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Adelic models of tensor-triangulated categories. (English) Zbl 1455.55008

“This paper is concerned with models for well-behaved tensor-triangulated categories”. The authors work with tensor-triangulated categories \(\overline{\mathcal{C}}\) which arise as the homotopy category of well-structured Quillen model categories \(\mathcal{C}\). They impose restrictions on these categories so that they have the same formal properties as categories of modules of finite dimensional Noetherian rings. The principal input is the Balmer spectrum of \(\overline{\mathcal{C}}\) which is the categorical analogue of the Zariski spectrum of a commutative ring. This allows one to apply methods from commutative algebra in more general contexts. For example to any prime in the Balmer spectrum, there are associated functors of localization and completion.
The main result of the paper shows that the model category \(\mathcal{C}\) is Quillen equivalent to a diagram of categories of modules over localized completed rings. There are two main examples: the derived category of a commutative Noetherian ring and the category of rational torus-equivariant cohomology theories. The key ingredient in the proof is the adelic approximation theorem which shows how to recover the monoidal unit of \(\mathcal{C}\) as a homotopy limit of a cubical diagram of products of localized completed rings. This is directly analogous to the Hasse square in commutative algebra.

MSC:

55P60 Localization and completion in homotopy theory
18N40 Homotopical algebra, Quillen model categories, derivators
13Dxx Homological methods in commutative ring theory
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References:

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