aa r X i v : . [ m a t h . K T ] J u l ON THE ALGEBRAIC K -THEORY OF DOUBLE POINTS NOAH RIGGENBACH
Abstract.
In this paper, we use trace methods to study the algebraic K -theory of ringsof the form R [ x , . . . , x d ]/( x , . . . , x d ) . We compute the relative p -adic K groups for R aperfectiod ring, p an odd prime. In particular, we get the integral K groups when R is afinite field, and the integral relative K -theory K ( R [ x , . . . , x d ]/( x , . . . , x d ) , ( x , . . . , x d )) when R is a perfect F p -algebra. We conclude the paper with some other notable compu-tations, including some rings which are not quite of the above form. Contents
1. Introduction 21.1. Outline 31.2. Acknowledgements 32. Reduction to TC calculation 33. Computation of THH as a T -spectrum 53.1. Weight and Cyclic Word Decomposition of B cy ( Π ) B cy ( Π , ω )
64. Topological negative cyclic and periodic homology 64.1. Recollection of the calculation of topological Hochschild homology of perfectoidrings 84.2. Case One: s even. 114.3. Case Two: m odd 125. TC calculation 135.1. Case One: s even 145.2. Case Two: m odd. 166. Examples 176.1. The case of R a perfect F p -algebra 176.2. The case of R = Z p [ ζ p ∞ ] . RIGGENBACH On The Algebraic K -Theory of Double Points Introduction
Recent years have seen a remarkable increase in interest in algebraic K -theory. The workof [BGT13] and [Bar16] have described a universal property of K -theory. On the morecalculational side, the work of Voevodsky and many others has lead to the solution to theQuillen-Lichtenbaum conjecture. We are able to say more than ever before what algebraic K -theory looks like.That being said, many aspects of K -theory remain mysterious. A key example of thisis what K -theory looks like at singular schemes. Many of the helpful calculational tools,such as the Quillen-Lichtenbaum conjecture and other tools coming from motivic homotopytheory, require the scheme to be regular. Once A -homotopy invariance fails, we lose many ofthese theorems. Despite this, there have been many successful computations of the K -theoryof singular schemes using trace methods, particularly in characteristic p > or rationally,e.g. [HM97a], [HM97b], [HN19], [LM08], [Spe19], and [Spe20].Define for a given ring R A d ∶ = R [ x , . . . , x d ]/( x , . . . , x d ) and m ∶ = ( x , . . . , x d ) ⊆ A d . Then the goal of the present paper is to compute K ( A d , m ) ∧ p . We manage to do this for R a perfectoid ring, and p an odd prime. In order to state the main theorem, we will needsome terminology from [Spe19]. This is the first of many connections to [Spe19] and [Spe20],which we explore in detail in Section 6.3.We take from [Spe19] two functions, t ev and t od . The function t ev = t ev ( p, r, m ′ ) is theunique positive integer, if it exists, such that m ′ p t ev − ≤ r < m ′ p t ev . If no such integerexists, t ev = . Similarly, t od = t od ( p, r, m ′ ) is, if it exists, the unique positive such that m ′ p t od − ≤ r + < m ′ p t od , and is zero if no such integer exists. We also define the sets ω s,d as the set of cyclic words, as defined in definition 3.1.1, of d letters, length s , and periodexactly s . In other words, the elements of ω s,d are strings of s letters from an alphabet of d letters such that the C s action of rotating the word acts freely. In her thesis, Rudman[Rud20] has recursively calculated the cardinality of this set as ∣ ω ,d ∣ = d and ∣ ω s,d ∣ = d s − ∑ u ∣ su ≠ s ∣ ω u,d ∣ s . Finally, let J p = { m ′ ∈ Z + ∣( m ′ , p ) = } .Our main result is now the following. Theorem 1.0.1.
Let p be an odd prime and let R be a perfectoid ring. Then, using thelanguage above, there are isomorphisms π ∗ ( K ( A d , m ) ∧ p ) ≅ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev − s even ∏ ω s,d W t ev − v p ( s ) ( R ) if ∗ = r ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod − ∏ ω s,d W t od − v p ( s ) ( R ) if ∗ = r + where W n ( R ) are the truncated p -typical Witt vectors of R , W ( R ) is understood to be zero,and v p is the p -adic valuation. Remark 1.0.1.
In Theorem 1.0.1, the products are finite. The inner two factors in bothcases are clearly finite, but as a soon as m ′ > r in the first case, or r + in the second, . RIGGENBACH On The Algebraic K -Theory of Double Points then t = . In particular, the second product is empty, and so all the factors contributed arezero. Additionally, this forces the groups to be zero for r ≤ . For the reader unfamiliar with perfectoid rings, the relevant theory is reviewed in Sec-tion 4.1. They key example to keep in mind is when R is a perfect F p -algebra, i.e., it isan F p -algebra where the Frobenius is an isomorphism. The main results cited and provenin Section 4.1 are lifts of results known for perfect F p -algebras to the setting of perfectoidrings.This work originally started as a revisit of [LM08], with the goal of reproducing thecalculation using the new methods introduced by Nikolaus and Scholze in [NS18]. Wediscuss how Theorem 1.0.1 relates to [LM08] briefly in Section 6.1, where we extend theirresult from perfect fields of characteristic p > to perfect F p -algebras. We have also managedto generalize some of the work of Speirs, as discussed in Section 6.3.1.1. Outline.
