description that completely characterizes the original concrete object. (As an example of the latter, it is very useful to think of the reals as a complete ordered field rather than say as a set of Dedekind cuts, even though there is only one complete ordered field. And it is sometimes helpful to think of the exponential function just as "a" function that satisfies \exp'(0)=1 and \exp(x+y)=\exp(x)\exp(y).)

If we do end up with two articles, there's no doubt that they will be closely related, and that this should be highlighted.

Post new comment

(Note: commenting is not possible on this snapshot.)

Before posting from this form, please consider whether it would be more appropriate to make an inline comment using the Turn commenting on link near the bottom of the window. (Simply click the link, move the cursor over the article, and click on the piece of text on which you want to comment.)