Redshift and multiplication for truncated Brown-Peterson spectra. (English) Zbl 1541.55010

Hahn, Jeremy; Wilson, Dylan

Ann. Math. (2) 196, No. 3, 1277-1351 (2022).

In the present paper, the authors prove that the algebraic \(K\)-theory of the height \(n\) truncated Brown-Peterson spectrum \(\mathrm{BP}\langle n\rangle\) has chromatic height \(n+1\). This produces the first examples of the Redshift Conjecture by Ausoni-Rognes at arbitrary chromatic heights [J. Rognes, Oberwolfach Rep. TB-2000-39, page 9, https://doi.org/10.14760/TB-2000-39 (2000) and “Algebraic \(K\)-theory of finitely presented ring spectra”, https://www.mn.uio.no/math/personer/vit/rognes/papers/red-shift.pdf].
In the chromatic picture of homotopy theory, the category of spectra \(\mathsf{Sp}\) is filtered by the height of \(1\)-dimensional formal group schemes. More precisely, Quillen’s theorem implies that the complex cobordism \(\mathrm{MU}\)-homology of a spectrum can be identified with an element in the derived category of the quasi-coherent sheaves on the moduli stack of formal groups. Under this correspondence, elements on the \(E_2\)-page of the \(\mathrm{MU}\)-based Adams spectral sequence (Adams-Novikov spectral sequence) are organized into families of different periodicities (wavelengths), corresponding to heights of formal groups – hence the name chromatic homotopy theory. For example, discrete rings (when identified with their Eilenberg-MacLane spectra), topological \(K\)-theory, and topological modular forms have heights \(0\), \(1\), and \(2\), respectively.
Localized at a prime \(p\), the spectrum \(\mathrm{MU}\) splits as a wedge sum of suspensions of the Brown-Peterson spectrum \(\mathrm{BP}\). The truncated Brown-Peterson spectrum \(\mathrm{BP}\langle n\rangle\) of height \(n\) is obtained by eliminating polynomial generators corresponding to heights greater than \(n\) from \(\mathrm{BP}_*=\mathbb{Z}_{(p)}[v_1,v_2,\dots]\). While \(\mathrm{MU}\) is an \(\mathbb{E}_\infty\)-ring spectrum, the existence of multiplicative structures on \(\mathrm{BP}\) and its truncations \(\mathrm{BP}\langle n\rangle\) is a very subtle question. The spectrum \(\mathrm{BP}\) is known to have a unique \(\mathbb{E}_4\)-ring structure by [M. Basterra and M. A. Mandell, J. Topol. 6, No. 2, 285–310 (2013; Zbl 1317.55005)] and its truncation \(\mathrm{BP}\langle n\rangle\) was only previously known to admit an \(\mathbb{E}_1\)-ring structure for a general height \(n\) by [A. Lazarev, \(K\)-Theory 24, No. 3, 243–281 (2001; Zbl 1008.55007); A. Baker and A. Jeanneret, Homology Homotopy Appl. 4, No. 1, 163–173 (2002; Zbl 1380.55009); V. Angeltveit, Geom. Topol. 12, No. 2, 987–1032 (2008; Zbl 1149.55006)]. At height \(1\), \(\mathrm{BP}\langle1\rangle\) admits an \(\mathbb{E}_\infty\)-ring structure at all primes since it is the Adams summand of \(p\)-local connective topological \(K\)-theory. At height \(2\), \(\mathrm{BP}\langle2\rangle\) has been equipped with an \(\mathbb{E}_\infty\)-ring structure at \(p=3\) by [M. Hill and T. Lawson, Adv. Math. 225, No. 2, 1013–1045 (2010; Zbl 1220.55006)] and at \(p=2\) by [T. Lawson and N. Naumann, J. Topol. 5, No. 1, 137–168 (2012; Zbl 1280.55007)]. However, \(\mathrm{BP}\) and \(\mathrm{BP}\langle n\rangle\) for any \(n\ge 4\) have no \(\mathbb{E}_{2(p^2+2)}\)-ring structure at \(p=2\) by [T. Lawson, Ann. Math. (2) 188, No. 2, 513–576 (2018; Zbl 1431.55011)] and at odd primes by [A. Senger, “The Brown-Peterson spectrum is not \(\mathbb{E}_{2(p^2+2)}\) at odd primes”, Preprint, arXiv:1710.09822].
The chromatic Redshift Conjecture predicts that the chromatic height of a ring spectrum increases by exactly \(1\) under algebraic \(K\)-theory, with the periodicities (wavelengths) of elements in the homotopy groups lengthening as the height increases. This is why the conjecture is named “redshift”.