We prove Theorem 1.0.1 using trace methods. Section 2 is the reduction toa TC calculation using trace methods. Section 3 then computes the topological HochschieldHomology as a T -equivariant spectrum. It turns out that this spectrum is comprised ofspectra with induced action, and so Section 4 is a computation of topological negative cyclicand periodic homology in terms of the homotopy fixed points and Tate construction for thecyclic subgroups of T . Section 5 combines this to get Theorem 1.0.1 using [NS18]. Finally,Section 6 contains some consequences of Theorem 1.0.1. For a similar strategy, see [HN19],and for a spectral sequence based approach to similar problems, see [Spe19] or [Spe20].1.2. Acknowledgements.
I would like to thank my advisor Michael Mandell for givingme useful feedback on this paper, and for being a helpful and supportive advisor. I wouldalso like to extend a special thanks to Ayelet Lindenstrauss and Emily Rudman, who bothhelped me work through many parts of this paper, pointed out multiple mistakes I made,and made the writing significantly better. I would also like to thank Sanjana Agarwal andMartin Speirs, for patiently helping me understand perfectiod rings, and for many otherhelpful conversations. Finally, I would like to thank Dylan Spence for giving an algebraicgeometer’s opinion of Section 4.1.While working on this paper I was supported by a Hazel King Thompson Scholarshipfrom the Mathematics Department at Indiana University.2.
Reduction to TC calculation The ring A d is particularly well suited for this computation, since on it we may apply theDundas Goodwillie McCarthy Theorem [DGM13, Theorem 7.0.0.2] (or for a more modernapproach and statement, [Ras18, Theorem 1.1.1]) to get a pullback square K ( A d ) TC ( A d ) K ( R ) TC ( R ) where the vertical maps are the quotient by the (nilpotent) ideal m . As noted in the diagram,the inclusion R → A d gives a splitting of the above maps, and hence we get an isomorphism(2.1) K ( A d ) ≅ K ( R ) ∨ K ( A d , m ) ≅ K ( R ) ∨ TC ( A d , m ) where the last equivalence comes from the above pullback square. . RIGGENBACH On The Algebraic K -Theory of Double Points The celebrated computation of Quillen [Qui72] gives the homotopy groups of the firstsummand in the case of R a finite field. In the more general case of R a perfectoid ring, wewill usually not be able to say more in the integral setting.Our main theorem is therefore equivalent to the following: Theorem 2.0.1.
For p an odd prime and R a perfectoid ring, there are isomorphisms TC r ( A d , m ) ∧ p ≅ ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev ( p,r,m ′)− s even ∏ ω s,d W t ev ( p,r,m ′ )− v p ( s ) ( R ) and TC r + ( A d , m ) ∧ p ≅ ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod ( p,r,m ′)− ∏ ω s,d W t od ( p,r,m ′ )− v p ( s ) ( R ) where v p (−) is the p -adic valuation, and W ( R ) is understood to be zero. Remark 2.0.1.
If we are willing to work rationally, Goodwillie’s theorem [Goo86] lets usreplace Equation (2.1) with K ( A d ) Q ≅ K ( R ) Q ∨ Σ HC ( A d ⊗ Q , m ⊗ Q ) . The second summand was studied in Rudman [Rud20] in the case of R = Z , which computes HC ( A d ) for more general R . In addition, Thomason [Tho85] showed that K (−) Q satisfiesétale descent for schemes under mild hypotheses, and is more amenable to computation thenordinary algebraic K -theory. For the remainder of this section we will work in the p -complete setting. In this setting wesee one of the first benefits of the recent advances in perfectoid rings and homotopy theory. Theorem 2.0.2 ([CMM18], Theorem B) . Let R be a ring henselian along ( p ) and suchthat R / p has finite Krull dimension. Let d = sup x ∈ Spec ( R / p ) log p [ k ( x ) ∶ k ( x ) p ] , where k ( x ) denotes the residue field at x . Then the map K ( R )/ p i → TC ( R )/ p i is an equivalence indegrees ≥ max ( d, ) for all i ≥ . As noted in [CMM18], their proof of this result specializes nicely to the case of R semiper-fectoid. For our purposes we will only need the following corollary. Corollary 2.0.1 ([CMM18]) . Let R be a perfectoid ring. Then the map K ( R ) ∧ p → TC ( R ) ∧ p exhibits the former as the connective cover of the latter. Hence going from a calculation of the relative K -theory K ( A d , m ) ∧ p to a computation of K ( A d ) ∧ p is a matter of computing TC ( R ) ∧ p for these rings. In principal, this is an easiercomputation. In particular, the recent work of Bhatt and Scholze in [BS19] introduces apromising calculational tool for R quasiregular semiperfectoid.To finish this section, we note that we do not lose any information in p -completion when R is a perfect F p -algebra. Proposition 2.0.1.
Let R be a perfect F p -algebra. Then all of the homotopy groups of TC ( A d , m ) are p -power torsion. In particular, TC ( A d , m ) is p -complete. Proof.
