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Abstract We construct a lift of the $$p$$-complete sphere to the universal height $$1$$ higher semiadditive stable $$\infty $$-category of Carmeli–Schlank–Yanovski, providing a counterexample, at height $$1$$, to their conjecture that the natural functor $$ _n \to \operatorname{\textrm{Sp}}_{T(n)}$$ is an equivalence. We then record some consequences of the construction, including an observation of Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg–MacLane spaces.
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The Arinkin–Gaitsgory temperedness conjecture
Abstract Arinkin and Gaitsgory defined a category oftempered‐modules on that is conjecturally equivalent to the category of quasi‐coherent (not ind‐coherent!) sheaves on . However, their definition depends on the auxiliary data of a point of the curve; they conjectured that their definition is independent of this choice. Beraldo has outlined a proof of this conjecture that depends on some technology that is not currently available. Here we provide a short, unconditional proof of the Arinkin–Gaitsgory conjecture.
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- PAR ID:
- 10418038
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 55
- Issue:
- 3
- ISSN:
- 0024-6093
- Page Range / eLocation ID:
- p. 1419-1446
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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