关于η-周期形式三元律

中文摘要 AI 辅助

我们研究了η-周期动机稳定同伦范畴SH(k)[η⁻¹]中Sp-定向背后的代数结构。博雷尔类确定了一个几何形式三元律，但HW-胡瑞维茨映射表明其泛系数生成一个真子环Λ⊊(MSp[η⁻¹])₊，不过Λ[1/2]=(MSp[η⁻¹])₊[1/2]，分类失败是纯2-主的。为捕捉部分缺失信息，引入了带框对合。谱MSp[η⁻¹]带有一个典范带框对合，产生一个具有望远镜MSL[η⁻¹]的奎伦型幂等元以及一个典范分裂(MSp[η⁻¹])₊≅ℛ_fr⊗ℤ(MSL[η⁻¹])₊，其中ℛ_fr是带框对合的泛环。然后对形式三元律进行公理化，构造泛沃尔特环𝒲^η，并证明𝒲^η在对2求逆后同构于拉扎德环L。若W(k)≅ℤ，泛几何形式三元律与典范带框对合诱导一个分类映射φ:𝒲^η→(MSp[η⁻¹])₊，它是单射且在对2求逆后成为同构。但在整数层面，需要额外的二次幂级数来恢复完整的定向数据。

英文摘要

We study the algebraic structure underlying Sp-orientations in the $η$-periodic motivic stable homotopy category $SH(k)[η^{-1}]$. Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper $W(k)$-subalgebra $Λ\subsetneq (MSp[η^{-1}])_*$, although $Λ[1/2]=(MSp[η^{-1}])_*[1/2]$. Thus the failure of generation is purely 2-primary. To capture part of the missing information, we introduce framed involutions. The spectrum $MSp[η^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[η^{-1}]$ and a canonical splitting $(MSp[η^{-1}])_* \cong \mathcal{R}_{fr} \otimes_{\mathbb{Z}} (MSL[η^{-1}])_*$, where $\mathcal{R}_{fr}$ is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring $\mathcal{W}^η$, and prove that $\mathcal{W}^η$ is isomorphic to the Lazard ring $L$ after inverting 2. If $W(k)\cong \mathbb{Z}$, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map $ϕ:\mathcal{W}^η\to (MSp[η^{-1}])_*$ that is injective and becomes an isomorphism after inverting 2. Integrally, however, additional secondary power series are needed to recover the full cobordism coefficient ring.

发表机构

• Université Grenoble-Alpes（格勒诺布尔阿尔卑斯大学）
• CNRS（法国国家科学研究中心）

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