Since p = in A d , m is p -power torsion. Applying [LT19, Theorem D] then givesthe result. (cid:3) . RIGGENBACH On The Algebraic K -Theory of Double Points Computation of
THH as a T -spectrum We begin by noting that A d = R ∧ Π , where Π = { , , x , . . . , x d } is the pointed monoidwith all products x i x j = . Since THH is symmetric monoidal [NS18, Section IV.2], we thenget that
THH ( A d ) ≃ THH ( R ) ∧ THH ( Σ ∞ Π ) . The latter term is equivalent to Σ ∞ B cy ( Π ) . We refer the reader to [Spe19, Sections 3.1and 3.2] for a comprehensive review of the relevant details about this decomposition. Inparticular, we have the following lemma. Lemma 3.0.1 ([HM97b], Theorem 7.1; [NS18], Section IV.2) . Let R be a ring, Π a pointedmonoid, and R [ Π ] the pointed monoid algebra. Then there is a natural T -equivariant equiv-alence THH ( R ) ∧ B cy ( Π ) ∼ Ð→ THH ( R [ Π ]) where the T action on the left is the diagonal action. Under this equivalence, the Frobeniusmap is induced by the Frobenius on THH ( R ) smashed with the unstable Frobenius on B cy ( Π ) . By naturality, we see that the map
THH ( R ) → THH ( R [ Π ]) is given by the composition THH ( R ) ≅ THH ( R ) ∧ S → THH ( R ) ∧ B cy ( Π ) . Consequently, the cofiber of this map is given by
THH ( R )∧ cofib ( S → B cy ( Π )) ≅ THH ( R )∧̃ B cy ( Π ) , where ̃ B cy ( Π ) is the cofiber of the map S → B cy ( Π ) in spaces.3.1. Weight and Cyclic Word Decomposition of B cy ( Π ) . As described in [Spe19,Section 3.2], B cy ( Π ) can be decomposed as a pointed T -space into a wedge of simplerspaces. In order to do this decomposition, we need to define the notions of word, wordlength, word period, and cyclic word. Definition 3.1.1.
Consider the set S = { x , . . . , x d } . We define a word ω of length m ≥ to be a mapping ω ∶ { , , . . . , m } → S. The cyclic group C m acts on the words of length m ;the orbit of a word ω is denoted ω and is called a cyclic word. The cardinality of ω is theperiod of ω which will we denote as s . For m > , note for any word of length m , s mustdivide m . For m = , s = . The n-simplicies B cy ( Π ) n of B cy ( Π ) correspond to the ( n + ) -tuples ( a , . . . , a n ) where a i ∈ { , x , . . . , x d } for i ∈ { , , . . . , n } . Each ( n + ) -tuple corresponds to exactly one cyclicword ω by ignoring any ’s in the sequence. The face maps d , . . . , d n − and degeneracy maps s , . . . , s n all will send the n -simplex corresponding to ( a , . . . , a n ) either to the basepoint0 or to a simplex corresponding the same reduced sequence. The n th face map d n and thecyclic operator t n will both send ( a , . . . , a n ) either to or to a simplex whose correspondingreduced cyclic word is ω . Since all the structure maps preserve ω we get a decomposition(3.1) B cy ( Π ) = ⋁ cyclic words ω B cy ( Π , ω ) where B cy ( Π , ω ) is the subspace of B cy ( Π ) whose cells all correspond to the same ω or to for any ω that is a cyclic word in the letters of S . Note that when m = , we are exactlygetting the simplicies with only zeros and ones. Hence the summand corresponding to m = ,i.e. the empty cyclic word, is homeomorphis to S . . RIGGENBACH On The Algebraic K -Theory of Double Points Equivariant Homotopy Type of B cy ( Π , ω ) .Lemma 3.2.1. Let ω be a cyclic word of cycle length s ≥ , with letters in the set S = { x , x , ⋯ , x d } and with length m = s ⋅ i . A choice of ω with cyclic word ω determines a T -equivariant homeomorphism S R [ C m ]− ∧ C i T + ∼ Ð→ B cy ( Π , ω ) where R [ C m ] − is the reduced regular representation of C m . Proof.
This follows almost directly from the proof of [Spe19, Lemma 9] noting the differ-ence in pointed monoids. In the paper [Spe19], Π d = { , , x , x , . . . , x , x , . . . , x d , x d , . . . } is the multiplicative monoid with basepoint 0 and multiplication x i x j = for i ≠ j . Hedefines a word of length m with no cyclic repetitions [Spe19, Definition 8] to be a word ω = w w . . . w m such that w i ≠ w i + for i = , , . . . , m − and w m ≠ w .The proof for [Spe19, Lemma 9] for cyclic words ω of length m with no cyclic repetitionsrelies on the following: all of the faces of any m -cell in B cy ( Π , ω ) with corresponding reducedword in ω are identified to the basepoint . In the pointed monoid used for the coordinateaxes paper, this is because attention was restricted to words with no cyclic repeats. However,for our pointed monoid, x i x j = for any i, j . Therefore, all faces of any m -cell in B cy ( Π , ω ) with corresponding reduced word in ω will be regardless of whether the word ω has cyclicrepetitions or not. The rest of the proof follows directly from the proof of [Spe19, Lemma9]. (cid:3) Lemma 3.2.2.
For a word ω be a cyclic word of cycle length s ≥ , with letters in the set S = { x , x , ⋯ , x d } and with length m = s ⋅ i .(1) For s even, there is a T -equivariant homeomorphism Σ B cy ( Π , ω ) ≅ S λ m / ∧ ( T / C i ) + . (2) For s and i both odd, there is a T -equivariant homeomorphism B cy ( Π , ω ) ≅ S λ ( m − )/ ∧ ( T / C i ) + . (3) For s odd and i even, there is a T -equivariant homeomorphism B cy ( Π , ω ) ≅ S λ ( m − )/ ∧ R P ( i ) . Here R P ( i ) is the cofiber of the map ( T / C i ) + → ( T / C i ) + in T -spaces, and λ n ≅ C ( ) ⊕ C ( ) ⊕ ⋯ ⊕ C ( n ) , where C ( j ) is the one-dimensional complex representation of T having z ∈ T act by multiplication by z j . Proof.
