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Chromatic homotopy theory is asymptotically algebraic. (English) Zbl 1442.55002
Localizing at the Johnson-Wilson theories \(E(n)\) provides a filtration of (\(p\)-local) spectra \[L_0\mathrm{Sp}_{(p)} \subset L_1\mathrm{Sp}_{(p)} \subset \cdots \subset L_n\mathrm{Sp}_{(p)}\subset \cdots \subset \mathrm{Sp}_{(p)}\] in which the bottom layer \(L_0\mathrm{Sp}_{(p)}\) is the category of rational spectra. Serre’s work shows that the category of rational spectra is equivalent to the derived category of \(\mathbb{Q}\), and one can ask if algebraic models can be found at higher chromatic heights. In fact, for all primes \(p\) and \(n>0\), there is no algebraic model for \(L_n\mathrm{Sp}_{(p)}\). Instead of fixing the prime \(p\) and varying the height \(n\), one can fix \(n\) and vary the prime. The main result of this paper states that at a fixed height \(n\), as \(p \to \infty\) chromatic homotopy theory admits a symmetric monoidal algebraic model. This algebraic model is built from the categories of twisted complexes of quasi-coherent sheaves on the moduli stack of formal groups introduced by Franke and further studied by M. Hovey [Contemp. Math. 346, 261–304 (2004; Zbl 1067.18012)] and D. Barnes and C. Roitzheim [Adv. Math. 228, No. 6, 3223–3248 (2011; Zbl 1246.55009)].
In order to make sense of such a statement, the authors develop a notion of ultraproduct for \(\infty\)-categories, motivated by the notion of an ultraproduct in model theory. This definition captures the asympotic behaviour of a collection of objects, and is likely to be of independent interest. The authors suggest many interesting applications of their main result to be returned to in future work, and give details of one particular application: the existence of multiplicative structures on local generalized Moore spectra.

MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P42 Stable homotopy theory, spectra
03C20 Ultraproducts and related constructions
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