Adams tower on bounded below spectra

Finally had an argument that I'm happy with

I've learned most techniques from Mathew–Naumann–Noel's Nilpotence and Descent paper [ND], as well as a central but mundane trick from Nikolaus–Scholze's Topological Cyclic Homology paper [TC]. Although the Adams spectral sequence is no stranger for the mathematical omnivore I have been, I've always assumed to some extent that they do what they claim to, and somewhat focused on what the machine produces, less so on what it computes.

References:

Auxiliary:

Caveat lector! For our purpose today, the Adams spectral sequence is the totalisation spectral sequence of a(n augmented) cosimplicial object associated to an \(\mathbb{E}_1\)-ring spectrum \(E\) and a spectrum \(X\). We won't discuss the spectral sequence, but rather the totalisation, which it morally computes. The totalisation spectral sequence comes from the tower of partial totalisations, also known as the coskeletal filtration, which is closely related to the titular Adams tower. The question of convergence is another topic [CC], and has to do with the right non-exactness of sequential limits (\(\operatorname{lim}^1\)). What will be addressed in this post is the vanishing of the (homotopy) limit of a certain tower, or conditional convergence in the terminology of [FS, Sec. 2.3.1].

The major examples we'd like to have in mind are \(E = \mathbb{F}_p\) and \(\mathrm{MU}\). Other common examples include Bott-periodic or connective topological \(\mathrm{K}\)-theory \(\mathrm{KU}\), \(\mathrm{ku}\), as well as Morava \(\mathrm{E}\)-theories. A well-known result (Devinatz–Hopkins cites Morava's change of rings theorem.) identifies the \(K(n)\)-local \(E_n\)-Adams spectral sequence with the homotopy fixed points spectral sequence for the Morava stabiliser group. We will not concern ourselves with these (extremely interesting) examples.

The situation we concern ourselves with in the current blog post is classical: When \(E\) is connective and \(\mathbb{S} \to E\) is surjective on \(\pi_0\). (A similar statement exists for \(\mathbb{Z} \subseteq \pi_0 E \subseteq \mathbb{Q}\).)

When \(E\) is a connective \(\mathbb{E}_1\)-ring whose unit morphism \(\mathbb{S} \to E\) is surjective on \(\pi_0\) and \(X\) is a bounded below spectrum, the \(E\)-descent object \(D_E X\) (in \(\mathrm{Sp}\)) for \(X\) (defined below) coincides with the Bousfield localisation of \(X\) at the Moore spectrum associated to \(\pi_0 E\).

For instance, if \(X\) is connective then \(D_{\mathrm{MU}} X \simeq X\) and \(D_{\mathbb{F}_p} X \simeq X_{\hat p}\) is the \(p\)-completion. This alludes to the classical case of Adams(-Novikov) spectral sequences.

The connectivity is crucial, both on \(E\) and on \(X\). At a fixed prime, the Morava \(K\)-theories are \(p\)-complete and mutually acyclic.

Before starting, we reiterate that Bousfield [LE, 6.5-7] already showed that the condition on \(\pi_0 E\) can be refined. Nevertheless, the examples we have in mind often don't require such generality.

The descent tower and the error

This section is closely following, if not taken from, [ND, Sec. 2.1].

For a spectrum \(X\), define its \(E\)-descent diagram \(D^\bullet_E X \coloneqq X \otimes E^{\otimes (1 + \bullet)}\) to be the (coaugmented) cosimplicial spectrum with face maps given by inserting units and degeneracies by multiplication of \(E\) (and constant on \(X\)). The total \(E\)-descent object of \(X\) is the totalisation \[ D_E X \coloneqq \lim_{\bullet : \Delta} D^\bullet_E X, \] which comes along with the coskeletal filtration \(D_E X \simeq \lim_{-1 \leq n \to \infty} D^{(n)}_E X\), where \[ D^{(n)}_E X \coloneqq \lim_{\bullet : \Delta_{\leq n}} X \otimes D^\bullet_E. \] We call this the descent tower. The associated spectral sequence is called the Adams spectral sequence.

