The derived category of a commutative ring is generated by residue fields under a sufficient condition with an obstruction, and the local-to-global principle holds if and only if the spectrum satisfies a topological property.
Support and vanishing for non-Noetherian rings and tensor triangulated categories
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abstract
We define and characterise small support for complexes over non-Noetherian rings and in this context prove a vanishing theorem for modules. Our definition of support makes sense for any rigidly compactly generated tensor triangulated category. Working in this generality, we establish basic properties of support and investigate when it detects vanishing. We use pointless topology to relate support, the topology of the Balmer spectrum, and the structure of the idempotent Bousfield lattice.
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2026 1verdicts
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The importance of being isolated
The derived category of a commutative ring is generated by residue fields under a sufficient condition with an obstruction, and the local-to-global principle holds if and only if the spectrum satisfies a topological property.