B\"okstedt periodicity and quotients of DVRs

In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for B\"okstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).


Introduction
Topological Hochschild homology (THH), together with its induced variant topological cyclic homology (TC), has been one of the major tools to compute algebraic K-theory in recent years. It also is an important invariant in its own right, due to its connection to p-adic Hodge theory and crystalline cohomology [BMS18,BMS19].
The key point is that THH * (R), as opposed to algebraic K-theory, can be completely identified for many rings R. Let us list some examples here.
(i) The most fundamental result in the field is Bökstedt periodicity, which states that THH * (F p ) = F p [x] for a class x in degree two. This is also the input for the work of Bhatt, Morrow and Scholze [BMS19]. (ii) The p-adic computation of THH * (Z p ) was also done by Bökstedt and eventually lead to the p-adic identification of K * (Z p ), see [BM94,Rog99]. (iii) More generally, Lindenstrauss and Madsen identify THH * (A) p-adically for a complete discrete valuation ring (CDVR) A with perfect residue field k of characteristic p [LM00]. This computation was one of the key inputs for Hesselholt and Madsen's seminal computation of K-theory of rings of integers in p-adic number fields. (iv) Brun computed THH * (Z/p n ) in [Bru00]. This gives some information about K * (Z/p n ), which is still largely unknown, see [Bru01].
In this paper, we revisit all the THH computations mentioned above from scratch, and give new, easier and more conceptual proofs. We go one step further and give a complete formula for THH * (A ) where A = A/π k is a quotient of a discrete valuation ring (DVR) A with perfect residue field of characteristic p. We identify THH * (A ) with the homology of an explicitly described differential graded algebra (DGA); see Theorem 5.2. This for example recovers the A. Krause and T. Nikolaus computation of THH * (Z/p k ) by Brun and also identifies the ring structure in this case (which was unknown so far). The result shows an interesting dichotomy depending on how large k is when compared with the p-adic valuation of the derivative of the minimal polynomial of a uniformizer of A (relative to the Witt vectors of the residue field), see § 6.
The main new idea employed in this paper is to first compute THH of A and A/π k relative to the spherical polynomial ring S[z]. This relative THH of A satisfies a form of Bökstedt periodicity, which was to the best of the authors' knowledge first observed by Lurie, Scholze and Bhatt. It appeared in work of Bhatt,Morrow and Scholze [BMS19] as well as in [AMN18]. However, the maneuver of working relative to the uniformizer is much older in the algebraic context, for example in the theory of Breuil-Kisin modules [Kat94, Bre99,Kis09]. 1 Finally, having computed THH relative to S[z] we use a descent style spectral sequence (see § § 4 and 5) to recover the absolute THH. In § 10 we also deduce the computation of logarithmic THH of CDVRs (due to Hesselholt and Madsen) from the computation of relative THH using a similar spectral sequence.

Conventions
We freely use the language of ∞-categories and spectra. For this we use [Lur17] as our main reference. Specifically we use the theory of algebras and modules in the symmetric monoidal ∞-category of spectra as discussed in § 7.1 of [Lur17]. The sphere spectrum is denoted by S. For a commutative ring R there is an associated commutative ring spectrum called the Eilenberg-MacLane spectrum of R which we abusively also denote by R.
In this situation we have the ring spectra HH(R) ('Hochschild homology') and THH(R) ('Topological Hochschild homology') defined as HH(R/Z) = R ⊗ R⊗ Z R R, THH(R) = R ⊗ R⊗ S R R, see [Lur17,§ 5.5 and Theorem 5.5.3.11] as well as [NS18,§ III.2]. We denote the homotopy groups of these spectra by THH * (R) and HH * (R). More generally there are relative versions for a ring R over a base ring (spectrum) S given as THH(R/S) = R ⊗ R⊗ S R R and similar for HH. Note that Hochschild homology as defined here is equivalent to THH(R/Z) and is automatically fully derived. It thus agrees with what is classically called Shukla homology.
We denote the p-completion of the spectrum THH(R) by THH(R; Z p ) and the homotopy groups accordingly by THH * (R; Z p ). Note that these are, in general, not the p-completions of the groups THH * (R), but in the case that the groups THH * (R) have bounded order of p ∞ -torsion this is true. There is the commonly used conflicting notation THH * (R; R ) for THH with coefficients in an R-algebra R , given by he homotopy groups of THH(R) ⊗ R R . To avoid confusion we do not use the notation THH(R; R ) in this paper. and some variants which are straightforward to prove and will be used frequently.

Bökstedt periodicity for F p
We want to give a proof of the fundamental result of Bökstedt, that THH of F p is a polynomial ring on a degree-two generator. The proof presented here is closely related to the Thom spectrum proof in [Blu10] based on a result of Hopkins and Mahowald, but in our opinion it is more direct, see Appendix A for a precise discussion. Let us first give a slightly more conceptual formulation of Bökstedt's result.
Theorem 1.1 (Bökstedt). The spectrum THH(F p ) is as an E 1 -algebra spectrum over F p free on a generator x in degree two, i.e. equivalent to F p [ΩS 3 ].
Here F p [ΩS 3 ] is the group ring of the E 1 -group ΩS 3 over F p , i.e. the F p -homology F p ⊗ S Σ ∞ + ΩS 3 . The equivalence between the two formulations relies on the fact that ΩS 3 is the free E 1 -group on S 2 , where S 2 is considered as a pointed space. The latter fact follows from the fact, due to Boardman-Vogt and May, that the functor Ω induces an equivalence between pointed connected spaces and E 1 -groups, see [Lur17,Theorem 5.2.6.10]. Thus, for any pointed space X (here X = S 2 ) maps of E 1 -groups from ΩΣX to any E 1 -group G = ΩY are indeed given by maps of pointed spaces ΣX → Y or equivalently maps of pointed spaces X → ΩY = G. A similar argument using Ω 2 and E 2 -groups shows that Ω 2 Σ 2 X is the free E 2 -group on any pointed space X.
Our proof of Theorem 1.1 relies on a structural result about the dual Steenrod algebra F p ⊗ S F p . We consider this spectrum as an F p -algebra using the inclusion into the left factor. 2 It is an E ∞ -algebra over F p , but has a universal description as an E 2 -algebra. This result seems to be known, at least to some experts, but we have not been able to find it written up in the literature. Theorem 1.2. As an E 2 -F p -algebra, the spectrum F p ⊗ S F p is free on a single generator of degree 1, i.e. it is as an We give a proof of Theorem 1.2 in the next section. However, let us first deduce Theorem 1.1 from it.
Proof of Theorem 1.1. We have an equivalence of E 1 -algebras The third equivalence uses that F p [−] sends products to tensor products and preserves colimits.
Remark 1.3. If one only wants to use that F p ⊗ S F p is free as an abstract E 2 -algebra and avoid space-level arguments, one can observe that in any pointed presentably symmetric monoidal ∞-category C one has for every object X ∈ C an equivalence This is proven in [Lur17,Corollary 5.2.2.13] for n = 0 and the case n > 0 can be reduced to this case using Dunn additivity by replacing C with the ∞-category of augmented 2 If we use the right factor this produces an equivalent Fp-algebra where the equivalence is the conjugation.

