Timeline for Matrix factorization categories beyond the isolated singularity case
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2011 at 8:57 | history | edited | Greg Stevenson | CC BY-SA 3.0 |
added 311 characters in body
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| Aug 29, 2010 at 8:03 | comment | added | Greg Stevenson | cont'd: - $i_*(O_Z)$ is not actually an object of $K_{\mathrm{ac}}(\mathrm{Inj}\;X)$ but there is a functor from the derived category making this homotopy category an infinite completion of the singularity category which one can apply. The reference for this is Krause's paper "The Stable Derived Category of a Noetherian Scheme". Also, thanks Daniel and Kevin - I'll try to remember to edit in a link when this stuff is ready (which will be soon). | |
| Aug 29, 2010 at 7:56 | comment | added | Greg Stevenson | @Daniel: The infinite completion I have in mind, for a noetherian separated scheme $X$, is $K_{\mathrm{ac}}(\mathrm{Inj}\;X)$ the homotopy category of acyclic complexes of quasi-coherent injective $O_X$-modules. What you say above is correct if $X$ is defined by a section of an ample line bundle on a noetherian regular separated scheme. For any noetherian separated scheme $X$ with hypersurface singularities there is a weaker sense in which $i_*(O_Z)$ generates $K_{\mathrm{ac}}(\mathrm{Inj}\;X)$, but I am not sure if it generates in the usual sense and kind of suspect it doesn't in general. | |
| Aug 29, 2010 at 7:35 | history | edited | Greg Stevenson | CC BY-SA 2.5 |
fixed latex to play nicely with mathjax
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| Aug 28, 2010 at 7:55 | vote | accept | Daniel Pomerleano | ||
| Aug 28, 2010 at 7:55 | comment | added | Daniel Pomerleano | Thanks Greg for sharing your work! I take it based upon your discussion that if X is a Noetherian scheme of finite type with enough locally frees(maybe seperated) and is a hypersurface, then assuming for simplicity that Z the singular locus is irreducible and let $i_*(O_Z)$ be the pushforward of the structure sheaf on Z with the reduced induced subscheme structure. This would be a compact generator for the some completion of $D_sg(X)$ too? | |
| Aug 28, 2010 at 0:51 | comment | added | Kevin H. Lin | This is awesome! Very geometric! | |
| Aug 28, 2010 at 0:32 | history | answered | Greg Stevenson | CC BY-SA 2.5 |