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1. Localization and nilpotent spaces in -homotopy theory

   https://doi.org/10.1112/S0010437X22007321  

   Asok, Aravind; Fasel, Jean; Hopkins, Michael J. (March 2022, Compositio Mathematica)

   Abstract For a subring $$R$$ of the rational numbers, we study $$R$$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in $${\mathbb {A}}^1$$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in $${\mathbb {A}}^1$$ -homotopy theory, paying attention to future applications for vector bundles. We show that $$R$$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $$BGL_n$$ is $${\mathbb {A}}^1$$ -nilpotent when $$n$$ is odd, and analyze the (more complicated) situation where $$n$$ is even as well. We establish analogs of various classical results about rationalization in the context of $${\mathbb {A}}^1$$ -homotopy theory: if $-1$ is a sum of squares in the base field, $${\mathbb {A}}^n \,{\setminus}\, 0$$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres. 

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2. 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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3. Classical Stable Homotopy Groups of Spheres via $$\mathbb {F}_2$$-Synthetic Methods

   https://doi.org/10.1007/s42543-025-00098-y  

   Burklund, Robert; Isaksen, Daniel_C; Xu, Zhouli (April 2025, Peking Mathematical Journal)

   Abstract We study the$$\mathbb {F}_2$$ $F_2$-synthetic Adams spectral sequence. We obtain new computational information about$$\mathbb {C}$$ $C$-motivic and classical stable homotopy groups. 

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4. Geometric Models for Algebraic Suspensions

   https://doi.org/10.1093/imrn/rnad094  

   Asok, A; Dubouloz, A; Østvær, P A (May 2023, International Mathematics Research Notices)

   Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $$X$$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the $${\mathbb P}^{1}$$ suspension of $$X$$; we also analyze a host of variations on this observation. Our approach yields many examples of $${\mathbb A}^{1}$$-$$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine $${\mathbb A}^{1}$$-contractible smooth schemes. 

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5. 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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