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Several related new entries: Gabriel-Rosenberg theorem, spectrum of an abelian category, local abelian category.
Can the spectrum of an abelian category be seen as some collection of (prime) ideals in that category along the lines of what Awodey takes as the ideals of a category in the work mentioned at ideal completion?
I think no.
There is a question about these entries here.
I thought I’d just open up the paper and answer their question, but I realized they’re right! These entries are fairly confusing on the point of removing the hypothesis of quasi-compactness. They all seem to say something like: Rosenberg proved you can reconstruct a scheme from its category of quasi-coherent sheaves in a more general situation by using a different spectrum.
spectrum of an abelian category has 8 eight papers listed by Rosenberg and its not clear that any of them are the one that contain this theorem. Does anyone know which paper has it and/or what the exact hypotheses are?
Look Prop. 8.2.1, p.32 in chapter II (page 65 of the file) in the book
This is in quasicompact case. Usage of another spectrum goes somewhat beyond. I will give the reference for that later, now I have to take the bus in few minutes.
I wrote the related entry spectrum of a triangulated category and updated related pages Alexander Rosenberg, Paul Balmer, Grigory Garkusha and Bondal-Orlov reconstruction theorem.
I added the references
Martin Brandenburg, Rosenberg’s reconstruction theorem (after Gabber), 2013, arXiv.
John Calabrese, Michael Groechenig. Moduli problems in abelian categories and the reconstruction theorem. 2013. arXiv.
to the page Gabriel-Rosenberg theorem.
I have expanded the Idea-section at spectrum of a triangulated category to provide a bit more of the basic background.
Also I cross-linked with prime spectrum of a symmetric monoidal stable (infinity,1)-category
Well spectrum of a triangulated category should eventually not redirect to spectrum of a tensor triangulated category as these are different notions (and the nontensor variant is more basic, logically). In general, from a triangulated category you can reconstruct much less than from a tensor triangulated category as it is the case in abelian case as well.
The existing article pointed to Balmer and followup work, which is about the monoidal case, I added the explanation for that. The non-monoidal case still appears, as it did, I just added a brief piece of glue in an attempt to further clarify.
But of course once we have some actual content in the entry, maybe we want to split it into two.
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