Dependencies

The \(d^{\delta }_r(x)\) we calculated in Proposition 4.5 may be inessential.

When \(d_r^{\delta _n}(x) = \lambda ^{r-n-1}y\) has no crossing, then \(d_r^{\delta _n}(\lambda ^a x) = \lambda ^{a+r-n-1}y\) also has no crossing.

From the proof above, we see that in (4.1), \(\lambda ^n\) is surjective or trivial, while \(\rho \) is injective or trivial.

For the convenience of readers to check gradings, whenever we write

for \(f: \Sigma ^{m,w} \nu X\to \nu Y\), \(x\in E_\infty ^{s_1,t_1,w_1}(\nu X)\) and \(y\in E_\infty ^{s_2,t_2,w_2}(\nu Y)\), the following conditions must hold:

The crossing defined here is opposite to crossings in Moss’s theorem.

For \(x,y,\delta \) in Proposition 4.5 we always have

if \(0\le a\le n\) and \(0\le a+r-n-1{\lt}m-n\) (the differential is trivial if \(a\) exceeds this range).

Suppose \(r\ge n+1\) and \(d_r(x)=y\), where

A crossing of \(d_r(x)=y\) on the \(E_{n+1}\)-page refers to an essential Adams differential

where

with \(0{\lt}a\le n-1\) and \(0\le b\le r-n-1\). See Figure 1.

From Definition 4.9, it immediately follows that there are no crossings of any differential on the \(E_2\)-page, as this would require \(0{\lt} a \le n-1 = 0\). According to Proposition 4.11, this reflects the fact that \(\delta _1\)-extensions have no crossings for degree reasons.

There are certain special maps between synthetic spectra that we want to consider. For any \(n{\lt}m\le \infty \) and a spectrum \(X\), we have the following distinguished triangles of \(\mathrm{H}{\mathbb F}_2\)-synthetic spectra

We simply write \(\rho =\rho _{n,m}\), \(\delta =\delta _{n,m}\) by abuse of notation if \(n,m\) is understood in the context. When \(m=\infty \), this sequence is interpreted as

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The synthetic Adams spectral sequence for \(\nu X\) is isomorphic to the \(\lambda \)-Bockstein spectral sequence:

1. \(E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\).

2. If there is a classical Adams differential \(d_r x = y\) for \(x \in E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\), then \(x\) admits a lift to \(\pi _{t-s,t}(\nu X / \lambda ^{r-1})\) whose image under the Bockstein \(\nu X / \lambda ^{r-1} \to \Sigma ^{1,-r+1} \nu X / \lambda \) equals \(d_r(x)\).

Consider a homotopy commutative diagram of converging spectral sequences \(V_1 \xrightarrow {p} V_3 \xrightarrow {g} V_4\) with \(q = g \circ p\). If \(d_m^p(x) = z\) and \(d_l^g(z) = w\) has no crossing, then \(d_{m+l}^q(x) = w\).

If \(g \circ f = 0\) (as morphisms of converging spectral sequences) and \(d_n^f(x) = y\), then \(y\) is a permanent cycle in the \(g\)-ESS: \(d_m^g(y) = 0\) for all \(m \ge 0\).

Consider a homotopy commutative diagram of converging spectral sequences \(V_1 \xrightarrow {f} V_2 \xrightarrow {q} V_3\) with \(p = q \circ f\). If \(d_n^f(x) = y\), \(d_m^p(x) = z\), and one of the two extensions has no crossing, then \(d_{m-n}^q(y) = z\).

The \(f\)-extension spectral sequence (\(f\)-ESS) is the spectral sequence

obtained by applying the filtered complex construction (Theorem 0.25) to the underlying filtered chain complex of Definition 0.34. By Theorem 0.26, if the filtrations on \(A_1\) and \(A_2\) are exhaustive and Hausdorff (which holds when they are bounded), then the \(f\)-ESS converges to the homology of the underlying complex, i.e., to \(\ker (f_A) \oplus \mathrm{coker}(f_A)\).

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. For \(r \ge 2\), the \(E_\infty \)-page of the synthetic Adams spectral sequence for \(\nu X / \lambda ^r\) is

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The \(E_\infty \)-page of the synthetic Adams spectral sequence for \(\nu X\) is

The Adams differential \(d_r(x)=y\) has a crossing on the \(E_{n+1}\)-page if and only if the corresponding \(\delta _n\)-extension

for

has a crossing.

Suppose in the classical Adams spectral sequence of \(X\) we have \(d_r(x)=y\), where \(x\in Z_{r-1}^{s,t}(X)\) and \(y\in Z_\infty ^{s+r,t+r-1}(X)\). Consider the map

1. If \(r\ge n+1\), then we view \(x\) as an element of

   \[ E_\infty ^{s,t,t}(\nu X/\lambda ^n)\cong Z_n^{s,t}(X), \]

   and \(\lambda ^{r-n-1} y\) as an element of

   \[ E_\infty ^{s+r,t+r-1, t+n}(\nu X/\lambda ^{m-n})\cong Z_{m-r+1}^{s+r,t+r-1}(X)/B_{r-n}^{s+r,t+r-1}(X). \]

   We then have

   \[ d^{\delta }_r(x)=\lambda ^{r-n-1}y, \]

   which is trivial if \(r{\gt}m\).

2. If \(r{\lt}n+1\), then we view \(\lambda ^{n+1-r}x\) as an element of

   \[ E_\infty ^{s,t,t-n-1+r}(\nu X/\lambda ^n)\cong Z^{s,t}_{r-1}(X)/B^{s,t}_{n+2-r}(X), \]

   and \(y\) as an element of

   \[ E_\infty ^{s+r,t+r-1, t+r-1}(\nu X)\cong Z_\infty ^{s+r,t+r-1}(X). \]

   In this case, we have

   \[ d^{\delta }_r(\lambda ^{n+1-r}x)=y. \]

The only nonzero differentials in the extension spectral sequences for the maps \(\lambda ^n\) and \(\rho \) from Notation 4.1 are the \(d_0\)’s:

As a result, these \(d_0\)’s have no crossings.

As indicated in the proof, the right-hand side of equation (4.4) (considered as a subset of \(Z_\infty ^{s+r,t+r-1}(X)\)) is a coset of

which is the same as the value of the classical Adams differential \(d_r(x)=y\). This implies that the equation (4.4) holds and is essential if and only if \(d_r(x)=y\) holds and is essential. Therefore the \(\delta \)-ESS encodes the same information as the classical Adams spectral sequence.