## 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

Funding Information:Received by the editors July 11, 2021, and, in revised form, January 17, 2022. 2020 Mathematics Subject Classification. Primary 14F42, 55S10, 55S91. The first author was supported by NSF grant DMS-2005476. The second and third authors were supported by NSF grant DMS-2003204.

Publisher Copyright:

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

## ASJC Scopus subject areas

- Mathematics (miscellaneous)

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