Extensive introduction in an incremental paper?

Question

Suppose I'm writting a (mathematics) paper proving that a famous conjecture is true in yet another special case. Should the introduction to the paper include extensive background explaining why this conjecture is important and what is already known?

On one hand, my first instinct is that, of course, if I'm writting a paper on a certain subject, I should explain the relevant background to my reader. If there were relatively few papers on this subject, there would not be any need for this question.

However, if I were to be totally honest, my introduction would go something like this:

A lot of people have worked on Conjecture X. For instance, in [1] {insert name of a famous mathematician} proves that Conjecture X is true for all flabby sheaves*. Here, I show the same conjecture for fine sheaves, for basically the same reasons. I could explain why, but you'll be better off reading the introduction to [1] instead, so I won't bother.

Is it a bad practice to write a more polite version of the above instead of a genuine introduction?

There is also a matter of citations, which work out very differently in the two sceniarios: in the latter case there would be much fewer. It's not the most crucial consideration, but I would be interested to hear if it is a separate reason to avoid minimalistic introductions like that.

---

*) The conjecture I'm talking about has nothing to do with algebraic geometry and I myself know nothing about algebraic geometry beyond the very basics. I'm only talking about sheaves because I'm enamoured with their terminology.

Answer 1

The answer fundamentally depends on who the audience for the result is.

If you are addressing people familiar with the conjecture, then your introduction makes sense, because it consisely conveys the information they would be looking for. In the rare case that a novice reader comes across your paper, they may be a bit put off, but can still follow your reference to read the introduction there.

If there is a reason why people not familiar with the conjecture should care about your paper, then you should give those people more. In particular, you'll need to tell them why exactly they should care.

Answer 2

Well, I would not write in the same style you wrote, but a similar, even if diluted a bit, message:

Conjecture X on flobby heaps is a major question in modern flap theory. Borkington (2014) proved that Conjecture X is true for all flabby shoves of discrete curvature. In the following, we use the terminology from Kal El (2015). The essential idea of this work generalised Kal El (2015): we postulate that flobby heaps are not only flabby on shoves, but also flippy for fine sheaves. The essential difference to prior works (Barkington 1888, Sockington 1999, Kal El, 2015) is that we iteratively construct a Noether pyramid of flobby flabs (Torkington, 2001) in slob space to show flippiness.

Notice that I would recycle the terminology (it still needs to be briefly introduces, but the lengths can be spared), but keep the motivation. You need to say what are you doing, how it is different from what others did, and why does it matter.