论文标题
$ \ mathbb {r} $ - 动机$ v_ {1} - $ of SECIDICITITITY $ 1 $
An $\mathbb{R}$-motivic $v_{1}-$self-map of periodicity $1$
论文作者
论文摘要
我们考虑了$ \ mathrm {C} _2 $的非平凡动作，$ 1 $ spectrum $ \ mathcal {y}：= \ m athrm {m} _2（1）\ wedge \ wedge \ mathrm {c}（c}（c}）结果有限$ \ MATHRM {C} _2 $ -Equivariant Spectrum $ \ Mathcal {y}^{\ Mathrm {C} _2} $也可以看作是有限$ \ Mathbb {r} $ - 动机$ \ Mathcal $ altacal $ nath y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y}在本文中，我们表明，$ 1 $ - 周期$ v_1- $ $ \ mathcal {y} $可以将$ \ Mathcal {y}^{y}^{\ mathrm {c} _2} $的自映）提升为$ \ Mathcal {y}^{\ mathrm {c} _2} $。此外，$ \ Mathcal {y}^{\ Mathbb {r}} $的自动图的cofiber实现了Subalgebra $ \ Mathcal {a}^\ Mathbb {r}（1）$ $ \ MATHBB {R} $ {R} $ - MOUITATION-ALDEENRED ALGE alge alge alge alge algebra。我们还表明，$ \ mathrm {c} _2 $ -equivariant自图在$ \ mathcal {y}^{\ mathrm {c} _2 _2} $的几何固定点上是nilpotent。
We consider a nontrivial action of $\mathrm{C}_2$ on the type $1$ spectrum $\mathcal{Y} := \mathrm{M}_2(1) \wedge \mathrm{C}(η)$, which is well-known for admitting a $1$-periodic $v_1-$self-map. The resultant finite $\mathrm{C}_2$-equivariant spectrum $\mathcal{Y}^{\mathrm{C}_2}$ can also be viewed as the complex points of a finite $\mathbb{R}$-motivic spectrum $\mathcal{Y}^\mathbb{R}$. In this paper, we show that one of the $1$-periodic $v_1-$self-maps of $\mathcal{Y}$ can be lifted to a self-map of $\mathcal{Y}^{\mathrm{C}_2}$ as well as $\mathcal{Y}^{\mathbb{R}}$. Further, the cofiber of the self-map of $\mathcal{Y}^{\mathbb{R}}$ is a realization of the subalgebra $\mathcal{A}^\mathbb{R}(1)$ of the $\mathbb{R}$-motivic Steenrod algebra. We also show that the $\mathrm{C}_2$-equivariant self-map is nilpotent on the geometric fixed-points of $\mathcal{Y}^{\mathrm{C}_2}$.