See Lemma 3.2.1 and [Spe19, Lemma 11]. (cid:3) Topological negative cyclic and periodic homology
In light of the previous section, we get the following decomposition of
THH ( A d ) :(4.1) THH ( A d ) ≃ THH ( R ) ∨ ⋁ m ∈ Z + ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⋁ s ∣ ms even (⋁ ω ∈ ω s,d Σ − THH ( R ) ∧ S λ m / ∧ ( T / C ms ) + ) ∨ ⋁ s ∣ mm odd (⋁ ω ∈ ω s,d THH ( R ) ∧ S λ m − / ∧ ( T / C ms ) + ) ∨ ⋁ s ∣ ms ≠ m mod (⋁ ω ∈ ω s,d THH ( R ) ∧ S λ m − / ∧ R P ( ms ))⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . RIGGENBACH On The Algebraic K -Theory of Double Points where ω s,d is the set of all cyclic words with cycle length actually equal to s (and not aproper divisor of s ) in d letters. In addition, the map THH ( R ) → THH ( A d ) giving thesplitting of TC is the inclusion as the first summand.Note also that each summand is of the form THH ( R ) ∧ S λ ∧ X for some representation λ and space X . The T -action on this can be quite complicated if we allow arbitrary E -ringspectra R , but we are only considering cases where R is a Z -algebra. Hence THH ( R ) is a Z -module since it is a module over THH ( Z ) , which is itself a Z -algebra. Thus THH ( R ) ∧ S λ ≃ ( THH ( R ) ∧ Z Z ) ∧ S λ ≃ THH ( R ) ∧ Z ( Z ∧ S λ ) . The spectrum Z ∧ S λ is, non-equivariantly, an Eilenberg-MacLane spectrum and thereforemust be equivalent as a Borel equivariant T -spectrum to one with trivial action by Lemma4.0.1. Hence THH ( R ) ∧ S λ ≃ Σ l THH ( R ) , where l = dim ( λ ) .The fact that Z ∧ S λ has trivial action is used in multiple papers, including [HN19] and[NS18], and is proven here for convenience: Lemma 4.0.1.
Let Σ n M be an Eilenberg-MacLane spectrum in S p B T , U ∶ S p → S p B T be theinclusion with trivial action functor, and F ∶ S p B T → S p the functor forgetting the T -action.Then U ○ F ( Σ n M ) ≅ Σ n M The following argument was pointed out to me by Michael Mandell.
Proof.
Assume without loss of generality that n = . Then M can be thought of as afunctor M ∶ B T → S p , and in particular a map B T → B haut ( F ( M )) , where haut ( X ) is thesubspace of maps S p ( X, X ) which are the connected components of weak equivalences.It is therefore enough to show that any map B T → B haut ( F ( M )) is null-homotopic. Infact, S p ( F ( M ) , F ( M )) is homotopy discrete (cohomology is in non-positive degrees). Hence B haut ( F ( M )) is a K ( π, ) , and all maps B T → B haut ( F ( M )) are contractable. (cid:3) Thus we may take the trivial representation sphere in all of the above formulas. Notethat since the representations λ i have complex dimension i (so real dimension i ), thesetrivial representation spheres are S n , where n is m − or m − , depending on the case.In particular, if we write THH ( A d ) = ⋁ m ∈ N THH ( m ) where THH ( m ) is the wedge of allsummands corresponding to cyclic words of length m , then the connectivity of THH ( m ) isat least m − and so(4.2) THH ( A d ) ≃ ⋁ m ∈ N THH ( m ) ≃ ∏ m ∈ N THH ( m ) . We may further decompose
THH ( A d ) into a product by noting that in Equation (4.1),each of the inner wedge sums are finite wedges so they are also equivalent to the respectiveproducts. Therefore, to compute TC − ( A d ) , and the relative TC − , it is enough to compute ( − ) h T on each of the the summands in Equation (4.1), and ( − ) h T for the summands where m ≥ , respectively.We may make a similar observation for the homotopy orbits as well. Since Equation (4.1)is given in terms of iterated wedge sums, ( − ) h T can be computed term by term. In ad-dition, homotopy orbits only increase connectivity, so in the decomposition THH ( A d ) h T ≃ ⋁ m ∈ N THH ( m ) h T , each term THH ( m ) h T is at least ( m − ) -connected. Consequently, wemay express ( THH ( A d )) h T as a product in the exact same way we did for TC − ( A d ) , and thecanonical map will respect this decomposition. In particular, we get the same decompositionof the topological periodic homology.Note that each summand in Equation (4.1) with s = m mod 2 is of the form X ∧ ( T / C n ) + .We use the following proposition to simplify the computation. . RIGGENBACH On The Algebraic K -Theory of Double Points Proposition 4.0.1 ([HN19], Proposition 3) . Let G be a compact Lie group. Let H ⊆ G be a closed subgroup, let λ = T H ( G / H ) be the tangent space at H = eH with the adjointleft H -action, and let S λ be the one-point compactification of λ . For every spectrum with G -action X , there are canonical natural equivalences ( X ∧ ( G / H ) + ) hG ≃ ( X ∧ S λ ) hH , ( X ∧ ( G / H ) + ) tG ≃ ( X ∧ S λ ) tH . In particular, ( X ∧ ( T / C n ) + ) h T ≃ ( Σ X ) hC n , and similarly for the Tate construction. Fur-thermore, the Tate construction is initial among functors under the homotopy fixed pointsvanishing on compact objects by [NS18, Lemma I.1.4(ii)], and so the canonical map mustbe sent to the canonical map under this equivalence. Remark 4.0.1.
The proof of Proposition 4.0.1 in [HN19] is from the point of view of infinitycategories, and uses constructions from [NS18]. While the proof using infinity categorieshas simplified things considerably, 4.0.1 was known before. It can be obtained from theWirthmüller isomorphism, which is proven in [May03] in a similar fashion as above, but interms of a six operation formalism of model categories from [FHM03].
Recollection of the calculation of topological Hochschild homology of per-fectoid rings.