Proof of Theorem 1.2
In order to prove this result we first recall from [BMMS86,III.3] that for every E 2 -ring spectrum R over F 2 there exist Dyer-Lashof operations for 2i ≤ k + 1. They satisfy the usual relations, except that for the top operations (where 2i = k + 1) there are correction terms in terms of the Browder bracket. In particular, they are not generally additive. See [BMMS86,III.3,Theorem], where the top operations are denoted by ξ 1 .
Note that for odd p, because Q i x = 0 when 2i < |x| and Q i x = x p when 2i = |x|, the only interesting operations are the top operations Q (|x|+1)/2 x for odd |x|. Similarly, for p = 2, the only interesting operations are Q |x|+1 x. The iterates of these operations describe generators of the homotopy of the free E 2 -algebra.
Proposition 1.4. Let R be the free E 2 -algebra over F p on a generator in degree one. Then Any E 2 -algebra R over F p whose homotopy ring together with the action of the Dyer-Lashof operations is of the above form, is also free on a generator in degree one.
Proof. The free E 2 -algebra over F p on a generator in degree one is given by F p [Ω 2 S 3 ], see the discussion after Theorem 1.1. Thus, we are simply describing H * (Ω 2 S 3 ; F p ) with the Pontryagin ring structure. The first part is due to Araki and Kudo [KA56,Theorem 7.1], the second part is due to Dyer and Lashof [DL62,Theorem 5.2]. These results are relatively straightforward computations using the Serre spectral sequence and the Kudo transgression theorem. Now for the last statement assume that we have given any such R and any non-trivial element x 1 ∈ π 1 (R). We get an induced map from the free algebra Free E 2 (x 1 ) → R. As this map is an E 2 -map the induced map on homotopy groups is compatible with the ring structure as well as the indicated Dyer-Lashof operations. Everything is generated from x 1 under these operations in the same way, so the map is an equivalence.
Proof of Theorem 1.2. By Proposition 1.4 we only have to verify that the homotopy groups of F p ⊗ S F p have the correct ring structure and Dyer-Lashof operations. This is a classical calculation due to Milnor for the ring structure and Steinberger [BMMS86, Chapter 3, 1686 Bökstedt periodicity and quotients of DVRs Theorems 2.2 and 2.3] for the Dyer-Lashof operations: at p = 2, the generator x i corresponds to the Milnor basis element ζ i , at p odd z i corresponds to the element ξ i and y i to τ i . Remark 1.5. We want to remark that Theorem 1.1 also implies Theorem 1.2. To see this, assume that Theorem 1.1 holds. We have that π 1 (F p ⊗ S F p ) is isomorphic to F p , generated by an element b. We can thus choose an E 2 -map which induces an equivalence on 1-types. 3 We can form the bar construction on these augmented F p -algebras, and the resulting map is an equivalence on π 2 , so by Theorem 1.1 it is an equivalence. Thus, Theorem 1.2 follows from the following lemma.
Proof. Assume A → B is not an equivalence. Let d denote the connectivity of the cofiber of A → B, i.e. π i (B/A) = 0 for i < d, but π d (B/A) = 0. The spectrum F p ⊗ A F p admits a filtration (obtained by filtering the bar construction over F p by its skeleta) whose associated graded is given in degree n by Σ n (A/F p ) ⊗ Fp n . Here A/F p is the cofiber of F p → A and 1-connective by assumption. The map Σ n (A/F p ) ⊗ Fp n → Σ n (B/F p ) ⊗ Fp n has (d + 2n − 1)-connective cofiber. Thus, the (d + 1)-type of the cofiber of F p ⊗ A F p → F p ⊗ B F p receives no contribution from the terms for n ≥ 2, and coincides with the (d + 1)-type of the cofiber of Σ(A/F p ) → Σ(B/F p ), which is Σ(B/A) and has non-vanishing π d+1 by assumption. Thus, F p ⊗ A F p → F p ⊗ B F p cannot have been an equivalence.

Bökstedt periodicity for perfect rings
Now we also want to recover the well-known calculation of THH for a perfect F p -algebra k. This can directly be reduced to Bökstedt's theorem. Let us first note that there is a morphism THH(F p ) → THH(k) induced from the map F p → k. Moreover, the spectrum THH(k) is a k-module, so that we obtain an induced map where the first term k[x] denotes the free E 1 -algebra on a generator in degree two.
Proposition 2.1. For a perfect F p -algebra k the map (1) is an equivalence.
Proof. Recall that for every perfect F p -algebra k there is a p-complete E ∞ -ring spectrum S W (k) , called the spherical Witt vectors, with π 0 (S W (k) ) ∼ = W (k) and which is flat over S p , see, e.g., [Lur18, Theorem 5.2.5 and Example 5.2.7]. It follows that the homology Z ⊗ S S W (k) is given by W (k) and, thus, the F p -homology F p ⊗ S S W (k) by k.

