AI 中文总结
本文在复动机稳定同伦范畴中利用滤子谱模型研究三类滤子，计算了多种谱的切片及有效切片谱序列，验证了相关猜想并得到新结果。
AI 中文摘要
我们在2-完备、胞腔化的复（$\boldsymbol{\reals^2}$）动机稳定同伦范畴中研究有效、连通及非常有效滤子，采用Gheorghe-Isaksen-Krause-Ricka提出的该范畴的滤子谱模型，特别是动机类比函子$\boldsymbol{\bigstar}$。我们可通过滤子谱表示由良好动机类比$\boldsymbol{\bigstar}(X)$的各滤子构成的覆盖，并用这些覆盖计算切片。将该方法应用于$X$为球谱、$\text{MU}$、$\text{ku}$或Eilenberg-MacLane谱的情形，可得到Voevodsky的多个猜想；应用于$\text{ko}$时，可得到Ananyevskiy-Röndigs-Østvær的计算结果；还可应用于$\text{tmf}$，计算动机模形式谱$\text{mmf}$的有效切片。此外，我们还研究$\boldsymbol{\bigstar}(X)$的有效切片谱序列，发现其包含与$X$的经典Adams-Novikov谱序列相同的信息。
英文摘要
We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $Γ_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $Γ_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-Röndigs-Østvær. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $Γ_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.
Comments43 pages. Comments welcome!