So far, with the exception of the end of Section 2, we have not used anyproperties of the spectrum R aside from it being an Eilenberg-MacLane spectrum of a ring;we only used the fact that R was a Z -module to simplify the representation spheres in 4.1.So we can run the arguments above to get similar results for a large class of ring spectrawithout needing to p -complete. If we are willing to work with nontrivial representationspheres, we have analogous results for all E -ring spectra.In order to do explicit calculations, we need to know more about THH ( R ) . For a reviewon the algebraic properties of perfectoid rings, we refer the reader to [BMS18]. We recallthe definition of perfectoid rings here for convenience: Definition 4.1.1 ([BMS18], Definition 3.5) . A ring R is perfectoid if there exists an element π ∈ R such that(1) π p divides p ;(2) R is π -adically complete, and is separated with respect to this topology (and thereforep-complete);(3) The Frobenius map ϕ ∶ R / p → R / p is surjective;(4) The kernel of the map θ ∶ A inf ( R ) → R is principal.where A inf ( R ) is Fontaine’s ring, W ( lim ( . . . ϕ Ð→ R / p ϕ Ð→ R / p )) that is: the ( p -typical) Wittvectors on lim ( . . . ϕ Ð→ R / p ϕ Ð→ R / p ) . Following [BS19], we will refer to a choice of element ξ ∈ A inf ( R ) , ( ξ ) = ker ( θ ) , as anorientation of A inf ( R ) . By abuse of notation, we will refer to such a ξ as an orientation of R as well. When there is no danger of confusion, we will drop the R in A inf .Notice that since A inf is defined as the Witt vectors of a perfect F p -algebra, it in particularcomes with a Frobenius automorphism ϕ ∶ A inf → A inf lifting the Frobenius ϕ ∶ R / p → R / p .The map θ ∶ A inf → R is one of a family of maps θ r ∶ A inf → W r ( R ) whose construction isreviewed in [BMS18]. These maps, along with the maps ˜ θ r = θ r ○ ϕ − r , are characterized bya universal property. . RIGGENBACH On The Algebraic K -Theory of Double Points As noted at the end of Section 2, many important theorems have been proven recentlyabout K -theory and TC of these rings, at least in the p -complete setting, such as theextension of Bökstedt’s periodicity result: Theorem 4.1.1 ([BMS19], Theorem 6.1) . For a perfectoid ring R , π ∗ ( THH ( R ) ∧ p ) ≅ R [ u ] isa polynomial ring, where u ∈ π ( THH ( R ) ∧ p ) ≅ π ( HH ( R ) ∧ p ) ≅ ker ( θ )/ ker ( θ ) can be chosento be any generator of ker ( θ )/ ker ( θ ) . It then follows that all the differentials in both the homotopy T -fixed point and Tate T -fixed point spectral sequences must be zero, since the nonzero elements are all concentratedin even dimensions. Both spectral sequences converge strongly, and the computation for R = F p by [NS18], [BMS19] extend to any perfectoid ring R . With some additional work,[BMS19] show the following. Proposition 4.1.1 ([BMS19], Proposition 6.2) . The commutative square TC − ( R ) ∧ p TP ( R ) ∧ p THH ( R ) ∧ p THH ( R ) tC p can ϕ h T p canϕ p gives, upon taking homotopy groups, the square A inf [ u, v ]/( uv − ξ ) A inf [ σ, σ − ] R [ u ] R [ σ, σ − ] ϕ -linear u ↦ σ,v ↦ ϕ ( ξ ) σ − u ↦ u,v ↦ σ ↦ σu ↦ σ where ξ is an orientation of R and the columns also apply θ ∶ A inf ( R ) → R . In addition to understanding TC − ( R ) ∧ p and TP ( R ) ∧ p , we also need to understand whathappens at the finite cyclic subgroups, i.e. ( THH ( R ) ∧ p ) hC pn and THH ( R ) tC pn . The rest ofthe section will be devoted to this calculation.Combining the calculation of the Tate construction from [BMS19, Remark 6.6] and[BMS18, Lemma 3.12] gives π ∗ (( THH ( R ) ∧ p ) tC pn ) ≅ W n ( R )[ σ, σ − ] ≅ A inf ( R )[ σ, σ − ]/( ˜ ξ n ) ≅ A inf ( R )[ σ, σ − ]/( ˜ ξ n σ − ) where ˜ ξ n = ϕ ( ξ ) ϕ ( ξ ) . . . ϕ n ( ξ ) . We use this to compute π ∗ (( THH ( R ) ∧ p ) hC pn ) , startingwith the following proposition. Proposition 4.1.2.
There is a choice of isomorphism such that the map τ ≥− ( THH ( R ) ∧ p ) hC pn ϕ p Ð→ τ ≥− ( THH ( R ) tC p ) hC pn ≃ τ ≥− THH ( R ) tC pn + (the last equivalence coming from [NS18, Lemma II.4.1]) gives on homotopy groups the map A inf [ u ]/( ˜ ξ n + ) Ð→ A inf [ σ ]/( ˜ ξ n + ) sending u ↦ σ . Under the same isomorphism, the map τ ≥− ( THH ( R ) ∧ p ) hC pn can hCpn − ÐÐÐÐÐÐ→ τ ≥− ( THH ( R ) tC p ) hC pn − ≃ τ ≥− THH ( R ) tC pn . RIGGENBACH On The Algebraic K -Theory of Double Points gives on homotopy groups the map W n + ( R )[ u ] Ð→ W n ( R )[ σ ] which reduces W n + ( R ) → W n ( R ) and sends u ↦ ˜ θ r ( ξ ) σ . Proof.
