What are some good examples of spectral sequences which degenerate after the first nontrivial differential?

Question

The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live with spectral sequences which never degenerate. I’m curious about spectral sequences with “intermediate” behavior.

Question 1: What are some examples of spectral sequences which have exactly one page of nonzero differentials?

Most interesting spectral sequences are sufficiently natural in their inputs that the first “globally” nonvanishing differential is some kind of cohomology operation. In a particular instance of the spectral sequence, the first nonvanishing differential might be this globally first nonvanishing differential, or it might come later.

Question 2: In these examples, does the unique nonvanishing differential coincide with the “globally first nonvanishing differential”?

For example, in the Atiyah-Hirzebruch spectral sequence for Morava $K$-theory $H^s(X;K(h)^t) \Rightarrow K(h)^{s+t}(X)$, the first globally nonvanishing differential is (a scalar multiple of) the $h$th Milnor primitive $Q_h : H^s(X) \to H^{s+2^p-1}(X)$ (at $p=2$). When $X = \mathbb R \mathbb P^\infty$, one can work out that this is the unique nonvanishing differential, so it’s an example answering to Question 1, and the answer to Question 2 in this case is “yes”.

Answer 1

Some examples with one nonzero family of differentials:

The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p=2$) and 5.14 (for $p$ odd) of Bruner-R. (Transactions AMS, 2022).

The local cohomology spectral sequence for the connective topological modular forms spectrum $tmf$, converging to the homotopy of its Brown-Comenetz dual $I(tmf)$, collapses at $E_3$ by Theorems 8.11 (at $p=2$) and 8.21 (at $p=3$) of Bruner-Greenlees-R. (Math. Z., 2022).

The motivic spectral sequence from syntomic cohomology to topological cyclic homology, for the connective real $K$-theory spectrum $ko$ with $A(1)$-coefficients, has nonzero $d_3$ and collapses at $E_4$, by Theorem 6.2 of Angelini--Knoll-Ausoni-R. (arXiv, 2023).

The Davis-Mahowald (CMS Conf. Proc., 1982) spectral sequence from Ext over $A(1)$ to Ext over $A(2)$ starts at $E_1$ and collapses at $E_2$.

Some examples with two or three nonzero differentials:

The Adams spectral sequence for $tmf/\eta$ collapses at $E_4$, with both $d_2$ and $d_3$ nonzero.

The Adams spectral sequences for $tmf$, $tmf/2$ and $tmf/\nu$ collapse at $E_5$, with nonzero $d_2$, $d_3$ and $d_4$.

Answer 2

The Serre spectral sequence for the path-loop fibration for an $n$-sphere is a positive answer to question 1, a negative answer to question 2. More generally, a fibration in which either the base or the fiber is an $n$-sphere will provide an example.

Answer 3

One example is a construction that is often used in the passage from smooth projective varieties to arbitrary varieties. There are various variants of this:

• For a proper variety $X$, take a hypercovering $X_\bullet \to X$ by smooth projective varieties (using alterations plus the inductive procedure of SGA 4$_{\text{II}}$, Exp. V$^{\text{bis}}$, §5). Then there is a hyperdescent spectral sequence $$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
• For a smooth variety $X$ with a smooth compactification $\bar X$ whose complement $Z = \bar X \setminus X$ is a simple normal crossings divisor $\bigcup_{i \in I} Z_i$, there is an excision spectral sequence $$E_1^{p,q} = \bigoplus_{\lvert J \rvert = p} H^q_{\text{ét}}\Big(\bigcap_{i \in J} Z_i,\mathbf Q_\ell\Big) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
• These methods can be combined: as I explain in [vDdB20, Thm. 6.6], for an arbitrary separated finite type $k$-scheme $X$, one may construct a simplicial variety $X_\bullet$ with smooth projective components $X_i$ and a spectral sequence $$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$ It has been suggested to me that one should be able to prove this using Voevodsky motives, but I don't know a reference. (My paper is written in the more classical language of Chow motives.)

There are also versions for singular cohomology if $k = \mathbf C$. For instance, the first example above is used to define the mixed Hodge structure on the cohomology of an arbitrary $\mathbf C$-variety.

Each of the above examples degenerates on the $E_2$ page for weight reasons: any $E_r^{p,q}$ is pure of weight $q$, so the differentials are forced to be $0$ once the source and target live in different rows. The observation that you only need the $E_1$ differentials is one of the key points of [vDdB20].

