I think everybody here has met lots of finiteness conditions, like those requiring a vector space to be finite dimensional, an abelian group to be finitely generated, a ring to be Noetherian, a manifold to be compact, a sheaf to be coherent, and a complex to be bounded. And there are lots of good theorems once you assume some finiteness condition. (e.g. the Serre duality and Hodge decomposition for compact Kähler manifolds.) And removing those finiteness conditions seems to be non-trivial and interesting. (This applies to e.g. the two theorems mentioned above.)
So my question is,
why are finiteness conditions so important?
This question baffled me for a long time. I remember before I learnt compactness, when doing proofs in calculus, I felt I needed some finite covering of the closed unit interval, but I somehow thought I should avoid using that in my proof. Even after I learnt compactness for a while, the only thing I felt it gives me, was some "combinatorial advantage" ---- I mean, I didn't understand the necessity of assuming compactness in many theorems in elementary analysis, although I was sure I used it in the proofs and I can make some tricky counterexample if compactness wasn't assumed. [I don't feel I got a better understanding on that even now, I can only say I got used to it, i.e. assuming compactness then good things happen.]
The remarks on compactness also apply to me when I first learn the condition of a ring being Noetherian. Somehow the condition looks unnatural to me at the beginning, although after getting used to it I felt examples of non-noetherian rings are crazy.
And one more thing, I think one thing that Hartshorne/EGA make (early-level) readers confused is that they spent lots of time proving finiteness conditions, like proper pushforward of a coherent sheaf is coherent, or the cohomology of a coherent sheaf on a proper scheme over A is a coherent A-module. One can only appreciate them if he/she is sophisticated enough. (If you are about to prove these theorem in your algebraic geometry class, how do you motivate them and describe why people care about them?)
A related question, maybe I should ask this in a separated thread, is, how do we recognize good finiteness conditions? Some are "easy", like compactness, and finite generation. But some are tricky, like the condition of a triangulated category being compactly generated. By recognizing good finiteness conditions one might hope to prove some good theorem, but how do we know whether the conditions are too restrictive or not? (I guess this requires hard work, but is there any convincing sign of a good condition before one dives into the details?) Anybody here knows the history of compactness (for topological spaces) and coherence (for sheave of modules)? [Judging by the name, coherent sheaves may come before quasi-coherent ones.]
Please re-tag it.