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1. Generalized $$\mathbb Z$$-homotopy fixed points of $$C_n$$ spectra with applications to norms of $$MU_{\mathbb R}$$

   Hill, Michael A; Zeng, Mingcong (January 2020, New York journal of mathematics)

   null (Ed.)

   We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations. 

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2. Equivariant Witt complexes and twisted topological Hochschild homology

   https://doi.org/10.1016/j.topol.2025.109444  

   Bohmann, Anna Marie; Gerhardt, Teena; Krulewski, Cameron; Petersen, Sarah; Yang, Lucy (May 2025, Topology and its Applications)

   The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra. 

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3. On realizations of the subalgebra 𝒜^{ℝ}(1) of the ℝ-motivic Steenrod algebra

   https://doi.org/10.1090/btran/114  

   Bhattacharya, P.; Guillou, B.; Li, A. (July 2022, Transactions of the American Mathematical Society, Series B)

   In this paper, we show that the finite subalgebra A R ( 1 ) \mathcal {A}^\mathbb {R}(1) , generated by S q 1 \mathrm {Sq}^1 and S q 2 \mathrm {Sq}^2 , of the R \mathbb {R} -motivic Steenrod algebra A R \mathcal {A}^\mathbb {R} can be given 128 different A R \mathcal {A}^\mathbb {R} -module structures. We also show that all of these A \mathcal {A} -modules can be realized as the cohomology of a 2 2 -local finite R \mathbb {R} -motivic spectrum. The realization results are obtained using an R \mathbb {R} -motivic analogue of the Toda realization theorem. We notice that each realization of A R ( 1 ) \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an R \mathbb {R} -motivic v 1 v_1 -self-map. The C 2 {\mathrm {C}_2} -equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the R O ( C 2 ) \mathrm {RO}({\mathrm {C}_2}) -graded Steenrod operations on a C 2 {\mathrm {C}_2} -equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C 2 {\mathrm {C}_2} -equivariant realizations of A C 2 ( 1 ) \mathcal {A}^{\mathrm {C}_2}(1) . We find another application of the R \mathbb {R} -motivic Toda realization theorem: we produce an R \mathbb {R} -motivic, and consequently a C 2 {\mathrm {C}_2} -equivariant, analogue of the Bhattacharya-Egger spectrum Z \mathcal {Z} , which could be of independent interest. 

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4. Dirac Geometry I: Commutative Algebra

   https://doi.org/10.1007/s42543-023-00072-6  

   Hesselholt, Lars; Pstra̧gowski, Piotr (June 2023, Peking Mathematical Journal)

   Abstract The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of differential graded algebras and, more generally, as the homotopy of algebras in spectra. The commutative algebras in question are those in the symmetric monoidal category of graded abelian groups, and, being commutative, they form the affine building blocks of a geometry, as commutative rings form the affine building blocks of algebraic geometry. We name this geometry Dirac geometry, because the grading exhibits the hallmarks of spin in that it is a remnant of the internal structure encoded byanima, it distinguishes symmetric and anti-symmetric behavior, and the coherent cohomology of Dirac schemes and Dirac stacks, which we develop in the sequel, admits half-integer Serre twists. Thus, informally, Dirac geometry constitutes a “square root” of$$\mathbb {G}_m$$ $G_m$-equivariant algebraic geometry. 

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5. The Chow t-structure on the ∞-category of motivic spectra

   https://doi.org/10.4007/annals.2022.195.25  

   Kong, Hana J.; Wang, Guozhen; Xu, Zhouli (February 2022, Annals of mathematics)

   We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k. 

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