We use the only reasonable convention that \(\Delta_{\leq -1}=\emptyset\). A few remarks are due (Most can be found in [ND] in a generalised setting):

  1. There is a natural map \(X \to D_E X\) induced by the coaugmentation, which we claimed and will know to be a localisation in the classical situation. However we might like this to be true, a priori the descent construction \(D_E(-)\) has no reason to be idempotent, let alone a localisation.
  2. If \(X\) has a right \(E\)-module structure, then the coaugmented cosimplicial object \(X \otimes D^\bullet_E\) has an extra degeneracy, and is hence a universal limit diagram. In this case, the map \(X \to D_E X\) is an isomorphism. This property propagates to the idempotent complete stable subcategory generated by \(E\)-modules.
  3. Consequently, if \(E\) is a finite \(E_1\)-ring spectrum, so that \(E \otimes -\) commutes with limits, we have that \(X \to D_E X\) is an \(E\)-equivalence into an \(E\)-local spectrum. The descent object thus coincides with the Bousfield \(E\)-localisation functor. Typical examples include the endomorphism rings of finite spectra, and by a recent result of Burklund, also \(\mathbb{S}/p^2\) for odd primes \(p\) and \(\mathbb{S}/2^3\) provide examples of \(\mathbb{E}_1\)-algebras. (I haven't got to read the paper yet, so I can't say how canonical or unique these rings are. These are almost surely not computable anyway.)
  4. The filtration is complete in the sense that its limit is \(D_E X\). Caution is due in translating this into computations on homotopy groups. We don't plan to cover this part in this blog post.
  5. Although \(\Delta^{\textrm{inj}}\) is coinitial in \(\Delta\) [HTT, 6.5.3.7], the same is not true for \(\Delta_{\leq n}\) (which isn't even weakly contractible, but see [HA, 1.2.4.17]). This is one reason we demanded that \(E\) be \(\mathbb{E}_1\).

Another reason for demanding a ring structure lies with the identification of the Adams \(E_2\)-page as \(\mathrm{Ext}\)-groups of graded comodules over a Hopf algebroid. This also depends on a flatness conditions which will not be discussed.

We'll make use of the following well-known description of the Adam tower [ND, Prop. 2.14]: (The proof is [HA 1.2.4.17]

Let \(I \coloneqq \mathrm{fib}(\mathbb{S} \to E)\) be the "kernel" of the ring's unit. Set iteratively \[ T^{(-1)}_E X \coloneqq X, T^{(n+1)}_E X \coloneqq T^{(n)}_E X \otimes I \to \mathbb{S} \otimes T^{(n)}_E X. \] So that \(T^{(n)}_E X \simeq X \otimes I^{\otimes (1 + n)}\).

Then \(T^{(n)}_E X \to X \to D^{(n)}_E X\) are (a tower of) fiber sequences, where the middle term is the constant tower, and the lateral terms are the towers described above. The limit of the tower \(T_E X\) is thus the fiber of the map \(X \to D_E X\).

Arguably, both the tower \(T^\bullet_E X\) or \(D^\bullet_E X\) deserve the name the Adams tower, although the literature has assigned the name to the former. We will call \(T^{\color{gray}(n)}_E\) the error of descent (tower).

Postnikov convergence

Recall that every spectrum \(X\) is the limit of its Postnikov truncations \(\tau_{\leq m} X\), and the only spectrum which is \(\infty\)-connective is zero. The following trick is extracted from [TC, Lem. I.2.6]

We say that an exact functor \(F : \mathrm{Sp} \to \mathrm{Sp}\) is bounded below if there is a constant \(c \in \mathbb{Z}\) such that \(F\) sends connective spectra to \(c\)-connective spectra.

If \(F\) is bounded below, then \(F(X) \simeq \lim_{m \to \infty} F(\tau_{\leq m} X)\) is the limit of its value on the Postnikov tower.