A. Krause and T. Nikolaus
In particular we get that where HH(k/F p ) is the Hochschild homology of k relative to F p . The result now follows once we know that this is given by k concentrated in degree zero. This immediately follows from the vanishing of the cotangent complex of k but we want to give a slightly different argument here. It suffices to show that the positive dimensional groups HH i (k/F p ) are zero. To see this it is enough to show that for every F p -algebra A the Frobenius ϕ : A → A induces the zero map HH i (A/F p ) → HH i (A/F p ) for i > 0, because for A = k perfect the Frobenius is also an isomorphism. Now for general A this follows because HH(A/F p ) is a simplicial commutative F p -algebra and the Frobenius ϕ acts through the levelwise Frobenius. However, the levelwise Frobenius for every simplicial commutative F p -algebra induces the zero map in positive dimensional homotopy. This follows because for every simplicial commutative F p -algebra R • the Frobenius can be factored as π n (R • ) → π n (R • ) ×p → π n (R • ) where the latter map is induced by the multiplication R ×p • → R • considered as a map of underlying simplicial sets. For n > 0 it follows by an Eckmann-Hilton argument that the multiplication map π n (R • ) × π n (R • ) → π n (R • ) is at the same time bilinear and linear, hence zero.
Remark 2.2. Note that the proof in particular shows that THH(S W (k) ) is p-adically equivalent to S W (k) as this can be checked on F p -homology. We also write THH(S W (k) ; Z p ) for the p-completion of THH(S W (k) ) so that we have Integrally this is not quite the case, as one encounters contributions form the cotangent complex L W (k)/Z which only vanishes after p-completion.
We also note that one can also deduce Proposition 2.1 from a statement similar to Theorem 1.2 which we want to list for completeness.
so the statement follows from base-changing the statement over F p .

Bökstedt periodicity for CDVRs
Now we want to turn our attention to CDVRs. We determine their absolute THH later, but for the moment we focus on an analogue of Bökstedt's theorem which works relative to the E ∞ -ring spectrum For a CDVR A we let π be a uniformizer, i.e. a generator of the maximal ideal, and consider it as a S[z]-algebra via z → π. Everything that follows will implicitly depend on such a choice. By assumption A is complete with respect to π. As π is a non-zero-divisor this is equivalent to A being derived π-complete. Moreover, A if has residue field of characteristic p, then A is also (derived) p-complete because p is contained in the maximal ideal.
The following result is, at least in mixed characteristic, due to Bhatt, Lurie, and Scholze, in a private communication; versions of it also appear in [BMS19] and in [AMN18].
Theorem 3.1. Let A be a CDVR with perfect residue field k of characteristic p. Then we have for x in degree two.
Proof. We distinguish the cases of equal and of mixed characteristic. In mixed characteristic we have the equation of ideals (p) = (π e ) where e is the ramification index. We deduce that where we in the third equivalence we used that the S[z]-module structure on THH(A/S[z]; Z p ) factors through the A-module structure, and in the fourth equivalence we used the fact that base-change from S[z]-modules to S-modules is symmetric monoidal and preserves colimits, thus commutes with relative THH (because it can be described as a cyclic bar construction).
By Proposition 2.1 this shows that the mod π reduction of THH(A/S[z]; Z p ) has homotopy groups given by an even-dimensional polynomial ring over k. Thus, from the long exact sequence associated with the cofiber sequence we see that the odd homotopy groups of THH(A/S[z]; Z p ) have vanishing mod π reduction. As they are also derived π-complete we deduce that they vanish. Then it follows that the even homotopy groups are π-torsion free. The result now follows by choosing a lift of the Bökstedt element x to THH 2 (A/S[z]; Z p ) and observing that the induced map is an isomorphism modulo π and, thus, an isomorphism.
If A is of equal characteristic p then A is isomorphic to the formal power series ring k [[z]]. We consider the E ∞ -ring S W (k) [[z]] obtained as the z-completion of S W (k) [z]. Then we have an equivalence A. Krause and T. Nikolaus which uses that F p is of finite type over the sphere. As a result, we obtain an equivalence is a pushout of commutative ring spectra, where ϕ is the Frobenius. This holds because induced from the square (2) is an equivalence because the square is a pushout. We claim again, as in the proof of Proposition 2.1, that this map is zero for i > 0. As ϕ : is an equivalence where A p is the p-completion of A. This is true for every ring A. Moreover, for a DVR the p-completion A p is the same as the completion of A with respect to the maximal ideal so that Theorem 3.1 applies to yield that For every prime = p we have that Remark 3.4. One can also deduce the mixed characteristic version of Theorem 3.1 from an analogue of Theorem 1.2 which under the same assumptions as Theorem 3.1 and in mixed characteristic states that A ⊗ S W (k) [z] A is p-adically the free E 2 -algebra on a single generator in degree one.
We also want to remark that there are some equivalent ways of stating Theorem 3.1 which might be a bit more canonical from a certain point of view.
Proposition 3.5. In the situation of Theorem 3.1 the map Proof. For the upper four maps this follows from the equivalences which can all be checked in F p -homology (see Remark 2.2 and the proof of Theorem 3.1). The last two vertical equivalences follows because THH(A/S W (k) [z]) and THH(A/S W (k) [[z]]) are already p-complete. If A is of equal characteristic this is clear anyhow (and in the whole diagram we did not need the p-completions). In mixed characteristic this follows from Lemma 3.6, because A is of finite type over S W (k) [z] and over S W (k) [[z]], which can be seen by the presentation where E is the minimal polynomial of the uniformizer π.
Recall that a connective ring spectrum A over a connective, commutative ring spectrum S is said to be of finite type if A is as an R-module a filtered colimit of perfect modules along increasingly connective maps (i.e. has a cell structure with finite 'skeleta').
Lemma 3.6. If A is p-complete and of finite type over R, then THH(A/R) is also p-complete.
Proof. We first observe that all tensor products A ⊗ R · · · ⊗ R A are of finite type over A (say by action from the right) which follows inductively. Thus, they are p-complete. Finally, the n-truncation of THH(A/R) is equivalent to the n-truncation of the restriction of the cyclic bar construction to Δ op ≤n+1 . This colimit is finite and the stages are p-complete by the above. We now consider quotients A of a CDVR A as in Theorem 3.1. Every ideal is of the form (π k ) ⊆ A and thus A ∼ = A/π k for some k ≥ 1.
Proposition 3.7. For a CDVR A with residue field of characteristic p, and a quotient A = A/(π k ) of A, where π denotes a choice of uniformizer, we have a canonical equivalence and on homotopy groups we obtain where y is a divided power generator in degree two.

