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Divided power rings play a pretty prominent role in a bunch of areas of math. For example, it’s used to define crystalline cohomology, which (after inverting the implicit prime) provides an example of a Weil cohomology theory; related to this, it shows up when studying the Hochschild homology of finite fields; and related to this, it naturally comes up when studying the cohomology of loop spaces. One natural reason why divided powers come up so often in algebraic topology is because the dual of a polynomial algebra in one generator, regarded as a bialgebra by letting its generator be primitive, is a divided power algebra. In algebraic geometry, the appearance of divided powers in crystalline cohomology is morally justified essentially because of Berthelot’s de Rham-crystalline comparison theorem.

Let’s recall the definition of divided powers. Let \(A\) be a ring and \(I\) an ideal. Then a divided power structure on \(I\) is a collection of maps \(\{\gamma_n: I \to I\}\) that’s roughly supposed to send \(x\) to “\(x^n/n!\)”; we won’t write down the axioms, but two important properties that are required are:

\[\gamma_n(x)\gamma_n(x) = \binom{m+n}{m} \gamma_{m+n}(x), \text{ and } \gamma_n(x+y) = \gamma_n(x) + \gamma_n(y) + \sum_{i=1}^{n-1} \gamma_i(x)\gamma_{n-i}(y).\]

For instance, if \(A\) is a DVR of mixed characteristic \((0,p)\), then the maximal ideal of \(A\) admits a divided power structure if and only if the ramification index \(e\) is at most \(p-1\); this is not hard to prove using the classical calculation of the \(p\)-adic valuation of \(n!\). In particular, the ideal \((p)\) inside \(\Z_p\) admits a divided power structure, since \(p^n/n!\in \Z_p\).

In basic algebraic topology, one source of divided power algebras comes from the cohomology of loop spaces of spheres. Let \(\Gamma[x]\) denote the divided power polynomial algebra on one generator (i.e., \(\Z[x_1,x_2,\cdots]/\left(\binom{m+n}{n} x_{m+n} = x_m x_n\right)\)) and \(\Lambda(x)\) the exterior algebra on one generator; then, a classical computation with the Serre spectral sequence shows that

\[\H^\ast(\Omega S^n) = \begin{cases} \Gamma[x_{n-1}] & \text{ if }n\equiv 1\pmod{2}\\ \Lambda(x_{n-1}) \otimes \Gamma[y_{2(n-1)}] & \text{ else}. \end{cases}\]

Here, the degrees of the generators are given in the subscripts. The reason for this is essentially due to the fact mentioned above about divided power algebras being dual to polynomial algebras, and that \(\H_\ast(\Omega S^n) \cong \Z[x_{n-1}]\) as bialgebras, where the generator \(x_{n-1}\) is primitive.

Things get a bit wonky in positive characteristic, though: by induction, we can prove that in \(\Gamma[x]\), we may identify \(x_n = x^n/n!\). We then conclude that if \(p=0\) in a base field \(k\), we’d have \(x^p = 0\). In fact, it’s not hard to show that the divided power algebra is \(\Gamma[x]\otimes k \cong k[x_1,x_p,\cdots]/(x_1^p, x_p^p, \cdots)\). That doesn’t look very nice…

Let’s trudge on anyway. I mentioned above that you can also get a divided power algebra by studying the Hochschild homology of a finite field, so let’s do that. We’ll do an explicit calculation: recall that the Hochschild homology of a commutative ring \(R\) is equivalent to the (everywhere derived) tensor product \(R\otimes_{R\otimes R} R\), so if we’d want to compute the Hochschild homology of \(\FF_p\), our first task would be to compute the derived tensor product \(\FF_p\otimes \FF_p\). To do this, we’d need to resolve \(\FF_p\) as a commutative DGA, but this is easy: we can just use \(\Z[t]/t^2\), where \(\d t = p\) and \(|t| = 1\). Our task is then reduced to understanding \(\FF_p\otimes_{\Z[t]/t^2} \FF_p\). We can now resolve \(\FF_p\) as a \(\Z[t]/t^2\)-algebra by considering the divided power algebra \(A[x_1,x_2,\cdots]/\left(\binom{m+n}{n} x_{m+n} = x_m x_n\right)\), where \(A = \Z[t]/t^2\), \(|x_n| = 2n\), and \(\d x_n = tx_{n-1}\). Base-changing this up to \(\FF_p\) shows that the Hochschild homology of \(\FF_p\) is a commutative DGA which is quasi-isomorphic to its homology.

