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I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.

In Wikipedia it has been stated that since different abelian categories can give rise to equivalent derived categories, so it is impossible to reconstruct $\mathcal{A}$ (an abelian category) from its derived category $D(\mathcal{A}).$

  1. To what extent it is possible to recover $\mathcal{A}?$ what would be the frame work for studying such questions?

  2. Is there any notion of moduli space of t-structures on the derived category of a given abelian category?

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2 Answers 2

up vote 10 down vote accepted

The derived category of an abelian category has a t-structure, so obviously that's the first thing you want. To a t-structure corresponds a heart, which is an abelian category, whose derived category might be different than the one you started with. To further complicate matters, as you noted, you could have a heart whose derived category is actually equivalent to the one you started with, but the heart itself is not the original abelian category.

It's not clear what you can recover from a triangulated category alone, or even from a triangulated category + t-structure. If you like algebraic geometry and are willing to consider additional structures then you can recover the abelian category. The derived category of a scheme, considered as a monoidal category (coming from tensor products) recovers the original scheme (and thus the abelian category). The same is true if you start with a variety with an ample canonical bundle, then the category plus the bundle do recover the variety.

Somehow this flexibility of derived categories is a nice feature, as it gives rise to interesting (and hidden?) "symmetries" and "relationships" between spaces.

As per the second question I can only think of what Sasha said, that is stability conditions. Given a stability condition one automatically gets a heart of a t-structure (which again may not have anything to do with the original abelian category) and the slices of the stability condition may be seen as a continuous family of t-structures. It would indeed be really nice to have such a thing as a moduli space of t-structures!

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Dear Jacob, you can recover a variety with ample or anti-ample canonical bundle from its derived category alone, you don't need the canonical bundle. – Piotr Achinger Feb 13 '13 at 22:55
To be precise, the varieties should be smooth and projective (Serre duality is used); and the reference for this is Bondal-Orlov, "Reconstruction of a variety from the derived category...". By the way, this uses only the graded structure, not the triangulated structure at all (!). For reconstructing a (Noetherian) scheme from the derived category considered as a monoidal triangulated category, see Balmer, "Presheaves of triangulated categories and reconstruction of schemes". – Adeel Khan Feb 13 '13 at 23:54
dear Piotr, thanks for the correction, I always get the Bondal-Orlov stuff wrong! – Jacob Bell Feb 14 '13 at 0:37
Dear Jacob, thank you, those algebraic geometry examples are indeed very good motivations. – Ehsan M. Kermani Feb 14 '13 at 5:12

There is a notion of "Bridgeland stability condition" which includes a t-structure. Those have a reasonable moduli space. See papers of Bridgeland for details.

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