Document Zbl 1483.55005 - zbMATH Open

In previous work [Invent. Math. 220, No. 3, 737-845 (2020; Zbl 1442.55002)], the authors proved that as the prime \(p\) tends to \(\infty\), the \(\infty\)-category \(\mathrm{Sp}_{n,p}\) of \(E_{n,p}\)-local spectra is “algebraic”. Here \(E_{n,p}\) is a height \(n\) Morava \(E\)-theory at the prime \(p\). To phrase this rigorously, they introduce an \(\infty\)-categorical ultraproduct construction and an algebraic category \(\mathrm{Fr}_{n,p}\) (inspired by unpublished work of Franke), and provide an equivalence of symmetric monoidal \(\infty\)-categories \[ \prod_\mathcal{F}^\mathrm{Pic} \mathrm{Sp}_{n,p}\simeq \prod_\mathcal{F}^\mathrm{Pic} \mathrm{Fr}_{n,p}, \] where \(\mathcal{F}\) is a non-principal ultrafilter on the set of primes. Explicit calculations in \(\mathrm{Sp}_{n,p}\) appear far less frequently in the literature when compared to their monochromatic analouges \(\widehat{\mathrm{Sp}}_{n,p}\) of \(K_p(n)\)-local spectra, where \(K_p(n)\). Moreover, \(\widehat{\mathrm{Sp}}_{n,p}\) admits no nontrivial localisation, hence is an essential building block of the stable homotopy category. The main result of the present paper is an equivalence of symmetric monoidal \(\infty\)-categories \[ \prod_\mathcal{F}^\mathrm{Pic} \widehat{\mathrm{Sp}}_{n,p}\simeq \prod_\mathcal{F}^\mathrm{Pic} \widehat{\mathrm{Fr}}_{n,p}, \] where \(\widehat{\mathrm{Fr}}_{n,p}\) is a monochromatic analogue of \(\mathrm{Fr}_{n,p}\). These two equivalences above are also shown to be compatible through the localisation functor \(\mathrm{Sp}_{n,p}\to \widehat{\mathrm{Sp}}_{n,p}\); interestingly enough, the authors mention that it is unknown if these equivalences are compatible with the canonical inclusions \(\widehat{\mathrm{Sp}}_{n,p}\to \mathrm{Sp}_{n,p}\).

It is noticable that the second equivalence above cannot be immediately deduced from the first. Indeed, the Pic-generated protoproduct introduced in [T. Barthel et al., Invent. Math. 220, No. 3, 737–845 (2020; Zbl 1442.55002)] requires that invertible objects inside the input symmetric monoidal \(\infty\)-categories are compact; it is well-known that the unit in \(\widehat{\mathrm{Sp}}_{n,p}\) is not compact; see [M. Hovey and N. P. Strickland, Morava \(K\)-theories and localisation. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0929.55010)]. In the present paper, an extension of the Pic-generated protoproduct is obtained which removes this compactness requirement, however, the output \(\infty\)-category only has the natural structure of a non-unital symmetric monoidal \(\infty\)-category. Overcoming these technical hurdles occupies much of the paper and is key to the proof of the main theorem.

MSC:

55P42	Stable homotopy theory, spectra
55P60	Localization and completion in homotopy theory
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology
55U35	Abstract and axiomatic homotopy theory in algebraic topology

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