Here are my preliminary thoughts on the problem. This is not a full solution but perhaps it may lead to one.
I think you can suppose without loss of generality that $A \subsetneqq R \subsetneqq \operatorname{Frac}(A)$. Now we have the inclusion map $\iota\:A \rightarrow R$. Therefore I think it ...
@AlexYoucis i started seeing 'geometry' as one big field rather than lots of smaller fields, and realised that pretty much everything that applies to it is interesting
@AlexanderAmenta I don't think I am ever going to be an analyst, but I know exactly what you mean. I am taking a course on Riemann surfaces next term that is going to mix algebra and analysis and it's amazing to me how cool analysis can be when applied to geometry--e.g. Dolbeault cohomology.
@BenjaminLim Yes, true. But before you start getting into more advanced algebraic topology it helps to know covering spaces and the basics of the fundamental group first.
@AlexanderAmenta Oh? Is it an undergraduate algebraic topology?
@AlexYoucis yeah, we don't have a dedicated 'first topology course', but we do a few weeks in our 'analysis 2' which is enough preparation for our algtop
@BenjaminLim That's a good (classic) book. I'm doing the exercises when I'm bored now. I would definitely take a look at Eisenbud, Matsumoro, and Reid though if you are really interested.
@BenjaminLim Good man! It's definitely not "perfect" (e.g. no tensor products, for example!) but still a good read (almost as good as his alg. geo book)
@AlexYoucis our 4th year courses are sometimes like a first graduate course - but typically the advanced students will take reading courses on top of the usual courses, and these can definitely be at 'graduate level'
@AlexanderAmenta Hmm, I see. I have done a hell of a lot of reading courses (I'm doing one right now!) but they are discouraged over taking graduate courses.
@BenjaminLim I wouldn't be put off by analysis. Analysis is actually very nice. Especially several complex variables and functional. I would try it a little more. Life doesn't end after baby Rudin.
@AlexanderAmenta Is Wells Differential Analysis on Complex Manifolds?
@AlexanderAmenta Yes, it definitely looks very neat. Next term I'll be doing a very analysis/differential geoemtry/PDE focused course though--it'll be refreshing. I've been doing nothing but group cohomology and "avanced" Galois theory all year.
Yeah, I know. In the course I am doing this term we are doing sheaf cohomology and I was looking forward to getting my hands a little more dirty with it.
also our number theory course is pretty weak, so i wasn't used to proving things by splitting into cases depending on what the prime was modulo 4 or 8...
I agree though, learning math is easy. In fact, do you ever feel like the only impediment is time? If you had infinite time you could learn everything.
so the original atiyah-singer proof (at least the original one that was published) requires 'pseudodifferential operators', which generalise differential operators
I want to search for questions involving the expression $n^n$, but it seems that the search box is disregarding the ^ symbol. It finds all the questions involving $n$, which is not useful. Is there any way to get what I want?
What? Wait, were you saying the tag was too specific to actually be a tag, or were you saying the wiki excerpt was too localized for specifying complex numbers?
I don't think that thread you were downvoted in was on the main page at the time of the vote. If you feel like it, perhaps you should keep some documentation of the times you are downvoted and (roughly) when the threads they're in are on the main page. If they don't match up at all it would be, in my opinion, mediocre evidence of a downvote stalker.
@anon He wrote a completely useless answer not answering even one part of OP's question and I left a comment and downvoted the post. Subsequently more followed and he deleted the post.