(To make this weight argument precise, either reduce to a finite type situation over $\operatorname{Spec} \mathbf Z$ and use Frobenius eigenvalues, or (if $\operatorname{char} k = 0$) reduce to a finite type situation over $\mathbf Q$, choose an embedding into $\mathbf C$, and use Hodge theory.)

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References.

[vDdB20] R. van Dobben de Bruyn, The equivalence of several conjectures on independence of $\ell$. Épijournal Géom. Algébrique 4, Art. nr. 16 (2020). ZBL1460.14053.

Answer 4

If $X$ is a smooth projective variety of dimension $r$ over $\mathbf{C}$ then the Leray spectral sequence of the (ordered) configuration space $F^n X$ of $n$ points on $X$ including into $X^n$ has only $d_{2r}$ nonvanishing.

Burt Totaro, "Configuration space of algebraic varieties", Topology, vol. 35, no. 4, pp. 1057–1067, Oct. 1996.

Answer 5

Any DG algebra $A$ over a field has a minimal model, which is a minimal $A_\infty$-algebra $(H,m_3,m_4,\dots)$. It consists of a graded algebra $B=H^*(A)$ equipped with multi-linear operations of degree $2-n$: $$m_n\colon B\otimes\stackrel{n}{\cdots}\otimes B\longrightarrow B,\qquad n\geq 3,$$ satisfying some equations. One can reconstruct the DGA $A$ out of the minimal model, up to quasi-isomorphism.

Triple Massey products are a well-known secondary cohomology operation in the cohomology $B=H^*(A)$ of a DG algebra $A$. They are related to Toda brackets in the derived category $D^c(A)$ of compact $A$-modules. These Toda brackets are just defined in terms of the triangulated category structure of $D^c(A)$, i.e. in terms of the exact triangles. The ternary operation $m_3\colon B\otimes B\otimes B\rightarrow B$ computes triple Massey products.

There’s the truncated notion of minimal $A_n$-algebra $(B,m_3,\dots,m_n)$. There’s also an obstruction theory to increase $n$. Classical obstructions live in Hochschild cohomology $HH^{\bullet,*}(B)$. This is not surprising because the $m_i$ are Hochschild cochains. If all obstructions vanish one can extend the minimal $A_n$-algebra to a minimal $A_\infty$-algebra, and then construct a DG algebra $A$ out of it. There are even obstructions to the uniqueness of $A$, up to quasi-isomorphism.

If $\mathcal{T}$ is an essentially small algebraic triangulated category over a field and $X\in\mathcal{T}$ is a generator, one wonders about how many essentially different DG algebras $A$ exist such that $D^c(A)\simeq \mathcal{T}$ in such a way that $A\mapsto X$. This would imply that $B=H^*(A)=\mathcal{T}^*(X,X)$, the graded endomorphism algebra of $X$ in $\mathcal{T}$. Moreover, the ternary operation $m_3$ of a minimal model $(\mathcal{T}^*(X,X),m_3,m_4,\dots)$ of $A$ should compute Toda brackets in $\mathcal{T}$.

If $X$ is an additive generator of $\mathcal{T}$, the operation $m_3$ is completely determined by the triangulated structure of $\mathcal{T}$, so one is interested in the obstruction theory to extend $(\mathcal{T}^*(X,X),m_3)$ to an $A_\infty$-algebra. Classical obstructions don’t vanish, but there’s an enhanced obstruction theory living in the pages of a spectral sequence with $E_2=HH(\mathcal{T}^*(X,X))$ (almost everywhere).

Fernando Muro, “Enhanced A∞-Obstruction Theory,” Journal of Homotopy and Related Structures 15, no. 1 (March 1, 2020): 61–112, https://doi.org/10.1007/s40062-019-00245-0.

The differential $d_2$ is highly non-trivial but $E_3$ is concentrated in a narrow band, so narrow that $d_3=0$ so the spectral sequence stops at $E_3$ and we deduce that $A$ is essentially unique.

Fernando Muro, “Enhanced Finite Triangulated Categories,” Journal of the Institute of Mathematics of Jussieu 21, no. 3 (May 2022): 741–83, https://doi.org/10.1017/S1474748020000250.

This extends to the case where $X$ is $d\mathbb{Z}$-cluster tilting. In this case, the spectral sequence is so sparse that $E_2=\cdots=E_{d+1}$ on the nose, $d_{d+1}$ is highly non-trivial, but $E_{d+2}$ is concentrated in a narrow band and the spectral sequence stops here.

Gustavo Jasso, Bernhard Keller, and Fernando Muro, “The Derived Auslander-Iyama Correspondence,” August 24, 2023, https://doi.org/10.48550/arXiv.2208.14413.