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Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is there a notion of an ideal class group of a Dedekind ring spectrum (Picard group)?

Thanks

PS This question was already asked on math.stackexchange, but I have heard there are some people on this site working on this sort of thing.

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They seem like you could just pretty much directly try to port the definitions. The definition of module seems clear. For an ideal you just need a concept of what counts as an inclusion, which might just be a set-theoretic inclusion (up to homotopy) but might be stranger. The definition of a principal ideal is obvious, and this gets you the Picard group. Prime ideal seems a bit subtle. I would say that, for any two submanifolds not in the ideal, there product is homotopic to a mnaifold not in the ideal. I doubt a ring of integers would work because defining the field of fractions seems hard. –  Will Sawin Dec 2 '11 at 20:27
    
I'm not sure I follow your use of manifolds? –  Jon Beardsley Dec 2 '11 at 20:48
    
@Will I mean, okay yes there is the notion of modules over ring spectra in homotopy theory ($E$ is a module over $R$ if there is some map $E\wedge R\to E$ satisfying some diagrams, etc.). So for ideal I'm guessing you just use subspectra because we can always localize at that subspectrum? How is the definition of a principal ideal obvious? –  Jon Beardsley Dec 2 '11 at 20:51
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Actually, I think it's much more complicated than Will's comment suggests. Every map is homotopic to an inclusion, so "subobjects" are really not the way to go if you want to define ideals. Jeff Smith had a clever idea to do this by looking in the arrow category Arr$(C$) where objects are morphisms in $C$ and morphisms are commutative squares. My advisor, Mark Hovey, has written an extensive unpublished paper on this subject and has been giving talks on the subject for the past year. He found the requisite model structure and discussed a homotopy theory of ideals. I guess you could email him –  David White Dec 2 '11 at 22:11
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Let me comment that I thought a bit about how to do things like principal ideals and even that was not at all obvious. Ideals act very nicely in some regards, but they are objects in a different category now and a lot of things become very hard to define or compute. Rings of integers and ideal class groups seem wildly out of reach right now. –  David White Dec 2 '11 at 22:13
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up vote 12 down vote accepted

In trying to generalize concepts from algebra to spectra, there are several issues that come into play.

In order for a concept in stable homotopy theory to be intrinsically meaningful it generally needs to be invariant under weak equivalence - whatever the appropriate notion of "weak equivalence" is (of spectra, of commutative ring spectra, etc). There are multiple reasons for this. On one hand, it's the homotopy category rather than the category of spectra that is "algebraic" enough to support generalizations like this. On the other hand, there is the practical consideration that there are many different models for spectra (e.g. symmetric spectra, orthogonal spectra, EKMM spectra, various diagram categories); if a concept isn't meaningful from the point of view of homotopy theory, it may have entirely different meanings in different models.

In addition, a concept may have several different directions of generalization. You could generalize an algebraic concept to one that's defined in terms of homotopy groups; this is easy to define and check, but tends to be less interesting and not satisfied in some principal cases of interest. You could try to phrase things in terms of categorical properties, and express a generalization that way; in order for this to be sensible you generally have to replace all concepts by their appropriate "derived" notions (derived pullback, derived invariants under a group action, etc), which makes it difficult to work with concepts that have almost no exactness properties. You could do something ad-hoc.

For these reasons, it's not a straightforward procedure. It's often a good idea to have some examples in mind or be looking for an application, rather than just generalizing for its own sake. A handy test for how difficult it will be is to try and determine a generalization for differential graded modules and algebras first.

Here are some of the pieces that show up in the definition of a Dedekind domain.

  • Integral domains. I don't really know a useful generalization that doesn't involve being an integral domain on homotopy groups. This leaves out a lot of interesting examples - there is a large zoo of regularity conditions in algebraic geometry are not satisfied by a large class of ring objects in homotopy theory.

  • Fields of fractions. Inverting elements - and localization in general - is something that works well in homotopy theory, and tends to give the expected results.

  • Integral closure. This one is much more difficult, because it involves solutions of an equation, and trying to "adjoin" elements in the fraction field. As David White mentioned, the concept of being a "subobject" is one that doesn't translate well, and so there's not a straightforward way to take an element in the homotopy of the fraction field and adjoin it to the base ring. In general it is very difficult to construct commutative ring objects with prescribed properties.

  • Rings of integers. See above. If you figure a out a useful notion for this, I'd love to hear from you.

  • Ideals. Again, there isn't an intrinsic meaning to "subobject" or "quotient". There are generalizations of the concept of an "ideal", but all the ones that I'm aware of boil down to an ideal being, by definition, something that gives you a map out to another ring. More problematically, because taking the "quotient by an ideal" usually involves a mapping cone/cofiber, being an ideal isn't a property of a map $I \to R$ of modules - it is all the extra data that allows you to construct a ring structure on $R/I$. In addition, for $R$ commutative, ideals as an associative ring and ideals as a commutative ring become separate concepts.

  • Principal ideals. A principal ideal, ideally, would be generated by an element in homotopy that you want to take the quotient by. Simply put, given an element in homotopy you may not be able to construct such an ideal even in cases that look amenable. If you can construct an ideal so that there is a quotient associative algebra, there are likely to be many different choices of quotient algebra structures. Being able to construct a quotient commutative algebra is an entirely different, much harder problem that often doesn't have a solution, and when we hope or expect it to have a solution we often can't prove it. There are decades-old conjectures about some of these.

  • Dimension. To define dimension you usually need prime ideals. There are useful definitions of dimension that use thick subcategories of the homotopy category of perfect complexes - see the work of Paul Balmer in particular. However, it is much harder to translate "deep" results about dimension into homotopy theory. More seriously, a heavy ratio of the interesting examples we know don't satisfy anything like a Noetherian property.

Having said all of this, the subject is in flux and we understand more as time goes on.

The Picard group exists very generally, for some notion of "exists". A general definition was given in a paper by Hopkins, Mahowald, and Sadofsky. For a strictly commutative ring spectrum $R$, an invertible module $M$ is one such that there exists an object $N$ such that $M \wedge_R N \simeq R$. The category of such $M$ is the Picard groupoid; if the homotopy category is essentially small then there is an associated Picard group. This has applications in a number of areas including computational applications. You can also interpret $RO(G)$-graded homotopy groups in equivariant homotopy theory in terms of some part of the Picard group - but $RO(G)$-graded homotopy predates this work by a very significant margin.

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@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. –  Jon Beardsley Dec 5 '11 at 17:35
    
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. –  Jon Beardsley Dec 5 '11 at 18:24
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@JBeardz: I'm happy to discuss anything about this kind of material; feel free to contact me. –  Tyler Lawson Dec 6 '11 at 2:41
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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). –  Tyler Lawson Dec 6 '11 at 2:44
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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 –  David White Dec 6 '11 at 14:39
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