At chromatic height \(0\), the redshift phenomenon is essentially the celebrated Lichtenbaum-Quillen Conjecture, proved by Rost-Voevodsky in [V. Voevodsky, Publ. Math., Inst. Hautes Étud. Sci. 98, 59–104 (2003; Zbl 1057.14028); Ann. Math. (2) 174, No. 1, 401–438 (2011; Zbl 1236.14026)]. It states that algebraic \(K\)-theory spectra of discrete rings are equivalent to their height \(1\) Bousfield localizations after taking high enough connective covers. At chromatic height \(1\), Ausoni-Rognes computed the mod \((p,v_1)\) algebraic \(K\)-theory of the Adams summand \(\ell^\wedge_p\) of \(p\)-adic topological \(K\)-theory at primes \(p\ge 5\) in [C. Ausoni and J. Rognes, Acta Math. 188, No. 1, 1–39 (2002; Zbl 1019.18008)]. Like the height \(0\) case, their computation implies that the height \(2\) Bousfield localization \(\mathrm{K}(\ell^\wedge_p)_{(p)}\to L_{2}^f\mathrm{K}(\ell^\wedge_p)_{(p)}\) is a \(\pi_*\)-isomorphism for \(*\gg 0\). In recent years, the Redshift Conjecture has been verified for more height \(1\) spectra in [A. J. Blumberg and M. A. Mandell, Acta Math. 200, No. 2, 155–179 (2008; Zbl 1149.18008); C. Ausoni, Invent. Math. 180, No. 3, 611–668 (2010; Zbl 1204.19002); C. Ausoni and J. Rognes, J. Eur. Math. Soc. (JEMS) 14, No. 4, 1041–1079 (2012; Zbl 1253.19001)]. Since the present paper, the Redshift Conjecture has been proved for \(\mathbb{E}_\infty\)-ring spectra by the works of [M. Land et al., “Purity in chromatically localized algebraic \(K\)-theory”, Preprint, arXiv:2001.10425; D. Clausen et al., “Descent and vanishing in chromatic algebraic \(K\)-theory via group actions”, Preprint, arXiv:2011.08233; A. Yuan, “Examples of chromatic redshift in algebraic \(K\)-theory”, Preprint, arXiv:2111.10837; R. Burklund et al., “The Chromatic Nullstellensatz”, Preprint, arXiv:2207.09929].
In the current paper, the authors first construct models of \(\mathrm{BP}\langle n\rangle\) as \(\mathbb{E}_3\)-\(\mathrm{BP}\)-algebras at an arbitrary height \(n\). It is with this particular model that the authors prove the Redshift Conjecture for \(\mathrm{BP}\langle n\rangle\), i.e., its algebraic \(K\)-theory spectrum has chromatic height \(n+1\), in the sense of fp-type of [M. Mahowald and C. Rezk, Am. J. Math. 121, No. 6, 1153–1177 (1999; Zbl 0942.55012)]. As a consequence, they establish a higher height analog of the Lichtenbaum-Quillen Conjecture showing that both maps \[ \mathrm{K}(\mathrm{BP}\langle n\rangle^\wedge_p)_{(p)}\longrightarrow L_{n+1}^f\mathrm{K}(\mathrm{BP}\langle n\rangle^\wedge_p)_{(p)},\qquad \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}\longrightarrow L_{n+1}^f\mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)} \] are equivalences after taking high enough connective covers.
To access \(\mathrm{K}(\mathrm{BP}\langle n\rangle)\), the authors use the trace method of Dundas-Goodwillie-McCarthy [B. I. Dundas et al., The local structure of algebraic K-theory. London: Springer (2013; Zbl 1272.55002)]. This relates algebraic \(\mathrm{K}\)-theory with the more computable topological cyclic homology \(\mathrm{TC}\), reducing the problem to proving \(\pi_*(F\wedge\mathrm{TC}(\mathrm{BP}\langle n\rangle))\) is bounded above for some finite complex \(F\) of type \(n+2\). By [T. Nikolaus and P. Scholze, Acta Math. 221, No. 2, 203–409 (2018; Zbl 1457.19007)], \(\mathrm{TC}\) is constructed from topological Hochschild homology \(\mathrm{THH}\) together with a cyclotomic Frobenius map and an \(S^1\)-action. The authors prove a Segal Conjecture for the cyclotomic Frobenius map and a canonical vanishing result for the \(S^1\)-action on \(\mathrm{THH}(\mathrm{BP}\langle n\rangle)\) after smashing with a finite complex of type \(n+2\). The two results collectively establish the Redshift Conjecture for \(\mathrm{BP}\langle n\rangle\).

Reviewer: Ningchuan Zhang (Bloomington)

MSC:

55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
19D55	\(K\)-theory and homology; cyclic homology and cohomology
18N70	\(\infty\)-operads and higher algebra

Keywords:

topological Hochschild homology; topological cyclic homology; redshift; \(K\)-theory; Lichtenbaum-Quillen conjecture; truncated Brown-Peterson spectrum

Citations:

Zbl 1317.55005; Zbl 1008.55007; Zbl 1380.55009; Zbl 1149.55006; Zbl 1220.55006; Zbl 1280.55007; Zbl 1431.55011; Zbl 1057.14028; Zbl 1236.14026; Zbl 1019.18008; Zbl 1149.18008; Zbl 1204.19002; Zbl 1253.19001; Zbl 0942.55012; Zbl 1272.55002; Zbl 1457.19007

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