As I said before, this appearance of divided powers in characteristic \(p\) is sort of pathological, and we’d like to resolve this. A philosophy that’s been becoming more prevalent recently is that the existence of divided powers is because one’s working with \(\H\Z\) as the base; if one instead regards the sphere spectrum as one’s base, then the divided power pathology vanishes. Let’s see this concretely in the case of Hochschild homology.

The topological Hochschild homology of an \(\E_\infty\)-ring \(A\) is the smash product \(A\wedge_{A\wedge A} A\), so we’d like to understand what happens when \(A = \H\FF_p\). See here. We first need to understand \(\H\FF_p\wedge \H\FF_p\). A theorem of Hopkins and Mahowald’s says that \(\H\FF_p\) is the free \(p\)-local \(\E_2\)-algebra with a nullhomotopy of \(p\), i.e., \(\H\FF_p\) is the smash product \(S^0 \otimes_{\F_{\E_2}(x)} S^0\), where one of the maps to \(S^0\) sends \(x\) to \(p\), and the other sends \(x\) to zero. We know that \(\H\FF_p\wedge \H\FF_p\) is therefore the free \(\E_2\)-\(\H\FF_p\)-algebra with \(p=0\). After smashing the entire diagram with \(\H\FF_p\) and taking the colimit, we obtain \(\H\FF_p\wedge_{\F_{\E_2\text{-}\H\FF_p}(x)} \H\FF_p\), where the maps \(\F_{\E_2\text{-}\H\FF_p}(x)\to \H\FF_p\) are the augmentations. It follows that the smash product above is \(\F_{\E_2\text{-}\H\FF_p}(\Sigma x)\). Using the bar construction, we find that the pushout \(\mathrm{THH}(A) = \H\FF_p\wedge_{\F_{\E_2\text{-}\H\FF_p}(\Sigma x)} \H\FF_p\) is \(\F_{\E_1\text{-}\H\FF_p}(\Sigma^2 x)\). The homotopy groups of this object is exactly \(\pi_\ast(\H\FF_p\wedge \Omega S^3) \cong \H_\ast(\Omega S^3; \FF_p) \cong \FF_p[u_2]\). The canonical map \(\pi_\ast\mathrm{THH}(\H\FF_p) \to \mathrm{HH}_\ast(\FF_p)\) sends \(u_2\) to the element denoted \(x_1\) above.

Morally speaking, why does working over the sphere resolve the issue of divided powers? The unit map \(S^0 \to \H\Z\) is the group-completion of the map \(\mathrm{FinSet}_\simeq \to \mathbf{N}\) of monoids in spaces, and this map quotients out symmetric groups; taking symmetric quotients is like working with divided powers. This might not be a satisfactory explanation, so let’s try for another.

