Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On realizations of the subalgebra $A^R(1)$ of the $R$-motivic Steenrod
  Algebra   In this paper, we show that the finite subalgebra
$\mathcal{A}^{\mathbb{R}}(1)$, generated by $\mathrm{Sq}^1$ and
$\mathrm{Sq}^2$, of the $\mathbb{R}$-motivic Steenrod algebra
$\mathcal{A}^{\mathbb{R}}$ can be given $128$ different
$\mathcal{A}^{\mathbb{R}}$-module structures. We also show that all of these
$\mathcal{A}^{\mathbb{R}}$-modules can be realized as the cohomology of a
$2$-local finite $\mathbb{R}$-motivic spectrum. The realization results are
obtained using an $\mathbb{R}$ -motivic analogue of the Toda realization
theorem. We notice that each realization of $\mathcal{A}^{\mathbb{R}}(1)$ can
be expressed as a cofiber of an $\mathbb{R}$-motivic $v_1$-self-map. The
$\mathrm{C}_2$-equivariant analogue of the above results then follows because
of the Betti realization functor. We identify a relationship between the
$\mathrm{RO}(\mathrm{C}_2)$-graded Steenrod operations on a
$\mathrm{C}_2$-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 $\mathrm{C}_2$-equivariant
realizations of $\mathcal{A}^{\mathrm{C}_2}(1)$. We find another application of
the $\mathbb{R}$-motivic Toda realization theorem: we produce an
$\mathbb{R}$-motivic, and consequently a $\mathrm{C}_2$-equivariant, analogue
of the Bhattacharya-Egger spectrum $\mathcal{Z}$, which could be of independent
interest.

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The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: cs