Proof. As π is a non-zero divisor we can write
where in the last step we have used that p is nilpotent in A and, thus, we are already p-complete. Finally HH((Z[z]/z k )/Z[z]) is given by a divided power algebra (Z[z]/z k ) y . To see this we first observe that Z[z]/z k ⊗ Z[z] Z[z]/z k is given by the exterior algebra Λ Z[z]/z k (e) with e in degree one. Then it follows that HH((Z[z]/z k )/Z[z]), which is the bar construction on that, is given by This implies the claim.

Absolute THH for CDVRs
For A a CDVR with perfect residue field of characteristic p we have computed THH relative to S[z]. In order to compute the absolute THH we are going to employ a spectral sequence which works very generally (see Proposition 7.1).
Proposition 4.1. For every commutative algebra A (over Z) with an element π ∈ A considered as a S[z]-algebra there is a canonical multiplicative, convergent spectral sequence Proof. This is a special case of the spectral sequence of Proposition 7.1. Now for A a CDVR we want to use this spectral sequence to determine THH * (A; Z p ). From Theorem 3.1 we see that this spectral sequence takes the form with |x| = (2, 0) and |dz| = (0, 1): Using the multiplicative structure one only has to determine a single differential In the equal characteristic case this has to vanish since x can be chosen to lie in the image of the map THH(F p ) → THH(A; Z p ) → THH(A/S[z]; Z p ) and, thus, has to be a permanent cycle. Thus, the spectral sequence degenerates and we get THH * (A) ∼ = A[x] ⊗ Λ(dz) as there can not be any extension problems for degree reasons. 5 Let us now assume that A is a CDVR of mixed characteristic. Once we have chosen a uniformizer π we get a minimal polynomial E(z) ∈ W (k)[z] which we normalize such that E(0) = p. Note that usually E is taken to be monic, of the form E(z) = z e + pθ(z). This differs from our convention by the unit θ(0).
For connectivity reasons, Comparing with the spectral sequence, this means that the image of d 2 : E 2 2,0 → E 2 0,1 is precisely the submodule of A{dz} generated by E (π) dz. As A is a domain, any two generators of a principal ideal differ by a unit and, thus, for any generator x in degree (2, 0), d 2 (x) differs from E (π) dz by a unit. In particular, we can choose x such that d 2 (x) = E (π) dz.
Remark 4.3. The generator x ∈ THH 2 (A/S[z]; Z p ) determined by Lemma 4.2 maps under basechange along S[z] → S to a generator of THH 2 (A/π; Z p ) ∼ = THH 2 (k). The choice of normalization of E with E(0) = p is chosen such that this is compatible with the generator obtained from the generator of THH 2 (F p ) under the map THH 2 (F p ) → THH 2 (k) induced by F p → k.
Lemma 4.2 implies that THH * (A, Z p ) is isomorphic to the homology of the DGA (A[x] ⊗ Λ(dπ), ∂), |x| = 2, |dπ| = 1 with differential ∂x = E (π) · dπ and ∂(dπ) = 0 as there are no multiplicative extensions possible. Here we have named the element detected by dz by dπ as it is given by the Connes operator A. Krause and T. Nikolaus d : THH * (A, Z p ) → THH * (A, Z p ) applied to the uniformizer π. This follows from the identification of the degree 1 part with Ω 1 A/W (k) as in the proof of Lemma 4.2. We warn the reader that we have obtained this description for THH * (A; Z p ) from the relative THH which depends on a choice of uniformizer. As a result the DGA description is only natural in maps that preserve the chosen uniformizer.
The homology of this DGA can easily be additively evaluated to yield the following result, which was first obtained in [LM00, Theorem 5.1], but with completely different methods.
Theorem 4.4 (Lindenstrauss-Madsen). For a CDVR A of mixed characteristic (0, p) with perfect residue field we have non-natural isomorphisms 6 where π is a uniformizer with minimal polynomial E.
In this case the multiplicative structure is necessarily trivial, so that we do not really get more information from the DGA description. However, we also obtain a spectral sequence analogous to that of Proposition 4.1 for p-completed THH of A with coefficients in a discrete A-algebra A , which is THH(A; Z p ) ⊗ A A . This takes the same form, just base-changed to A . Thus, we obtain the following result, which was of course also known before.

Proposition 4.5. For a CDVR A of mixed characteristic and any map of commutative algebras
with ∂x = E (x) dπ and ∂(dπ) = 0.

Absolute THH for quotients of DVRs
Now we return to the case of quotients of DVRs. Thus, let A = A/m k ∼ = A/π k where A is a DVR with perfect residue field of characteristic p. Recall that in Proposition 3.7 we have shown that We want to consider the spectral sequence of Proposition 4.1, which in this case takes the form with |x| = (2, 0), |y| = (2, 0) and |dz| = (0, 1): Here we write y [n] for the nth divided power of y. The reader should think of y [n] as 'y n /n!'. 6 In the sense that they are only natural in maps that preserve the chosen uniformizer.

1694
Bökstedt periodicity and quotients of DVRs Lemma 5.1. We can choose the generator y and its divided powers in such a way that in the associated spectral sequence, d 2 (y [i] ) = kπ k−1 · y [i−1] dz. In particular, the differential is a PD-derivation, i.e. satisfies d 2 (y [i+1] ) = d 2 (y)y [i] for all i ≥ 0. 7 Proof. The construction of the spectral sequence of Proposition 4.1 (given in the proof of Proposition 7.1) applies generally to any HH(Z[z]/Z)-module M to produce a spectral sequence

As we can write
Here π ∈ A is a uniformizer and E its minimal polynomial.
Proof. This follows immediately from Lemma 5.1 together with the fact that there are no extension problems for degree reasons.

