In [5], Hopkins, Kuhn, and Ravenel discovered a generalized character theory that proved useful in studying cohomology rings of the form E * (BG). In this paper we use the geometry of p-divisible groups to describe a sequence of "intermediate" character theories that retain more information about the cohomology theory E and yield the related result of [5] as a special case.ii Dedicated to my Wife and my Parentsiii
We analyze the ring tmf * tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E2-term of the Adams spectral sequence for tmf ∧ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf * tmf in TMF * TMF Q admits a description in terms of 2-variable modular forms, and (3) modulo v2-torsion, tmf * tmf injects into a certain product of copies of π * TMF0(N ), for various values of N . We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf * tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ∧ tmf gives a connective cover of TMF0(3), and show that another piece gives a connective cover of TMF0(5). To help motivate our methods, we also review the existing work on bo * bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.for bo ∧ bo (respectively, tmf ∧ tmf) splits as a direct sum of Ext-groups for the integral (respectively, bo) Brown-Gitler spectra. Section 2.4 recalls some exact sequences used in [9] which allow for an inductive approach for computing Ext of bo-Brown-Gitler comodules, and introduces related sequences which allow for an inductive approach to Ext groups of integral Brown-Gitler comodules. Section 3This section is devoted to the motivating example of bo ∧ bo. Sections 3.1-3.3 are primarily expository, based upon the foundational work of Adams, Lellmann, Mahowald, and Milgram. We make an effort to consolidate their theorems and recast them in modern notation and terminology, and hope that this will prove a useful resource to those trying to learn the classical theory of bo-cooperations and v 1 -periodic stable homotopy. To the best of our knowledge, Sections 3.4 and 3.5 provide new perspectives on this subject. Section 3.1 is devoted to the homology of the HZ i and certain Ext A(1) * -computations relevant to the Adams spectral sequence computation of bo * bo.We shift perspectives in Section 3.2 and recall Adams's description of KU * KU in terms of numerical polynomials. This allows us to study the image of bu * bu in KU * KU as a warm-up for our study of the image of bo * bo in KO * KO.We undertake this latter study in Section 3.3, where we ultimately describe a basis of KO 0 bo in terms of the '9-Mahler basis' for 2-adic numerical polynomials with domain 2Z 2 . By studying the Adams filtration of this basis, we are able to use the above results to fully describe bo * bo mod v 1 -torsion elements.In Section 3.4, we link the above two perspectives, studying the image of bo * HZ i in KO * KO. Theorem 3.6 provides a complete description of this image (mod v 1 -torsion) in terms of the 9-Mahler basis.We conclude with Section 3.5 which studies a certain mapconstructed from Adams operations. We show that this map is an injection after applying π * and exhibit how it interacts with the Brown...
For a finite abelian group A, we determine the Balmer spectrum of Sp ω A , the compact objects in genuine A-spectra. This generalizes the case A = Z/pZ due to Balmer and Sanders [BS17], by establishing (a corrected version of) their logp-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions [Kuh04].
We modify the transchromatic character maps to land in a faithfully flat extension of Morava E-theory. Our construction makes use of the interaction between topological and algebraic localization and completion. As an application we prove that centralizers of tuples of commuting prime-power order elements in good groups are good and we compute a new example
In previous work, we used an ∞-categorical version of ultraproducts to show that, for a fixed height n, the symmetric monoidal ∞-categories of En,p-local spectra are asymptotically algebraic in the prime p. In this paper, we prove the analogous result for the symmetric monoidal ∞-categories of Kp(n)-local spectra, where Kp(n) is Morava K-theory at height n and the prime p. This requires ∞-categorical tools suitable for working with compactly generated symmetric monoidal ∞-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the En,p-local ∞-categories.
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