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1. Stratified Noncommutative Geometry

   https://doi.org/10.1090/memo/1485  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (May 2024, Memoirs of the American Mathematical Society)

   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

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2. The duality resolution at $n=p=2$

   https://doi.org/10.1007/s00209-025-03754-2  

   Beaudry, Agnès; Bobkova, Irina; Henn, Hans-Werner (July 2025, Mathematische Zeitschrift)

   Abstract Working at the prime 2 and chromatic height 2, we construct a finite resolution of the homotopy fixed points of MoravaE-theory with respect to the subgroup$$\mathbb {G}_2^1$$ $G_2^1$of the Morava stabilizer group. This is an upgrade of the finite resolution of the homotopy fixed points ofE-theory with respect to the subgroup$$\mathbb {S}_2^1$$ $S_2^1$constructed in work of Goerss–Henn–Mahowald–Rezk, Beaudry and Bobkova–Goerss. 

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3. The stable Galois correspondence for real closed fields

   https://doi.org/10.1090/conm/707/14250  

   Heller, J.; Ormsby, K. (January 2018, Contemporary mathematics - American Mathematical Society)

   In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a finite Galois extension of fields with Galois group G, there is a functor c∗L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗L/k(G/H+) = Spec(LH)+. The main theorem of [7] says that when k is a real closed field and L = k[i], the restriction of c∗L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c∗L/k itself. Our main tools are Bachmann’s theorem on the (2,η)- periodic stable motivic homotopy category and an isomorphism range for the map πRS → πC2 S induced by C2-equivariant Betti realization. 

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4. Transchromatic extensions in motivic bordism

   https://doi.org/10.1090/bproc/108  

   Beaudry, Agnès; Hill, Michael; Shi, XiaoLin Danny; Zeng, Mingcong (March 2023, Proceedings of the American Mathematical Society, Series B)

   We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MUR. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the π_{**}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_{\star}MUR-module structure on the Real Morava K-theories. 

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5. Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

   https://doi.org/10.1007/s00222-022-01120-1  

   Beaudry, Agnès; Goerss, Paul G.; Hopkins, Michael J.; Stojanoska, Vesna (June 2022, Inventiones mathematicae)

   The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum 𝐸𝑛, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that 𝐸𝑛 is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group 𝔾𝑛 of the formal group in question. In this paper we find that the 𝔾𝑛-equivariant dual of 𝐸𝑛 is in fact 𝐸𝑛 twisted by a sphere with a non-trivial (when 𝑛>1) action by 𝔾𝑛. This sphere is a dualizing module for the group 𝔾𝑛, and we construct and study such an object 𝐼𝒢 for any compact p-adic analytic group 𝒢. If we restrict the action of 𝒢 on 𝐼𝒢 to certain type of small subgroups, we identify 𝐼𝒢 with a specific representation sphere coming from the Lie algebra of 𝒢. This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of 𝐸ℎ𝐻𝑛 for select choices of p and n and finite subgroups H of 𝔾𝑛. 

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