Evaluation of the result
In this section we want to make the results of Theorem 5.2 explicit. We start by considering the case of the p-adic integers Z p , in which Theorem 5.2 reduces additively to Brun's result, but gives some more multiplicative information. We note that all the computations in this section depend on the presentation A = A/π k and are, in particular, highly non-natural in A .
Example 6.1. We start by discussing the case A = Z p and k ≥ 2. We pick the uniformizer π = p. The minimal polynomial is E(z) = z − p, and A = Z/p k . The resulting groups THH * (Z/p k ) were additively computed by Brun [Bru00].
We have ∂(y [i] ) = kp k−1 y [i−1] dπ, and because the minimal polynomial is given by z − p we obtain ∂x = dπ. If k ≥ 2, then y = y − kp k−1 x still has divided powers, given by which makes sense because v p (l!) < l/(p − 1) ≤ l(k − 1) by Lemma 6.6. Now ∂((y ) [i] ) = 0, and we get a map of DGAs which is an isomorphism by a straightforward filtration argument. By Proposition 4.5, the homology of ((Z/p k )[x] ⊗ Λ(dπ), ∂) coincides with π * (THH(Z p ) ⊗ Zp Z/p k ). Thus, applying the Künneth theorem we obtain THH * (Z/p k ) ∼ = π * (THH * (Z p ) ⊗ Zp Z/p k ) ⊗ Z Z y as rings. Concretely, we obtain Thus, in the case k ≥ 2, we can replace the divided power generator of our DGA by one in the kernel of ∂. We contrast this with the case k = 1. In this case, of course, we expect to recover Bökstedt's result THH * (Z/p) ∼ = (Z/p)[x], but it is nevertheless interesting to analyze this result in terms of Theorem 5.2 and observe how this differs from Example 6.1.
Example 6.2. For A = Z p with uniformizer p and k = 1, i.e. A = Z/p, we have ∂x = dπ and ∂y = dπ. Here, we can set x = x − y to obtain an isomorphism of DGAs As ∂y [i] = y [i−1] dz, and thus the homology of the second factor is just Z/p in degree zero, Künneth applies to show that THH * (Z/p) ∼ = (Z/p)[x ].
The two qualitatively different behaviors illustrated in Examples 6.1 and 6.2 also appear in the general case: for sufficiently big k, we can modify the divided power generator y to a y that splits off, and obtain a description in terms of THH(A; A ) (Proposition 6.7). For sufficiently small k, we can modify the polynomial generator to an x that splits off, and obtain a description in terms of HH(A ) (Proposition 6.4). In the general case, as opposed to the case of the integers, these two cases do not cover all possibilities, and for k in a certain region the homology groups of the DGA of Theorem 5.2 are possibly without a clean closed form description.
Recall that, in the DGA of Theorem 5.2, we have ∂x = E (π) dπ and ∂y = kπ k−1 dπ. Keep in mind that the description depends on a choice of uniformizer and Eisenstein polynomial, but then everything is unambiguously defined, in particular the elements x and y. The behavior of the DGA depends on which of the two coefficients has greater valuation. Lemma 6.3. In mixed characteristic, we have (i) If p|k and π k |E (π), we can take as generators (ii) If p|k and E (π)|π k , we can take as generators (iii) If p k and E (π)|π k−1 , we can take as generators (iv) If p k and π k−1 |E (π), we can take as generators Proof. By Theorem 5.2, THH 2 (A ) is isomorphic to the kernel of the map where ∂x = E (π) dπ, ∂y = kπ k−1 dπ. In the first case, both coefficients vanish in A , because both are divisible by π k , so in that case the kernel is free on x, y as claimed.
In the second case, ∂y = 0 in A , and ∂x ∈ A {dπ} has annihilator ideal generated by π k /E (π). Thus, the kernel is generated by y and (π k /E (π))x, and they generate a submodule of the form A ⊕ (A/E (π)).
In the third case, observe first that the indicated y and x together form another basis of A {x, y}. We have ∂y = 0, so it is contained in the kernel. The annihilator ideal of ∂x ∈ A {dπ} is as in the second case, so we again see that the kernel is of the form A ⊕ (A/E (π)), but this time the first summand is generated by y .
In the fourth case, we again observe that the indicated x and y form another basis of A {x, y}, and ∂x = 0. The annihilator ideal of ∂y ∈ A {dπ} is (due to p k) generated by π, and so the kernel is generated by x and πy. They generate a submodule of the form A ⊕ (A/π k−1 ).
We now want to discuss the structure of THH * (A ) in the cases appearing in Lemma 6.3. We start with the simplest case, which is analogous to Example 6.2.
Proposition 6.4. Assume we are in the situation of Theorem 5.2 and that either A is of equal characteristic, or A is of mixed characteristic and we are in case (i) or (iv) of Lemma 6.3, 1697 A. Krause and T. Nikolaus i.e. p|k and π k |E (π), or p k and π k−1 |E (π). Then, we have Proof. We set x = x if A is of equal characteristic or if p|k and π k |E (π), and x = x − (E (π)/kπ k−1 )y if p k and π k−1 |E (π). Then ∂x = 0. We obtain a map of DGAs which is an isomorphism by a straightforward filtration argument. By Künneth, we obtain an isomorphism The additive description of the homology is easily seen from the fact that ∂y [i] Remark 6.5. In fact, we can identify H * (A y ⊗ Λ(dπ), ∂) with the Hochschild homology Essentially, the takeaway of Proposition 6.4 is that in cases (i) and (iv) of Lemma 6.3 we can modify the polynomial generator x to a cycle which splits a polynomial factor off THH(A ).
One would hope that, complementarily, in cases (ii) and (iii), we can split off a divided power factor. This is only true under more restrictive conditions. To formulate those, we require the following lemma on the valuation of factorials.
Lemma 6.6 (Legendre). For a natural number l ≥ 1 and a prime p we have v p (l!) < l p − 1 .
Proof. We count how often p divides l!. Every multiple of p not greater than l provides a factor of p, every multiple of p 2 provides an additional factor of p, and so on. We obtain the following formula, due to Legendre: where − denotes rounding down to the nearest integer. In particular,