One philosophy adopted in chromatic homotopy theory is that the sphere spectrum should be the global sections of some sheaf of \(\Eoo\)-ring spectra on the moduli stack \(\Mfg\) of formal groups. In particular, a natural algebraic approximation to the sphere spectrum is \(\Mfg\); so instead of considering the derived tensor product \(\FF_p\otimes_{\FF_p\otimes_\Z \FF_p} \FF_p\), let us consider the fiber product \(\spec \FF_p\times^\mathbf{L}_{\spec \FF_p \times_{\Mfg} \spec \FF_p} \spec \FF_p\). Let’s specialize to the case when \(p=2\); then, \(\spec \FF_2 \times_{\Mfg} \spec \FF_2 \cong \Aut(\hat{\GG}_a) \cong \spec A_\ast\), where the last isomorphism was discussed in the beginning of this post. In particular, \(\spec \FF_2\times^\mathbf{L}_{\spec \FF_2 \times_{\Mfg} \spec \FF_2} \spec \FF_2 \cong \mathrm{Tor}_{A_\ast}(\FF_2, \FF_2) \cong \Lambda_{\FF_2}[\sigma \zeta_1, \sigma \zeta_2, \cdots]\), where \(|\sigma \zeta_n| = |\zeta_n| + 1 = 2^n\); this is the \(E_2\)-page of the Kunneth spectral sequence for \(\mathrm{THH}(\H\FF_2)\). (More precisely, the algebra above is generated by \(I/I^2\), where \(I\) is the augmentation ideal of the dual Steenrod algebra.) You could do this at any odd prime, too, and you’d still get an exterior algebra. I’m not sure if one should consider this to be “pathological” in the same way that the ordinary Hochschild homology of \(\FF_p\) is pathological: the Kunneth spectral sequence degenerates at the \(E_2\)-page, and the only data required to solve the extension problems is precisely the data of the Dyer-Lashof operations (for instance, \(Q^2(\zeta_1) = \zeta_2\), so the generator \(\sigma \zeta_1\) should square to \(\sigma \zeta_2\)).

We can run this argument more generally by replacing \(\spec \FF_2\) with \(\spec \pi_0 A\) for any Landweber-exact even-periodic \(\Eoo\)-ring \(A\). Suppose that \(\GG\) is the associated formal group over \(\spec \pi_0 A\); then, the fiber product \(\spec \pi_0 A\times^\mathbf{L}_{\spec \pi_0 A \times_{\Mfg} \spec \pi_0 A} \spec \pi_0 A\) is isomorphic to \(\spec \pi_0 A\times^\mathbf{L}_{\Aut_{\pi_0 A}(\GG)} \spec \pi_0 A\). This is the derived loop space \(L\Aut_{\pi_0 A}(\GG)\) (in the sense of derived, not spectral, algebraic geometry) of the \(\pi_0 A\)-group scheme \(\Aut_{\pi_0 A}(\GG)\). The global sections of this derived stack over \(\pi_0 A\) is the \(E_2\)-page of the Kunneth spectral sequence computing \(\mathrm{THH}(A)\); the difference between \(L\Aut_{\pi_0 A}(\GG)\) and \(\mathrm{THH}(A)\) is precisely the difference between \(L\Aut_{\pi_0 A}(\GG)\) and the spectral loop space \(L\Aut_A(\wt{\GG})\), where \(\wt{\GG}\) is the spectral formal group living over \(A\).

Let’s look at another natural source of divided powers: the representation theory of the additive group scheme \(\GG_a\). Since \(\GG_a \cong \spec \Z[x]\), we find that representations of \(\GG_a\) are equivalent to comodules over \(\Z[x]\), which in turn are equivalent to modules over the divided power ring \(\Gamma_\Z[x]\). Given this, one might ask: what are the representations of the “spectral additive group”?

There are two different notions of what one might mean by the phrase “spectral additive group”. Let’s first consider the spectral affine line (because the underlying scheme of the additive group is the affine line). In the classical world, the affine line \(\AA^1_R\) over a discrete ring \(R\) is defined as \(\spec\) of the polynomial ring \(R[x]\) in one generator. In the spectral setting, a natural replacement for the polynomial ring over an \(\Eoo\)-ring \(R\) might therefore be the free \(\Eoo\)-ring \(R\{x\}\) on one generator. We almost immediately run into issues: the free \(\Eoo\)-ring on one generator is not flat over \(R\); in fact, it’ll be flat if and only if \(R\) is a \(\QQ\)-algebra.

Let’s make a brief digression. In the classical setting, a map \(X\to Y\) is smooth if, Zariski-locally, it admits an etale cover by affine space. In the spectral setting, one might want to imitate this definition. Since there is a reasonable definition of etaleness, and the phrase “Zariski locally” makes sense, we only need to decide on a definition of affine space. In particular, the notion of smoothness resulting from using the free \(\Eoo\)-ring as the affine line, termed “differentially smooth” by Lurie, does not reduce to the classical definition in the discrete case (so, for example, \(\PP^1_R\) is not differentially smooth when \(R\) is discrete). Moreover, \(\pi_0 R\{x\}\) need not look like a polynomial ring on one generator over \(\pi_0 R\). This suggests a natural replacement: there is an \(\Eoo\)-ring \(\Sigma^\infty_+ \mathbf{N}\), and we can define this to be a polynomial ring \(S[t]\) over the sphere spectrum. Working with the resulting notion of smoothness (“fiber smoothness”) affords a lot of rigidity not available with differentially smooth morphisms.

