Timeline for Dedekind spectra
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2011 at 19:53 | comment | added | Tyler Lawson | @JBeardz: "Original poster" (in this case, you). | |
| Dec 6, 2011 at 22:10 | comment | added | Jonathan Beardsley | @David So this is probably a ridiculous question, but I've seen it everywhere, and I'll probably feel stupid when I find out, but what does OP stand for? | |
| Dec 6, 2011 at 14:39 | comment | added | David White | This is a really great answer, thanks! That said, I wanted to mention that some notions of dimension do seem to work well, namely homological dimensions like right global dimension or weak dimension. This won't help with the OP's question, where he really needs Krull dimension (for sufficiently nice rings the global dimension agrees with Krull dimension, but his rings won't be that nice), but it might help someone else who stumbles upon this thread. Here's a paper on the subject: arxiv.org/abs/1001.0902 | |
| Dec 6, 2011 at 2:44 | comment | added | Tyler Lawson | Rognes has the notion of a Galois extension of commutative ring spectra (see his memoir, "Galois extensions of structured ring spectra"), and you can indeed use rings of integers in number fields to produce examples. One thing to note is that Rognes' definition will not apply to examples that have ramified (finite) primes, and for example there are no Galois extensions of the sphere spectrum or HZ without inverting some primes first. There are some specific reasons for this (for example, it's not possible to adjoin a p'th root of unity to the sphere without inverting p first). | |
| Dec 6, 2011 at 2:41 | comment | added | Tyler Lawson | @JBeardz: I'm happy to discuss anything about this kind of material; feel free to contact me. | |
| Dec 5, 2011 at 18:24 | comment | added | Jonathan Beardsley | I have heard that Rognes has a concept of field extensions in the homotopy category? If we wanted to be somewhat closed minded, I suppose couldn't we look at "extensions" of HQ as "number fields" Hk? And then somehow look at finite wedges of HZ (from the maximal order point of view)? This idea is probably rather silly and useless. I imagine we'd lose a lot of specificity in some sense, but I guess we might have some general framework within which HO_k might be an example. I guess what I'd really like to see is some use of homotopy theory to answer number theory questions in greater generality. | |
| Dec 5, 2011 at 17:35 | comment | added | Jonathan Beardsley | @Tyler Thanks very much for this exposition. I will probably spend some time thinking about it. I'm extremely interested in such issues, at least, in the questions, though I'm not sure my education has progressed far enough yet for me to really talk about answers. My advisor recommended asking you specifically to find out what sort of things made such generalizations difficult, so I was delighted to see that you had responded. | |
| Dec 5, 2011 at 17:27 | vote | accept | Jonathan Beardsley | ||
| Dec 4, 2011 at 15:11 | history | answered | Tyler Lawson | CC BY-SA 3.0 |