Bökstedt periodicity and quotients of DVRs
Then we have an isomorphism of rings In particular, we get additively Proof. If p|k, all y [i] are cycles, and we set y x. In either case, (y ) admits divided powers, defined in the first case just by (y ) [i] = y [i] , and in the second case by by assumption and Lemma 6.6. We obtain a map of DGAs which is an isomorphism by a straightforward filtration argument. By Proposition 4.5 and Künneth, we then obtain Finally, we want to illustrate that the case 'in between' Propositions 6.7 and 6.4 is more complicated and probably does not admit a simple uniform description.
Example 6.8. For a mixed characteristic CDVR A with perfect residue field and A = A/m k = A/π k , Theorem 5.2 implies that the even-degree part of THH * (A ) is given by the kernel of ∂ in the DGA (A [x] y ⊗ Λ(dπ), ∂). We can thus consider THH 2n (A ) as a subring of A [x] y . Suppose we are in the situation of case (iii) of Lemma 6.3. Then a basis for THH 2 (A ) is given by Now suppose the valuations of the coefficients kπ k−1 /E (π) and π k /E (π) are positive, but small, say smaller than 1/p. Then observe that A. Krause and T. Nikolaus in particular, under our assumptions, (y − (kπ k−1 /E (π))x) p is divisible by π but not p. Similarly, is divisible by π but not p. Thus, both of our generators of THH 2 (A ) are nilpotent, but cannot admit divided powers. It is not hard to see that this holds more generally for any element of THH 2 (A ) that is non-zero mod π. Thus, in this situation, THH * (A ) cannot admit a description similar to Proposition 6.4 or 6.7. One example for A fulfilling the requirements used here is given by A = Z p [ e √ p] with uniformizer π = e √ p, and k = e + 1, as long as p e, k and e > 2p.

The general spectral sequences
We now want to establish a spectral sequence to compute absolute THH from relative ones of which Proposition 4.1 is a special case. This will come in two slightly different flavors. We let R → A be a map of commutative rings and let S R be a lift of R to the sphere, i.e. a commutative ring spectrum with an equivalence The example that will lead to the spectral sequence of Proposition 4.1 is Recall that for every commutative ring R we can form the derived de Rham complex LΩ R/Z , which has a filtration whose associated graded is in degree * = i given by a shift of the nonabelian derived functor of the i-term of the de Rham complex Ω i R/Z (considered as a functor in R). Concretely this is done by simplicially resolving R by polynomial algebras Z[x 1 , . . . , x k ], taking Ω i •/Z levelwise and considering the result via Dold-Kan as an object of D(Z). This derived functor agrees with the ith derived exterior power Λ i L R/Z of the cotangent complex L R/Z . For R smooth over Z this just recovers the usual terms in the de Rham complex. In general one should be aware that LΩ R/Z is a filtered chain complex, hence has two degrees, one homological and one filtration degree. We only need its associated graded LΩ * R/Z which is a graded chain complex. We warn the reader that the homological direction comes from deriving and has nothing to do with the de Rham differential.
Proposition 7.1. In the situation described above there are two canonical multiplicative, convergent spectral sequences Here we use homological Serre grading, i.e. the displayed bigraded ring is the E 2 -page and the d r -differential has (i, j)-bidegree (−r, r − 1). A similar spectral sequence with all terms p-completed (including the tensor products) exists as well.
Proof. We consider the lax symmetric monoidal functor where we have used the equivalence HH(R/Z) THH(S R ) ⊗ S Z to obtain the HH(R/Z)-module structure on THH(A).

Now we filter HH(R/Z) by two different filtrations: either by the Whitehead tower
· · · → τ ≥2 HH(R/Z) → τ ≥1 HH(R/Z) → τ ≥0 HH(R/Z) = HH(R/Z) or by the Hochschild-Kostant-Rosenberg (HKR) filtration [NS18,Proposition IV.4.1] · · · → F 2 HKR → F 1 HKR → F 0 HKR = HH(R/Z). The HKR filtration is, in fact, the derived version of the Whitehead tower, in particular for R smooth (or more generally ind-smooth) the filtrations agree. Both filtrations are complete and multiplicative, in particular they are filtrations through HH(R/Z) modules. On the associated graded pieces the HH(R/Z)-module structure factors through the map HH(R/Z) → R of ring spectra. This is obvious for the Whitehead tower and thus also follows for the HKR filtration. Thus the graded pieces are only R-modules and as such given by HH i (R) in the first case and by Λ j L R/Z in the second case.
After applying the functor (3) to this filtration we obtain two multiplicative filtrations of THH(A): which are complete because the connectivity of the pieces tends to infinity. Let us identify the associated gradeds for the HKR filtration, the case of the Whitehead tower works the same: Thus by the standard construction we obtain conditionally convergent, multiplicative spectral sequences which are concentrated in a single quadrant and, therefore, convergent.
If R is smooth (or more generally ind-smooth) over Z then the spectral sequences of Proposition 7.1 agree and take the form . In general, the HKR spectral sequence seems to be slightly more useful even though the other one looks easier (at least easier to state). We explain the difference in the example of a quotient of a DVR in § 8 where R = Z[z]/z k and S R = S[z]/z k .
Remark 7.2. With basically the same construction as in Proposition 7.1 (and if R ⊗ Z A is discrete in the first case) one obtains variants of these spectral sequences which take the form . These spectral sequences agree with those of Proposition 7.1 as soon as A is flat over R or R is smooth over Z, which covers all cases of interest for us. These modified spectral sequences are probably, in general, the 'correct' ones, but we have decided to state Proposition 7.1 in the more basic form.
Finally we end this section by constructing a slightly different spectral sequence in the situation of a map of rings A → A . This was constructed in Theorem 3.1 of [Lin00]. See also 1701 A. Krause and T. Nikolaus Brun [Bru00], which contains the special case A = Z p . We explain how it was used by Brun to compute THH * (Z/p n ) in the next section and compare that approach with ours.