By Proposition 4.1.1, the Frobenius map
THH ( R ) ∧ p → THH ( R ) tC p is the con-nective cover map. In addition, since both have π − = , the above map is an isomorphismon τ ≥− covers. The functor ( − ) hG for any Lie group G preserves coconnectivity, that is:it takes maps that induce isomorphisms on homotopy groups above a given dimension tomaps that induce isomorphisms on homotopy groups above that dimension. So we applythis to ϕ hC pn p ∶ ( THH ( R ) ∧ p ) hC pn → THH ( R ) tC pn + to see that it is also an equivalence on τ ≥− -covers. Furthermore, the Frobenius is an E ∞ -ring map, and so we get the claimed ringstructure on π ∗ ( τ ≥− THH ( R ) hC pn ) . In addition, we may use this presentation to take theFrobenius to be the isomorphism sending u ↦ σ on homotopy groups.It remains to study what the canonical map does. Consider the following commutativediagram: τ ≥− TC − ( R ) ∧ p τ ≥− TP ( R ) ∧ p τ ≥− ( THH ( R ) ∧ p ) hC pn τ ≥− THH ( R ) tC pn can h T can hCpn Applying π ∗ , it gives the commutative diagram A inf [ u ] A inf [ σ ] W n + ( R )[ u ] W n ( R )[ σ ] u ↦ ξσ ˜ θ n The upper horizontal map is identified by [BMS19, Proposition 6.3], and the right ver-tical map is identified by [BMS19, Remark 6.6]. Since we are using the presentation of π ≥− (( THH ( R ) ∧ p ) hC pn ) coming from the equivalence with the Tate construction above, itfollows that the left vertical map in the above diagram must be ˜ θ n + ○ ϕ . All the maps aresurjective on π , so there is a unique map making the diagram commute. From [BMS18,Lemma 3.4], we see that the map on π must be the restriction, and then the polynomialgenerator does as claimed. (cid:3) Since there are suspensions in Equation (4.1), we need to understand the what happenson the negative homotopy groups as well.
Proposition 4.1.3.
The map τ < ( THH ( R ) ∧ p ) hC pn can hCpn − ÐÐÐÐÐÐ→ τ < ( THH ( R ) tC p ) hC pn − ≃ τ < THH ( R ) tC pn induces an isomorphism on homotopy groups. Proof.
By the Tate orbit lemma [NS18, Lemma I.2.1] and induction, as in [NS18,Lemma II.4.1], we see that (( THH ( R ) ∧ p ) hC p ) hC pn − ≃ ( THH ( R ) ∧ p ) hC pn . Homotopy orbitsfor any compact Lie group preserve connectivity, so ( THH ( R ) ∧ p ) hC pn is connective. On the . RIGGENBACH On The Algebraic K -Theory of Double Points other hand, (( THH ( R ) ∧ p ) hC p ) hC pn − is the homotopy fiber of the map can hC pn − . It followsthat this map is an equivalence below degree zero, as claimed. (cid:3) Corollary 4.1.1.
The map TC − ( R ) → ( THH ( R ) ∧ p ) hC pn on homotopy groups is the quotientmap A inf [ u, v ]/( uv − ξ ) → A inf [ u, v ]/( uv − ξ, ˜ ξ n v ) Proof.
From the computations in Proposition 4.1.2 and Proposition 4.1.3, we see thatthe map TC − ( R ) ∧ p → ( THH ( R ) ∧ p ) hC pn must be surjective on homotopy groups. Further, theelement ˜ ξ n v ∈ π − ( TC − ( R ) ∧ p ) must be mapped to zero in π − (( THH ( R ) ∧ p ) hC np ) since it goesto zero under the canonical map, which is an isomorphism in this degree. Consequently, weget a factorization A inf [ u, v ]/( uv − ξ ) π ∗ ( THH ( R ) ∧ p ) hC pn A inf [ u, v ]/( uv − ξ, ˜ ξ n v ) Composing the dotted map with ϕ hC pn is then an isomorphism in non-negative degrees bythe computation in Proposition 4.1.1. Since ϕ hC pn p is an isomorphism in these degrees byProposition 4.1.2, it follows that the dotted map must also be an isomorphism.Similarly, composing with can hC pn is an isomorphism on negative homotopy groups.Therefore, the dotted map is also an isomorphism on negative homotopy groups by Propo-sition 4.1.3. (cid:3) Note that this description, when combined with the isomorphism π ∗ ( THH ( R ) tC pn ) ≅ A inf [ σ, σ − ]/( ˜ ξ n σ − ) , completely determine what the Frobenius and the canonical maps mustbe in this presentation.4.2. Case One: s even. Using the above, we are now ready to evaluate TP ( A d , R ) ∧ p and TC − ( A d , R ) . By the decomposition in Equation (4.2), we may break up these computationsinto the cases in Equation (4.1). In this subsection, we will compute these invariants for theterms with s even. Lemma 4.2.1.
Let X ( m ) = ∏ s ∣ ms even ∏ ω ∈ ω s,d ( Σ m − THH ( R ) ∧ ( T / C ms ) + ) , so that X = ∏ m ∈ Z + X ( m ) is the summand of s even elements in THH ( A d , R ) . Then there are iso-morphisms π ∗ ( TC − ( X ( m ) ∧ p )) ≅ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∏ s ∣ ms even ∏ ω ∈ ω s,d W v p ( ms )+ ( R ) if ∗ > m − is even ∏ s ∣ ms even ∏ ω ∈ ω s,d W v p ( ms ) ( R ) if ∗ < m − is even otherwiseand π ∗ ( TP ( X ( m )) ∧ p ) ≅ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∏ s ∣ ms even ∏ ω ∈ ω s,d W v p ( ms ) ( R ) if ∗ is even otherwisewhere W ( R ) is understood to be zero, and v p is the p -adic valuation. . RIGGENBACH On The Algebraic K -Theory of Double Points Proof.
