Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this seemingly naïve question: Are regular local rings the same objects as local rings that are regular? At first, I thought, "My mind must be acting stupid again." However, upon further analysis, it turned out that the answer to my question was non-trivial after all.
One direction, namely proving that a local ring that is regular is actually a regular local ring, is not very hard to establish. Indeed, it can be assigned as a homework problem in an undergraduate abstract algebra course. The key observation is that for a local ring $ (R,{\frak{m}}) $, upon localization at $ {\frak{m}} $, we obtain $ R_{\frak{m}} = R $. This is because $ R \setminus {\frak{m}} $ is precisely the set of units of $ R $. Hence, by the definition of regular ring, we see that $ (R,{\frak{m}}) $ is a regular local ring.
The other direction is a well-known (in my opinion, highly) non-trivial result in homological algebra, which states that the localization of a regular local ring at any prime ideal is still a regular local ring. By the definition of regular ring once again, regular local rings are therefore local rings that are regular.
I am wondering, are there any pairs of concepts in other areas of mathematics that look so similar that their similarity may be mistaken for tautology but, in reality, can only be established by a hard proof?
Best Answer
$f:\mathbb R^2\to \mathbb R$ is $C^\infty$.
$f:\mathbb R^2\to \mathbb R$ is $C^\infty$ along each $C^\infty$-curve $c:\mathbb R\to \mathbb R^2$; i.e., $f\circ c$ is $C^\infty$ for each such $c$.
Equivalence was proved only in 1979 by Jan Boman. EDIT: It was 1967, sorry for being careless.
EDIT: Using "general abstract nonsense" and functional analysis, one push this result from $\mathbb R^2$ to Frechet spaces. Beyond Frechet spaces, the notions start to divergence. Analysis based on (2) is called convenient analysis, since it leads to a diffeomorphism $$C^\infty(U,C^\infty(V,W)) \cong C^\infty(U\times V, W)$$ and a monoidally closed category.
See:
A.Frölicher and A.Kriegl: Linear spaces and differentiation theory. John Wiley & Sons Ltd., Chichester, 1988.
Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. (pdf)