## Abstract

In this paper, we show that the finite subalgebra A^{ℝ}(1), generated by Sq1 and Sq2, of the R-motivic Steenrod algebra A^{ℝ} can be given 128 different A^{ℝ}-module structures. We also show that all of these A-modules can be realized as the cohomology of a 2-local finite R-motivic spectrum. The realization results are obtained using an ℝ-motivic analogue of the Toda realization theorem. We notice that each realization of A^{ℝ} (1) can be expressed as a cofiber of an ℝ-motivic v_{1}-self-map. The C_{2}-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the RO(C_{2})-graded Steenrod operations on a 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 C2-equivariant realizations of A^{C2} (1). We find another application of the ℝ-motivic Toda realization theorem: We produce an ℝ-motivic, and consequently a C_{2}-equivariant, analogue of the Bhattacharya-Egger spectrum Ƶ, which could be of independent interest.

Original language | English |
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Pages (from-to) | 700-732 |

Number of pages | 33 |

Journal | Transactions of the American Mathematical Society Series B |

Volume | 9 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Publisher Copyright:© 2022 by the author(s) under Creative Commons Attribution 3.0 License.

## ASJC Scopus subject areas

- Mathematics (miscellaneous)

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