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.

]]>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.

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*

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 wrote the related entry spectrum of a triangulated category and updated related pages Alexander Rosenberg, Paul Balmer, Grigory Garkusha and Bondal-Orlov reconstruction theorem.

]]>Look Prop. 8.2.1, p.32 in chapter II (page 65 of the file) in the book

- Alexander Rosenberg,
*Geometry of Noncommutative ’Spaces’ and Schemes*, MPIM 2011-68, pdf

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.

]]>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?

]]>I think no.

]]>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?

]]>Several related new entries: Gabriel-Rosenberg theorem, spectrum of an abelian category, local abelian category.

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