Timeline for Geometric interpretation of nonconnective, non-coconnective chain complexes / spectra?
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21 events
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| Apr 3, 2020 at 16:02 | comment | added | skd | So, in this setting, the picture is somewhat close to behaving like "usual" connective algebraic geometry (but, as pointed out in the link you mentioned in the question, is still quite different): examples of such spectra arise naturally from algebro-geometric considerations. However, the sort of E_oo-ring spectra (or, more generally, spectral schemes) which arise from algebro-topological (I'm thinking "chromatic") considerations are unbounded. This is getting dangerously close to philosophy, but my intuition is that the "source" of these unbounded (periodic) E_oo-rings is just different. | |
| Apr 3, 2020 at 16:01 | comment | added | skd | As for question 2: I agree with the interpretation of positive homotopy/homology groups as nilpotent fuzz, and negative homotopy groups as cohomology (concentrated in positive cohomological degree). One comment: while one will get examples of spectra/chain complexes with homotopy concentrated in positive and negative dimensions, most natural algebro-geometric objects have finite cohomological dimension, so the resulting spectra will be bounded below. | |
| Apr 3, 2020 at 15:52 | comment | added | skd | ... perhaps this is acceptable as a motivator for "unbounded" algebraic geometry if you believe that even-periodic E_oo-rings are sufficiently geometrically motivated. | |
| Apr 3, 2020 at 15:52 | comment | added | skd | The first is the world of algebraic geometry modeled on connective E_oo-rings as the affines; this is what Lurie's books study. The other is the world of algebraic geometry modeled on even-periodic E_oo-rings as the affines (as Denis mentioned); this is motivated by the chromatic picture, and Lurie's approach to this in Elliptic-N is by developing the connective theory and then localizing. The output of taking global sections on a spectral scheme in this latter world will only yield periodic things, so ... | |
| Apr 3, 2020 at 15:52 | comment | added | skd | Question 3 can be interpreted in at least two ways. The first is as you said in your comment; namely, whether KU is the global sections of a sheaf of connective E_oo-rings on a scheme. I do not know the answer to this, but I would not believe any such statement to be true. The second is whether there is some derived scheme whose global sections is KU. The answer to this question is yes: there is a sheaf of even-periodic E_oo-rings on Spec(Z) whose global sections is KU. In general, it seems that the realm of spectral algebraic geometry is divided into two kingdoms. | |
| Apr 3, 2020 at 15:26 | comment | added | Denis Nardin | @TimCampion My intuition is that affine should mean "even periodic" rather than "connective" (this is not true in SAG at the moment). | |
| Apr 3, 2020 at 15:17 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 14:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 13:57 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 13:51 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 13:37 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 13:16 | comment | added | Tim Campion | @skd You know what -- if I had a good "geometric" way to think about the fact that a sheaf of connective $E_\infty$ ring spectra can have nonconnective global sections, I think that would satisfy me. Actually, what's a good example of this? Eg is $KU$ the global sections of a natural sheaf of connective ring spectra on something? The sense I got here is that nonconnective $E_\infty$ rings inevitably take one away from geometric intuition. Should I resign myself to this? | |
| Apr 3, 2020 at 13:15 | comment | added | Tim Campion | @HarryGindi Well, I'm finding nonconnective spectra to be just as mysterious as nonconnective chain complexes. So if you could convince me that spectrum objects -- "combinatorial" or otherwise -- are "geometric" in the super-imprecise sense that I'm failing to make clear, then sure. I don't know what an $(\infty,\mathbb Z)$-category is -- I think I get $\infty$-groupoids via the homotopy hypothesis -- if there's a "geometric picture" for what it means to pass to $\mathbb Z$, that would be cool. | |
| Apr 3, 2020 at 12:51 | comment | added | Harry Gindi | Are you happy with the answer that the unbounded chain complexes are combinatorial spectrum objects in Abelian groups? Also, there's this paper of Paul Lessard where he identifies spectra with locally finite (∞,Z)-categories: arxiv.org/abs/1812.00122 . Somehow the identification between simplicial abelian groups and bounded below complexes is kind of too strict to be a homotopy-invariant statement. It's an equivalence of presentations of homotopy theories rather than an equivalence of homotopy theories. | |
| Apr 3, 2020 at 12:41 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 3, 2020 at 12:33 | comment | added | Tim Campion | @skd Sure, but again there's the issue that cohomology theories live "one category level up", as functors defined on the category of spaces. I'm hoping for something which "lives at category level zero" in some sense. | |
| Apr 2, 2020 at 23:49 | comment | added | skd | One could then interpret your question as asking for natural sources of derived/spectral schemes whose derived ring of functions is concentrated in unbounded degrees. Highly structured periodic cohomology theories provide the most natural example of such objects (e.g., BZ/2 giving rise to KO). Not sure if this counts as "geometric" to you. | |
| Apr 2, 2020 at 23:49 | comment | added | skd | Somewhat stupid tongue-in-cheek response: if you were to allow a slight broadening of your question to asking about spectra which have unbounded (positive and negative) homotopy, then you get examples from the myriad of periodic cohomology theories. This actually isn't as facetious as it might seem to be at first sight. A broadening of the perspective you mentioned on nonpositively graded chain complexes is as the derived global sections of the structure sheaf on a (derived connective) scheme: cohomology is concentrated in negative homological degree. | |
| Apr 2, 2020 at 18:43 | comment | added | Tim Campion | @DenisNardin Hmmm... Maybe? I guess my issue is that a functor is typically an object that lives a category level up. I suppose we've learned to think of sheaves as first-class geometric objects -- so maybe if you could convince me that reduced excisive functors are as "geometric" as sheaves are? | |
| Apr 2, 2020 at 18:20 | comment | added | Denis Nardin | Are "reduced excisive functors" an acceptable answer? | |
| Apr 2, 2020 at 17:52 | history | asked | Tim Campion | CC BY-SA 4.0 |