Since both the homotopy fixed points and the Tate construction will commutewith the products in our decomposition, we may compute each term wise. We will computethis for homotopy fixed points, the Tate computation will follow similarly. Thus we needonly compute (( Σ m − THH ( R ) ∧ ( T / C ms ) + ) ∧ p ) h T For any spectrum X ∈ S p B T , the map X ∧ ( T / C ms ) + ( p -completion )∧ id ÐÐÐÐÐÐÐÐÐÐ→ X ∧ p ∧ ( T / C ms ) + mod p becomes up to equivalence X / p ∧ ( T / C ms ) + id Ð→ X / p ∧ ( T / C ms ) + . Consequently, ( X ∧ ( T / C ms ) + ) ∧ p ≃ ( X ∧ p ∧ ( T / C ms ) + ) ∧ p . This equivalence is equivariant, so we have that theabove computation reduces to computing (( Σ m − THH ( R ) ∧ p ∧ ( T / C ms ) + ) ∧ p ) h T ≃ Σ m − (( THH ( R ) ∧ p ∧ ( T / C ms ) + ) h T ) ∧ p . (Here, to get the p -completion inside the Tate construction, we would use [NS18, LemmaI.2.9], along with the p-adic equivalence BC p ∞ → B T .). By Proposition 4.0.1, we in turnget the equivalence Σ m − ( THH ( R ) ∧ p ∧ ( T / C ms ) + ) h T ≃ Σ m − ( S ∧ THH ( R ) ∧ p ) hC ms ≃ p -equiv Σ m ( THH ( R ) ∧ p ) hC pvp ( ms ) . The last equivalence comes from the fact that the map C r → C p vp ( r ) is a p -adic equivalence.The homotopy groups of the third term are completely described by Corollary 4.1.1. Inparticular, the third term is already p -complete. (cid:3) Note that both TC − ( X ) and TP ( X ) commute with the product. Thus the computationabove also lead to a computation of TC − ( X ) ∧ p and TP ( X ) ∧ p . Remark 4.2.1.
It is easy to check that the maps in Lemma 4.2.1 are isomorphisms ofgraded A inf -modules. Since TC − ( A d ) and TP ( A d ) are E ∞ ring spectra and X is a summandof THH ( A d ) , the graded A inf -modules above should have a product. It would be interestingto find out what that product is. This seems tractable, but it is not needed for the results ofthis paper. Case Two: m odd. In this section, we compute the topological negative cyclic andperiodic homology in the the second case in Equation (4.1).
Lemma 4.3.1.
Let X ( m ) = ∏ s ∣ m ∏ ω ∈ ω s,d ( Σ m − THH ( R ) ∧ ( T / C ms ) + ) , so that if we let X = ∏ m ∈ Z + ∖ Z + X ( m ) , then X is the summand where s and m are odd in Equation (4.1).Then there are isomorphisms π ∗ ( TC − ( X ( m )) ∧ p ) ≅ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∏ s ∣ m ∏ ω ∈ ω s,d W v p ( ms )+ ( R ) if ∗ > m − is odd ∏ s ∣ m ∏ ω ∈ ω s,d W v p ( ms ) ( R ) if ∗ < m − is odd otherwise . RIGGENBACH On The Algebraic K -Theory of Double Points and π ∗ ( TP ( X ( m )) ∧ p ) ≅ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∏ s ∣ m ∏ ω ∈ ω s,d W v p ( ms ) ( R ) if ∗ is odd otherwisewhere W ( R ) is understood to be zero, and v p is the p -adic valuation. Proof.
There was nothing essential about the parity of m and s used in the proof ofLemma 4.2. Hence the same proof will go through in this case, the only difference being theparity of m is odd, so the homotopy groups will be concentrated in odd degree instead ofeven degree. (cid:3) TC calculation We can now calculate π ∗ ( TC ( A d ) ∧ p ) . To do this, we will use the technique introduced byNikolaus and Scholze. Theorem 5.0.1 ([NS18], Theorem II.4.11) . Let X be a genuine cyclotomic spectrum suchthat the underlying spectrum is bounded below. Then TC ( X ) ≃ hofib ⎛⎝ TC − ( X ) ( ϕ p − can ) p ∈ P ÐÐÐÐÐÐÐ→ ∏ p ∈ P TP ( X ) ∧ p ⎞⎠ . Taking the p -completion of the above diagram, or using the p -adic equivalence BC p ∞ → B T and [NS18, Theorem II.4.10], we get the following TC ( X ) ∧ p ≃ hofib ( TC − ( X ) ∧ p ϕ p − can ÐÐÐÐ→ TP ( X ) ∧ p ) ≃ hofib ( TC − ( X ∧ p ) ϕ p − can ÐÐÐÐ→ TP ( X ∧ p ) ∧ p ) For X = THH ( A d ) , the splitting of THH ( A d ) in Equation (4.1) can be written as a productas explained below Equation (4.2) and since we are p -completing and p ≠ , ( R P ) ∧ p ≃ ∗ andso the factors with s odd and m even collapse. This gives a T -invariant decomposition(5.1) THH ( A d ) ∧ p ≃ ∏ m ∈ N ∏ s ∣ ms ≡ m mod ∏ ω ∈ ω s,d ( Σ m − THH ( R ) ∧ ( T / C ms ) + ) ∧ p with actions as explained in the previous sections.We know what TC − and TP look like on each of the factors: Lemma 4.2.1 deals with thefactors where s and m are both even and Lemma 4.3.1 deals with the factors where theyare both odd.We can determine what ϕ p and the canonical map can do in these terms. Note that byLemma 3.0.1, the Frobenius on THH ( R ) ∧ B cy ( Π ) ≃ THH ( R [ Π ]) is induced by the smashproduct of the Frobenius maps on each part. The decomposition of Equation (3.1) is one of T -spaces, and underlies the decompositions of TC − and TP that we are using. The canonicalmap goes from the homotopy fixed points to the homotopy Tate construction of a particular T -space, so it sends the summand corresponding to a particular ω (which determines its m and s ) to itself. However, the Frobenius map on B cy ( Π ) sends the summand correspondingto ω (of length m and cycle length s ) to the concatenation ω ⋆ p of ω p times (of length pm and with the same cycle length s ). . RIGGENBACH On The Algebraic K -Theory of Double Points Before considering the two cases, we identify the Frobenius in terms of the Frobenius on
