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Stratifying triangulated categories. (English) Zbl 1239.18013

Earlier work by the authors, which classified the localization subcategories of the stable module category of representations of a finite group, introduced a notion of stratification for tensor triangulated categories. This work defines a notion of stratification for compactly generated triangulated categories and investigates the implications of the existence of such a structure for classification and other problems.
The context is a compactly generated triangulated category \(T\) with the action of a graded commutative noetherian ring \(R\) taking the form of a graded ring homomorphism from \(R\) to the graded center of \(T\). Unions \(V\) of Zariski closed subsets of Spec\(R\) have an associated localizing subcategory of \(V\)-torsion objects from which a local cohomology functor \(\Gamma_V : T \to T\) can be extracted. Of particular interest is the case \(V(p) = \{ a \in \mathrm{Spec}R \mid p \subset a \}\). The support of an object \(X \in T\) is the set of prime ideals of \(R\) for which \(\Gamma_p X \neq 0\).
If \(T\) satisfies the local-global principle, meaning that the localizing subcategory generated by each object \(X\) is the same as the localizing subcategory generated by the \(\Gamma_p X\), then the problem of classifying localizing subcategories may be reduced to the problem of classifying localizing subcategories supported by a single prime. We say \(T\) is stratified by \(R\) if it satisfies this condition and if for each prime \(p\) the subcategory of objects supported at \(p\) is either zero or contains no proper localizing subcategories. In this case, localizing subcategories correspond to subsets of supp\(_RT\). Examples of such triangulated categories include the derived category of modules over a commutative noetherian ring.
A number of other results are also presented. In the presence of a further noetherian condition, a localizing functor that preserves coproducts has a compactly generated kernel. The special case of tensor triangulated categories with a compatible \(R\)-action is also discussed. A notable feature is that the local-global principle is automatically satisfied, so the stratification hypothesis is simplified. An analog of the tensor product theorem of modular representation theory is proven when this condition holds.

MSC:

18G99 Homological algebra in category theory, derived categories and functors
13D45 Local cohomology and commutative rings
18E30 Derived categories, triangulated categories (MSC2010)
20J06 Cohomology of groups
55P42 Stable homotopy theory, spectra
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