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nLab > Latest Changes: quasicoherent oo-stack

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    I started quasicoherent infinity-stack. Currently all this contains is a summary of some central definitions and propositions in Toen/Vezzosi's work. I tried to list lots of direct pointers to page and verse, as their two articles tend to be a bit baroque as far as notation and terminology is concerned.

    This goes parallel with the blog discussion here.

    In the process I also created stubs for SSet-site and model site. These are terms by Toen/Vezzosi, but I think these are obvious enough concepts that deserve an entry of their own. Eventually we should also have one titled "(oo,1)-site", probably, that points to these as special models.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Do you realise that you've created both quasicoherent infinity-stack and quasicoherent ?-stack?

    (Well, the forum software just destroyed one of those links, but you can probably guess what it should be.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Hm, strange. I started with "quasicoherent oo-stack" and then hit "change entry name". But something went wrong.

    Sorry for that, no sure what happened. Now I renamed "quasicoherent oo-stack" to "quasicoherent oo-stack > history". But I suppose the cache bug also still has a word to say...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Here is a remark that I would like to bounce off in particular Zoran:

    If we use, for definiteness, the projective model structure on complexes, then the cofibant complexes are those that are degreewise projective (and every object is fibrant). That means (doesn't it?) that in the model category of quasicoherent modules, the fibrant-cofibrant objects are in fact the projective chain complexes. But aren't this the complexes of vector bundles?

    In other words, I am wondering if in the oo-setting the oo-category of derived quasicoherent modules isn't much closer to being the oo-category of oo-vector bundles than it may seem.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    I assume you are talking about bounded complexes.

    If you ask weather the algebraic vector bundle is the same as a projective 0-module, then the answer is yes and no. Literally the Serre's theorem (one half of Serre-Swan) is about qcoh sheaves of O-modules on an affine scheme (and even Noetherianess is assumed in Serre's paper what is less essential). For complex analytic manifolds, the Serre's GAGA is extending this projectivity philosophy with really talking about complexes of vector bundles and more generally coherent sheaves in the statements. If you sheafify the condition, you come to local/affine situation, namely under mild assumptions on a scheme, locally free qcoh sheaves and locally projective qcoh sheaves are the same. But algebraic vector bundles are finite-dimensional by definition, what has many reasons. Vector bundles should properly generalize to the sub-infinity-category of perfect complexes (roughly: of compact objects) in infinity setup and this is exactly what Ben Zvi and others are really careful about: to state fine conditions on the subcategory of perfect complexes when relevant. Now it is essential that for the smooth schemes the infinity subcategory of perfect complexes tells you all, while the nonsmooth part (singularities) really need the rest of the information. Look for Orlov's paper on "triangulated categories of singularities" to get convinced into that.

    I just created a stub Serre-Swan theorem.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Thanks, Zoran. I need to have a closer look at all this.

    Now that you mention it: what does the homotopy category of modules over a simplicial ring actually give: the derived category of bounded or of unbounded complexes? (I should look at Toen's lectures...)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    I just created a stub Serre-Swan theorem.

    Thanks! I added some links.

    we should write algebraic vector bundle at some point...

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJan 18th 2010

    I think bounded from one side only. I just created entry Max Karoubi.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeJan 18th 2010

    @ Urs #3

    Caches cleared.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Thanks, Toby!

1 to 10 of 10

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