Proposition 7.3. In general, for a map of rings A → A there is a multiplicative, convergent spectral sequence
Proof. We filter THH(A) ⊗ A A =: T by its Whitehead tower τ ≥• T and consider the associated filtration This filtration is multiplicative, complete and the colimit is given by THH(A ). The associated graded is given by where we have again used various base change formulas for THH.

Comparison of spectral sequences
Let us consider the situation of § 5, i.e. A = A/π k is a quotient of a DVR A with perfect residue field of characteristic p. We want to compare four different multiplicative spectral sequences converging to THH(A ) that can be used in such a situation. They all have absolutely isomorphic (virtual) E 0 -pages give by A[x] y ⊗ Λ(dz) but totally different grading and differential structure.
(i) In § 5 we have constructed a spectral sequence which ultimately identifies THH * (A ) as the homology of a DGA (A [x] y ⊗ Λ(dπ), ∂), see Theorem 5.2. This spectral sequence takes the form i.e. we have both x and y along the lower edge, and they both support differentials hitting certain multiples of dz (here dz corresponds to dπ). The main point is that it suffices to determine the differential on x and y and the rest follows using multiplicative and divided power structures. There is no space for higher differentials.
We interpret ∂ as the d 0 -differential 8 and obtain the following picture: This spectral sequence behaves well and degenerates in the 'big k' case discussed in Proposition 6.7, because then we have divided power elements (y ) [i] ∈ THH * (A ) that are detected by the y [i] , but this is not obvious from this spectral sequence, and Brun [Bru00] has to do serious work to determine its structure in the case A = Z/p k for k ≥ 2.
In fact, for A = Z/p k with k = 1 the spectral sequence becomes highly non-trivial. After d 0 , determined by d 0 (x) = dz, the leftmost column consists of elements of the form x ip and x ip−1 dz. From Example 6.2, we know that THH * (F p ) is polynomial on x = x − y. This is detected as y in this spectral sequence. As y is a divided power generator, its pth power is zero on the E ∞ -page. However, p k−1 (x − y) p = p k−1 x p and, thus, there is a multiplicative extension. In addition, the elements x kp−1 dz and the divided powers of y cannot exist on the E ∞ -page, so there are also longer differentials.
Although these phenomena might seem like a pathology in the case A = Z p (after all, we knew THH(Z/p) before) qualitatively, they generally appear whenever we are not in the 'big k' case discussed in Proposition 6.4.
(iii) We can also consider the first spectral sequence constructed in Proposition 7.1, which takes the form and consider ∂ as a d 0 differential. Then the spectral sequence visually looks as follows: This spectral sequence behaves well and degenerates in the 'small k' case discussed in Proposition 6.4, because then we have a polynomial generator x ∈ THH * (A ) whose powers are detected by the x i . If we are not in this case, we generally have non-trivial extensions. For example, let A be chosen such that p k, and πE (π)|π k−1 . In this case, THH 2 (A ), using Theorem 5.2, is of the form In this spectral sequence, the E ∞ page consists in total degree 2 of a copy of (A/π k−1 ){πy} in degree (0, 2), and a copy of (A/πE (π)){(π k−1 /E (π))x} in degree (2, 0). The element y ∈ THH 2 (A ) is detected as a generator of the degree (2, 0) part, but it is not actually annihilated by πE (π). Rather, πE (π)y agrees with πE (π)y, detected as a E (π)-multiple of the generator in degree (0, 2) and non-zero under our assumption πE (π)|kπ k−1 .
(iv) Finally we can consider the second spectral sequence constructed in Proposition 7.1 which takes the form

One again has THH
and LΩ A /A is computed as the homology of the DGA (A y ⊗ Λ(dz), ∂) where this time y sits in grading one and homological degree one (recall that LΩ has a grading and a homological degree). Thus, our virtual E 0 -term this time takes the form y ⊗ Λ(dz), |y| = (1, 1), |x| = (2, 0), |dz| = (0, 1) and the differential ∂ becomes a d 1 . The spectral sequence looks graphically as follows: This spectral sequence is a slightly improved version of spectral sequence (iii) as there are way less higher differentials possible. The whole wedge above the diagonal line through 1 on the j-axis is zero. Again this spectral sequence behaves well and degenerates in the 'small k' case Proposition 6.4, but behaves as badly in the other cases. Essentially, one should view Proposition 6.4 as degeneration result for the spectral sequences (iii) and (iv), and Proposition 6.7 as a degeneration result for the Brun spectral sequence (ii). By putting both the Bökstedt element x and the divided power element y (coming from the relation π k = 0) in the same filtration, the spectral sequence (i) that we have used allows us to uniformly treat both of these cases, as well as still behaving well in the cases not covered by Propositions 6.7 and 6.4 (like Example 6.8), where the homology of the DGA of Theorem 5.2 becomes more complicated and all of the three alternative spectral sequences discussed here can have non-trivial extension problems, seen in our spectral sequence in the form of cycles which are interesting linear combinations of powers of x and y.

Bökstedt periodicity for complete regular local rings
In this section we include a very brief discussion of the more general case of a complete regular local ring A, that is, a complete local ring A whose maximal ideal m is generated by a regular sequence (a 1 , . . . , a n ), see [Sta19, Tag 00NQ] and [Sta19, Tag 00NU]. Assume furthermore that A/m = k is perfect of characteristic p. We focus on the mixed characteristic case, because by a result of Cohen [Coh46], A agrees with a power series ring over k in the equal characteristic case.
We can regard A as an algebra over S[z 1 , . . . , z n ] = S[N × · · · × N]. We then have the following generalization of Theorem 3.1: Theorem 9.1. For a complete regular local ring A of mixed characteristic with perfect residue field of characteristic p we have with x in degree two.
We give a proof which is completely analogous to that of Theorem 3.1. We first need the following Lemma. Note that by perfectness of A/m = k we get a canonical map W (k) → A and, thus, together with the choice of generators a 1 , . . . , a n an algebra structure over W (k) [ Thus, the image of d 2 in degree (0, 1) has to agree with the ideal generated by i (∂E/∂z i ) dz i . Up to a unit, we thus have For n = 2, this differential again completely determines THH * (A; Z p ), because d 2 in degrees (2k, 0) → (2k − 2, 1) is injective, and therefore the E 3 -page is concentrated in degrees (0, 0), (2k, 1) and (2k, 2) for k ≥ 0 and the spectral sequence degenerates thereafter without potential for extensions. For n ≥ 3, there could be extensions, and for n ≥ 4, there could be longer differentials, both of which we do not know how to control.
Finally , and HH * (A /A) is easily seen to be a divided power algebra on d generators. Thus, the spectral sequence is even and cannot have non-trivial differentials. However, there is potential for multiplicative extensions. We have been informed by Guozhen Wang that these do indeed show up, as shown in joint work with Ruochuan Liu in [LW22] which appeared after our paper was finished.