The upshot of the above discussion is that we have two different candidates for the spectral affine line: \(\AA^1_\top = \spec S^0\{x\}\) and \(\AA^1_\der = \spec \Sigma^\infty_+ \mathbf{N}\). Both admit the structure of group schemes, so we’ll denote these group schemes by \(\GG_a^\top\) and \(\GG_a^\der\). We can then consider representations of these algebraic groups. Representations of \(\GG_a^\der\) are the same as modules over the dual of the \(\Eoo\)-algebra \(\Sigma^\infty_+ \mathbf{N}\). By the discussion in Section 1.4 here, these are the same as modules over the free smooth coalgebra over the sphere spectrum on one generator. This is a flat \(S^0\)-algebra \(A\) such that \(\pi_0 A \cong \Gamma_\Z[x_0]\).

Let’s now instead consider representations of \(\GG_a^\top\). These are the same as modules over the dual of the \(\Eoo\)-algebra \(S^0\{x\}\). By the Barratt-Priddy-Quillen theorem, \(S^0\{x\} \cong \coprod_{n\geq 0} B\Sigma_n\) (where \(B\Sigma_n\) really means the unpointed supension spectrum). Since \(B\Sigma_n = S^0_{h\Sigma_n}\), we find that the dual of \(S^0\{x\}\) is \(\prod_{n\geq 0} (S^0)^{h\Sigma_n}\). This a priori has nothing to do with divided powers, but it’s also just a hopelessly unapproachable object. To attempt to understand this beast, let’s \(K(n)\)-localize the entire situation (for some implicit prime \(p\) and a fixed height \(n\)), and consider the \(K(n)\)-local dual of \(L_{K(n)} (S^0\{x\}) \simeq (L_{K(n)} S^0)\{x\} =: L_{K(n)} S^0\{x\}\). By the discussion above, we are reduced to understanding the \(K(n)\)-local dual of \(L_{K(n)} B\Sigma_m\) for all integers \(m\).

We claim that the \(K(n)\)-local dual of \(L_{K(n)} B\Sigma_m\) is equivalent to \(L_{K(n)} B\Sigma_m\). A consequence of the classical work of Hovey and Sadofsky and Example 5.1.10 of Hopkins and Lurie is that the \(K(n)\)-local Spanier-Whitehead dual of \(L_{K(n)} B\Sigma_m\) is equivalent to a shift of \(L_{K(n)} B\Sigma_m\) by an element of the Picard group of \(L_{K(n)} \mathrm{Sp}\). Let’s not bother ourselves with this Picard shift (see here, though); then, we find that the \(K(n)\)-local Spanier-Whitehead dual of \(L_{K(n)} S^0\{x\}\) is equivalent to the infinite product \(\prod_{m\geq 0} L_{K(n)} B\Sigma_m\). In fact, I think that the “\(K(n)\)-local divided power algebra” given by the Spanier-Whitehead dual of \(L_{K(n)} S^0\{x\}\) is equivalent to the completion of \(L_{K(n)} S^0\{x\}\) at the ideal generated by \((x)\); see Example 3.14 here.

This is very much analogous to the situation involving divided power algebras in characteristic zero: in this case, we can simply divide by \(n!\), so the free divided power algebra \(\Gamma_k[x]\) over a field of characteristic \(k\) is simply the polynomial algebra \(k[x]\). There’s no pathological behavior here! The upshot of this entire discussion is that if we regard “divided powers” as arising via duals of polynomial algebras, then upon passing to the sphere spectrum, there should be no distinction between divided powers and free \(\Eoo\)-algebras at all the “finite primes” of \(S^0\); the only divergence occurs at the “infinite prime” \((p,v_1,v_2,\cdots)\).