Intuitively, this is a condition similar to "finite cohomological dimension" assumptions.

Proof. As \(F\) is exact, there are fiber sequences \[ F(\tau^{\gt m} X) \to F(X) \to F(\tau_{\geq m} X). \] We just have to show that the limit \(\lim_{m} F(\tau^{\gt m} X)\) is zero. The assumption implies that each \(F(\tau^{\gt m} X)\) is \((m-c)\)-connective, and a sequential limit of \(r\)-connective spectra is \((r-1)\)-connective, so the fiber is \(\infty\)-connective.

As evident from the argument, the assumption on \(F\) can be relaxed, and there is a similar generalisation, replacing \(\mathrm{Sp}\) with stable categories with \(t\)-structure. The interested reader will have already taken this as an exercise.

Evidently the identity functor is bounded below. We verify that the classical descent functors, as well as the \(p\)-completion functor, are bounded below.

If \(E\) is a connective \(\mathbb{E}_1\)-ring whose \(\pi_0\) is a quotient or localisation of \(\mathbb{Z}\), then the error of descent \(T_E\) and descent \(D_E\) are bounded below functors.

Proof For a connective ring spectrum \(E\), the fiber \(I\) of the unit morphism is \((-1)\)-connective, and \(\pi_{-1} I\) is the cokernel of \(\mathbb{Z} \to \pi_0 E\). In the stated case, \((\pi_{-1}I)^{\otimes 2} = 0\), so all the \(I^{\otimes(1+n)}\) are \((-1)\)-connective. The functor \(T_E\) decreases connectivity by at most \(2\), and is thus bounded below. It follows that the descent object \(D_E\) is also bounded below.

The \(p\)-adic completion, which is the Bousfield localisation of spectra at \(\mathbb{S}/p\), is given as a limit as follows: \[ X_{\hat p} \simeq \lim_{n \to \infty} X/p^n. \] As a result, \(p\)-completion decreases connectivity by at most \(1\).

Proof The \(p\)-completion is part of the recollement on \(\mathrm{Sp}\) induced by the idempotent algebra \(\mathbb{S}[1/p]\). More precisely, let \(C \coloneqq \mathrm{fib}(\mathbb{S} \to \mathbb{S}[1/p])\) be its complementary idempotent coalgebra.

We have (see the previous post on inverting endomorphisms) \[ \begin{gather*}\mathbb{S}[1/p] \simeq \operatorname*{colim} \left(\mathbb{S} \xrightarrow{p} \mathbb{S} \xrightarrow{p} \ldots\right)\\ C \simeq \operatorname{\Omega} \operatorname*{colim}_n \mathbb{S}/p^n \simeq \operatorname*{colim}_n (\mathbb{S}/p^n)^\vee \eqqcolon \operatorname{\Omega}\mathbb{S}/p^\infty,\end{gather*} \] and the \(p\)-completion is given by the mapping spectrum \(X_{\hat p} \simeq \mathrm{map}(\operatorname{\Omega}\mathbb{S}/p^\infty, X)\). The above formula follows.

The comparison of \(D_E\) with completion

Assume for this section that \(\pi_0 E \simeq \mathbb{Z}/n\). The classical situation is concerned with the discrete ring spectrum \(E = \mathbb{F}_p\), but everything we said above about idempotent algebras applies to \(\mathbb{S}[1/n]\). We call the \(\mathbb{S}/n\)-localisation \(n\)-(adic )completion.

Every \(\mathbb{S}/n\)-acyclic spectrum is \(E\)-acyclic. Thus every \(E\)-local spectrum, in particular every spectrum which admits an \(E\)-module structure, is \(n\)-complete.

Proof. The \(\mathbb{S}/n\)-acyclic spectra are exactly the \(\mathbb{S}[1/n]\)-modules, and \(\mathbb{S}[1/n] \otimes E = 0\).