Logarithmic THH of CDVRs
In this section we want to explain how to deduce results about logarithmic THH from our methods. This way we recover known computations of Hesselholt and Madsen [HM03] for logarithmic THH of DVRs. We thank Eva Höning for asking about the relation between relative and logarithmic THH, which inspired this section.
First we recall the definition of logarithmic THH following [HM03, Lei18] and [Rog09]. In practice, we only need the case M = N with the map S[N] = S[z] → R given by sending z to an element π ∈ R. In this case, we also denote THH(R | N) by THH(R | π). We use this lemma to obtain a spectral sequence similar to that of Proposition 7.1. To this end let us introduce some further notation. We set

A. Krause and T. Nikolaus
Proof. Let us first phrase Theorem A.1 a bit more conceptually following [AB19]. We can view Ω 2 S 3 → BGL 1 (S p ) as the free E 2 -monoid on in the category (S * ) / BGL 1 (Sp) of pointed spaces over BGL 1 (S p ). The Thom spectrum functor S / BGL 1 (Sp) → Mod Sp is symmetric-monoidal and, thus, the Thom spectrum of Ω 2 S 3 can equivalently be described as the free E 2 -algebra over S p on the pointed S p -module obtained as the Thom spectrum of S 1 1−p −−→ BGL 1 (S p ). This is easily seen to be S p → S p /p. As the free E 2 -S-algebra on the pointed S-module S → S/p is already p-complete, it also agrees with this Thom spectrum. We write this as Free E 2 (S → S/p). There is a map S/p → F p of pointed S-modules which induces an isomorphism on π 0 . We obtain an induced map Free E 2 (S → S/p) → F p .
(A.1) Theorem A.1 is now equivalently phrased as the statement that the map (A.1) is an equivalence. As both sides are p-complete, this is equivalent to the claim that the map is an equivalence after tensoring with F p . This is the map Free E 2 Fp (F p → F p ⊗ S/p) → F p ⊗ S F p induced by the map F p ⊗ S/p → F p ⊗ S F p of pointed F p -modules. It follows by elementary connectivity arguments that this map is an isomorphism on π 0 and π 1 . Now we have an equivalence F p ⊗ S/p F p ⊕ ΣF p as pointed F p -modules. Thus, we can also write Free E 2 Fp (F p → F p ⊗ S/p) as the free E 2 -algebra on the unpointed F p -module ΣF p . Thus, the Hopkins-Mahowald result is seen to be equivalent to the claim that the map Free E 2 Fp (ΣF p ) → F p ⊗ S F p induced by a map ΣF p → F p ⊗ S F p which is an isomorphism on π 1 , is an equivalence. This is precisely Theorem 1.2.
In § 1 we have deduced Bökstedt's theorem (Theorem 1.1) directly from Theorem 1.2. Blumberg, Cohen and Schlichtkrull deduced an additive version of Bökstedt's theorem in [BCS10,Theorem 1.3] from Theorem A.1. A variant of this argument is also given in [Blu10, § 9]. We note that the argument that they use only works additively and does not give the ring structure on THH * (F p ). We explain this argument now and also how to modify it to give the ring structure as well.
Proof of Theorem 1.1 from Theorem A.1. The Thom spectrum functor S * / BGL 1 (Sp) → Mod Sp preserves colimits and sends products to tensor products, and thus sends the unstable cyclic bar construction of Ω 2 S 3 to the cyclic bar construction of F p . This identifies THH(F p ) as an E 1 -ring with a Thom spectrum on the free loop space LBΩ 2 S 3 LΩS 3 . Now, using the natural fiber sequence of E 1 -monoids in S / BGL 1 (Sp) , Ω 2 S 3 → LΩS 3 → ΩS 3 , one can identify THH(F p ) with F p [ΩS 3 ]. For example, because this is a split fiber sequence of E 1 monoids, one obtains an equivalence LΩS 3 Ω 2 S 3 × ΩS 3 and, thus, an identification of THH(F p ) as a tensor product of the Thom spectrum on Ω 2 S 3 (i.e. F p ) and the Thom spectrum on ΩS 3 . Thus, a Thom isomorphism yields an equivalence THH(F p ) F p [ΩS 3 ]. However, the equivalence LΩS 3 Ω 2 S 3 × ΩS 3 is not an E 1 -map, so this argument only describes THH(F p ) additively.
One can fix this as follows. The Thom spectrum can be interpreted as the colimit of the functor LΩS 3 → Sp obtained by postcomposing with the functor BGL 1 (S p ) → Sp that sends the point to S p . Instead of passing to the colimit directly, one can pass to the left Kan extension along the map LΩS 3 → ΩS 3 . This yields a functor ΩS 3 → Sp which sends the basepoint of ΩS 3 to the colimit along the fiber, i.e. the Thom spectrum over Ω 2 S 3 , which is precisely F p . We thus obtain a functor ΩS 3 → BGL 1 (F p ), whose colimit is the Thom spectrum of LΩS 3 . As the original functor LΩS 3 → Sp was lax monoidal, because it came from an E 1 map, the Kan extension ΩS 3 → BGL 1 (F p ) is also an E 1 map. The space of E 1 maps ΩS 3 → BGL 1 (F p ) agrees with the space of maps S 2 → BGL 1 (F p ) and is, thus, trivial. Thus, the resulting colimit THH(F p ) is, as an E 1 ring, given by F p [ΩS 3 ].