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Atrium > Preprints & Publications: t-structures and factorization systems

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  1.  
    • CommentRowNumber1.
    • CommentAuthordomenico_fiorenza
    • CommentTimeAug 12th 2014
    • (edited Aug 12th 2014)

    With Fosco Loregian we are now fine tuning a short note on t-structures and factorization systems in ∞\infty-stable categories. In case you’d like to have a preview of it before we post it to the arXiv, any suggestion, comment or criticism is welcome.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeAug 12th 2014
    • (edited Aug 12th 2014)

    What is the time span of posting ? I mean I am very interested to read this one and comment as I worked in localization theory for a while (and revived that this Spring), but this current week is rather busy to get to a serious new paper, beyond some minutes in public transport…

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 12th 2014
    • (edited Aug 12th 2014)

    I have just updated torsion theory for the full reference to Belgiannis-Reiten article which to my surprise is not cited in your paper.

    • Apostolos Beligiannis, Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. pdf

    Regarding that the classical theory of torsion theories is developed somewhat beyond the abelian case, I would expect that in ∞\infty-categories one could go somewhat beyond the stable case…

  4. Hi Zoran, glad you are interested in this! No hurry: we were planning to upload on arXiv around September 1st or so.

  5. Thanks for the reference! I now see that’s indeed extremely relevant to what we did, we were not aware of it. Thanks a lot!

  6. Ah, no, I was misatken here: the Belgiannis-Reiten work you are referring to is what we are citing as [BR07]. It appears already on page 1 of our note. Maybe I’m confused here and we are pointing to two different articles?

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 12th 2014
    • (edited Aug 12th 2014)

    Oh, no, I was mistaken. The reference is indeed in your list but not at alphabetic place and having being very well skilled in alphabetic search I dismissed as Belgiannis is not before Borges :) Sorry for confusing you…

    • CommentRowNumber8.
    • CommentAuthorFosco
    • CommentTimeAug 12th 2014
    • (edited Aug 12th 2014)

    You have no reason to be sorry: on the contrary, I thank you for pointing out the thing. Improving readability implies avoiding alphabetical mistakes. And I plan to re-read Beligiannis-Reiten first chapter and some others works on which our preprint is based in order to refer to them more precisely.

    I am struggling with some minor corrections precisely in the bibliography, but at the moment I have no idea how to prevent BibTeX to put the two references in different order.

  9. Hi Fosco, not sure the references should be in a different order: a casual reader is not expected to guess from [BR07] what reference is that, and so will search for [BR07] in the bibliography after [Bor]. In this case the problem seems quit striking since [Bor] is a non-math reference. But it could have been Borcherds here :) So in the end I’d trust BibTeX choice: I don’t see a solution which will be optimal from any point of view.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2014
    • (edited Aug 18th 2014)

    Sorry for the slow reaction. I had been somewhat absorbed with other things, and still am.

    This here just to say what I believe I had already said when you announced this elsewhere, earlier: it seems to me a really nice result. I’ll look into it, but not this week (am at a meeting once more, and am way behind with my required preparation).

    Meanwhile, this statement certainly deserves an nLab entry. Maybe somebody likes to create one with a summary of the main statement and a pointer to the preprint?

    • CommentRowNumber11.
    • CommentAuthorFosco
    • CommentTimeAug 18th 2014

    I was planning to do it as soon as possible! Maybe in the following days, certainly for when we will upload on arxiv the paper. Thank you a lot for your interest

    • CommentRowNumber12.
    • CommentAuthorFosco
    • CommentTimeAug 23rd 2014
    • (edited Aug 24th 2014)

    I’m back home, finally. I plan to add a link to the statement to the existing page about t-structures, and then create a new page, linking the existing reflective factorization system and adding informations on normal simple and semi-left/right-exact factorization system, and on how the three notions coincide in the stable ∞\infty-categorical setting.

    Until the stub of this page is ready I’d like to hear your opinion on a couple of things I’m thinking about since Domenico opened this thread:

    1. I think that a section about examples would really improve readability and the percolation of the main result of the paper: t-structures pop out in algebraic topology (Bousfield localization of spectra), algebraic geometry (perverse sheaves) representation theory and general homological algebra in various ways. Hence working out in detail which factorization systems give rise to these t-structures seems to be of some interest for the generic reader. And this is certainly the right place to ask for advices on where to look to find other examples of t-structures in Algebra and Geometry!
    2. (semi?)orthogonal decompositions are “well-behaved” t-structures, where the reflection and coreflection associated to the factorization system are exact functors; there is a restriction of the correspondence we outlined in the paper, that associates to every semiorthogonal decomposition (D ≥0,D <0)(\mathbf{D}_{\ge 0}, \mathbf{D}_{\lt 0}) a stable factorization system. We are working out the details and this stuff will (maybe) appear in a subsequent paper, especially due to the importance semiorthogonal decompositions seem to have in Algebraic Geometry (see for example Kuznetsov-Lunts “Resolution of singularities”).

  13. With Fosco we are going to upload on arXiv so the note should appear there on September 1st. In case you have any comments/suggestions, no hurry at all: we will be leaving it on the arXiv for at least a month before submitting it to a journal.

1 to 13 of 13

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