Wiki. “Adams 谱序列” [Adams谱序列]
Wiki. “Adams 谱序列” [Adams谱序列]
Adams 谱序列是计算配边不变量的一种方法, 另一种是指标定理. — 万喆彦, BIMSA 公开课 2023-09-18
Adams 谱序列是计算谱的同伦群的有力工具.
$$ \operatorname{Ext}_{\mathcal A_p}^{s,t}(H^*(X,\mathbb{Z}/p),\mathbb{Z}/p) \Rightarrow \pi_{t-s}(X)_p^\wedge $$ 其中 $\mathcal A$ 为对应素数 $p$ 的 Steenrod 代数.
一般而言, 对于环谱 $E$, 有 $E$-Adams 谱序列, 其第二页为 $E$-(上) 同调. 它是非稳定同伦论中 Bousfield–Kan 谱序列的类比.
For $E$ a ring spectrum, hence an E-∞ ring, the totalization of its Amitsur complex cosimplicial spectrum $E^{\wedge\bullet}$ is really the algebraic dual incarnation of the $1$-image factorization of the terminal morphism $$ \operatorname{Spec}E \to\operatorname{Spec}(\operatorname{Tot}(E^{\wedge\bullet})) \overset{p}{\to} \operatorname{Spec}\mathbb S. $$ Moreover, a spectrum $X$ is equivalently a quasicoherent sheaf on $\operatorname{Spec}(\mathbb S)$ and $E^{\wedge\bullet}\wedge X$ is accordingly the Sweedler coring that expresses the descent property of $X$ pullled back along the cover $p$, dually the $E$-localization of $X$ (Bousfield 局部化). The Adams spectral sequence may then be seen to be the computation of the homotopy groups of the $E$-localization of $X$ in terms of its restriction to that cover. In general, notably for $E=H\mathbb F_p$, the 1-image of $\operatorname{Spec}(E)\to \operatorname{Spec}(\mathbb S)$ is smaller than $\operatorname{Spec}(\mathbb S)$ and therefore this process computes not all of $X$, but just the restriction to that one image (for instance just the $p$-local component). Examples of ring spectra which are “complete” with respect to the sphere spectrum in that the above 1-image coincides with $\operatorname{Spec}(\mathbb S)$ notably includes the complex cobordism cohomology spectrum $E=\mathrm {MU}$. That explains the relevance of the Adams-Novikov spectral sequence (noticing that the wedge summands of $\mathrm MU(p)$ are the BP-spectra) and the close interplay between the ANSS and chromatic homotopy theory.