The \(E\)-locality of \(E\)-modules follows from \[\mathrm{map}_{\mathrm{Sp}}(X, M) \simeq \mathrm{map}_{\textrm{Mod-}E}(X \otimes E, M)\]

Thus the descent diagram \(D^\bullet_E X\) consists of \(E\)-local spectra, so the limit \(D_E X\) is also \(E\)-local. We thus obtain a unique factorisation \[ X \to X_{\hat n} \to D_E X. \]

If \(E\) and \(X\) are connective, then \(X\to D_E X\) is an \(n\)-adic equivalence. In other words, \(X_{\hat n} \xrightarrow{\sim} D_E X\) is an equivalence.

Proof. As we have seen, both sides are limits of their values on truncations, and the spectra for which the conclusion holds is stable (and idempotent complete). Thus it suffices to prove the conclusion when \(X\) is discrete (i.e. an Eilenberg–Mac Lane spectrum). We can tensor the \(E\)-descent diagram with \(\mathbb{S}/n\) (a finite spectrum!), which commutes with totalisation. The goal is to show that \(X/n \to (D_E X)/n \simeq D_E(X/n)\) is an isomorphism.

We show that \(X/n\) admits an \(E\)-module structure. As \(X\) is discrete, it admits a \(\mathbb{Z}\)-module structure, and \(X/n\) admits a module structure over \(\mathbb{Z} \otimes \mathbb{S}/n \simeq \mathbb{Z}/n \simeq \tau_{\leq 0} E\). Since \(E\) was assumed to be connective, the truncation \(E \to \tau_{\leq 0} E\) is a morphism of rings, and we're done.

The comparison of \(D_E\) with localisation

Assume now that \(\pi_0 E \simeq \mathbb{Z}[T^{-1}]\), then \(E\) is an algebra over the idempotent algebra of the Moore spectrum \(\mathbb{S}[T^{-1}]\). Hence there is a factorisation \[ X \to X[T^{-1}] \to D_E X. \]

If \(E\) and \(X\) are connective, then \(X[T^{-1}] \simeq D_E X\).

Proof. As before, we reduce to the case where \(X\) is discrete. Since \(E \simeq \mathbb{S}[T^{-1}] \otimes E\), we have \(D_E X \simeq D_E(X[T^{-1}])\). Again, the \(E\)-descent diagram for \(X[T^{-1}]\) splits since it admits a module structure over \(\tau_{\leq 0} E\).

Remark. If \(\tau_{\leq 0} E \simeq \mathbb{Z}\), then \(I\) is \(1\)-connective, so already the error of descent \(T_E\) vanishes on bounded below spectra.

Bonus: a few examples of \(\operatorname*{lim^1}\) to keep in mind

I don't have a specific example of a connective spectrum whose Adams spectral sequence fails to converge for this reason, but instead I'll list a few examples of non-exactness of limits as a sketch of the issue which may arise.

Limit of surjections need not be surjection

The limit as \(n \to \infty\) of the exact sequences \[ 0 \to \mathbb{Z} \xrightarrow{p^n} \mathbb{Z} \to \mathbb{Z}/ p^n \to 0 \] (constant in the middle) is only left exact: \[ 0 \to 0 \to \mathbb{Z} \to \mathbb{Z}_{p}. \] One can see that \[ \operatorname*{lim^1} (\ldots \to \mathbb{Z} \xrightarrow{p} \mathbb{Z} \xrightarrow{p} \mathbb{Z} \to\ldots) \simeq \mathbb{Z}_{p}/\mathbb{Z} \]

Remark The limit of a diagram consisting of surjective maps of sets surjects onto the diagram.

Limit and image don't play along

In order to show that the \(E_\infty\)-page computes the graded pieces of a filtration on the colimit, one needs to look at certain intersections of images. This is an issue even when the tower conditionally converges, as in our cases above.

As \(N \to \infty\), each of the maps \[ \bigoplus_{p \geq N \text{ prime}} p\mathbb{Z} \xrightarrow{\nabla} \mathbb{Z} \] is surjective, so that the limit of image is \(\mathbb{Z}\), but the image of the limit map is of course \(0\).