For each prime $p$, we define a $t$‑structure on the category $\,\widehat{\!S^{0,0}}/\tau\text{-}\mathbf{Mod}_{\mathrm{harm}}^b$ of harmonic $\mathbb{C}$-motivic left-module spectra over $\,\widehat{\!S^{0,0}}/\tau$, whose MGL-homology has bounded Chow–Novikov degree, such that its heart is equivalent to the abelian category of $p$‑completed $\mathrm{BP}_*\mathrm{BP}$-comodules that are concentrated in even degrees. We prove that $\,\widehat{\!S^{0,0}}/\tau\text{-} \mathbf{Mod}_{\mathrm{harm}}^b$ is equivalent to $\mathcal{D}^b({\mathrm{BP}_*\mathrm{BP}\text{-}\mathbf{Comod}}^{\mathrm{ev}})$ as stable $\infty$-categories equipped with $t$‑structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\,\widehat{\!S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\,\widehat{\!S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams–Novikov $E_2$-page for the sphere spectrum $\,\widehat{\!S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the $90$-stem, with ongoing computations into even higher dimensions.

中文翻译：

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稳定同伦范畴的运动变形的特殊纤维是代数的

对于每个素数 $p$，我们在类别 $\,\widehat{\!S^{0,0}}/\tau\text{-}\mathbf{Mod}_{\ $\,\widehat{\!S^{0,0}}/\tau$ 上的谐波 $\mathbb{C}$-motivic 左模谱的 mathrm{harm}}^b$，其 MGL-同源性有有界的 Chow-Novikov 度，使得它的核心等价于 $p$-completed $\mathrm{BP}_*\mathrm{BP}$-comodules 的阿贝尔范畴，它们集中在偶数度上。我们证明 $\,\widehat{\!S^{0,0}}/\tau\text{-} \mathbf{Mod}_{\mathrm{harm}}^b$ 等价于 $\mathcal{ D}^b({\mathrm{BP}_*\mathrm{BP}\text{-}\mathbf{Commod}}^{\mathrm{ev}})$ 作为稳定的 $\infty$-categories 配备 $ t$-结构。作为一个应用，对于每个素数$p$，我们证明$\,\widehat{\!S^{0,0}}/\tau$的动机亚当斯谱序列收敛到$的动机同伦群\,\widehat{\!S^{0, 0}}/\tau$, 与代数 Novikov 谱序列同构，收敛于经典的 Adams–Novikov $E_2$-page 用于球谱 $\,\widehat{\!S^0}$。谱序列的这种同构性允许 Isaksen 和第二和第三作者计算至少到 $90$-stem 的稳定球体同伦群，并持续计算到更高的维度。