THH ( R ) ∧ p . Indeed, we have the following commutative diagram: THH ( R ) ∧ p ∧ B cy ( ω ) THH ( R ) tC p ∧ B cy ( ω ) THH ( R ) tC p ∧ B cy ( ω ∗ p ) tC p ( THH ( R ) ∧ p ∧ B cy ( ω ∗ p )) tC p ϕ THH ( R ) p ∧ idϕ p id ∧ ϕ Bp l where l is the lax-monoidal map. Then, by a similar argument as in [HN19, Lemma 2], thevertical maps in the above diagram are equivalences. Thus, up to a canonical isomorphism,we may take ϕ p = ϕ THH ( R ) p ∧ id .5.1. Case One: s even. By Lemmas 4.2.1 and 4.3.1, the homotopy groups of both targetand source are concentrated in even degrees, so we have π r ( TC ( X ) ∧ p ) ≅ ker ( π r ( TC − ( X ) ∧ p ) ( ϕ p − can ) ∗ ÐÐÐÐÐÐ→ π r ( TP ( X ) ∧ p ) and π r − ( TC ( X ) ∧ p ) ≅ coker ( π r ( TC − ( X ) ∧ p ) ( ϕ p − can ) ∗ ÐÐÐÐÐÐ→ π r ( TP ( X ) ∧ p ) . Where X = ∏ m ∈ Z + ∏ s ∣ ms even ∏ ω ∈ ω s,d ( Σ m − THH ( R ) ∧ ( T / C ms ) + ) . Since the Frobenius fixes s , and only changes m , we will rewrite X = ∏ m ′ ∈ J p ∞ ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω ∈ ω s,d Σ m − THH ( R ) ∧ ( T / C ms ) + where J p is the set of positive integers coprime to p (as in [Spe19]). Both ϕ p and thecanonical map can will respect the outside product, so it is enough to consider X ( m ′ ) forfixed m ′ ∈ J p , X ( m ′ ) being the m ′ term in X . For the rest of this subsection, we will have m ′ , p , and r fixed.We see from 4.2.1 that there are two distinct cases to consider. We will split themup similar to the argument in [HN19]. Recall that the function t ev = t ev ( p, r, m ′ ) is theunique positive integer, if it exists, such that m ′ p t ev − ≤ r < m ′ p t ev . If no such positiveinteger exists, t ev ( p, r, m ′ ) = . Then we have the following commutative diagram, wherethe columns are the exact sequences: . RIGGENBACH On The Algebraic K -Theory of Double Points ∞ ∏ ν = t ev ∏ s ∣ m ′ p ν s is even ∏ ω s,d W ν − v p ( s ) ( R ) ∞ ∏ ν = t ev ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s ) ( R ) TC − r ( X ( m ′ )) ∧ p TP r ( X ( m ′ )) ∧ pt ev − ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s )+ ( R ) t ev − ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s ) ( R ) ϕ p − canϕ p − canϕ p − can Recall that by Proposition 4.1.2 and Proposition 4.1.3, we know what the Frobenius andthe canonical map look like on each component, at least enough to do computations. On thetop line, the canonical map is an isomorphism. In the bottom components, the Frobeniuscan be taken to be the identity, and the canonical map is the reduction in length andmultiplication by ˜ θ r ( ξ ) on the polynomial generator. Lemma 5.1.1.
The top horizontal map in the above diagram is an isomorphism.
Proof.
In this range, the canonical map is an isomorphism. Let x ∈ ∞ ∏ ν = t ev ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s ) ( R ) , and let p t ev , p t ev + , . . . be the projection maps of the outermost product. Define the degree of x to be the minimum n such that p n ( x ) ≠ , which exists if and only if x ≠ . Then can ( x ) also has degree n since can is an isomorphism on the product terms. On the other hand, ϕ p must increase the degree by at least one, or send x to zero. In either case, p n ( ϕ p ( x )) = ,so p n ( can ( x ) − ϕ p ( x )) ≠ , and can − ϕ p is injective.To see that can − ϕ p is surjective, we write x n ∶ = p n ( x ) . We will define a pre-image y inductively as y t ev = can − ( x t ev ) , and y k = can − ( x k − ϕ p ( y k − )) . It is then straightforwardto check that ( can − ϕ p )( y ) = x , as desired. (cid:3) Hence we get by the snake lemma applied to the diagram above that π r ( TC ( X ( m ′ )) ∧ p ) ≅ ker ( can − ϕ p ) , and π r ( TC r − ( X ( m ′ )) ∧ p ) ≅ coker ( can − ϕ p ) . Once again, to compute can − ϕ p , there are two cases to consider: when ν = t ev − and when it does not.For convenience, we write TC − r ( + ) ∶ = t ev − ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω ∈ ω s,d W ν − v p ( s )+ ( R ) . RIGGENBACH On The Algebraic K -Theory of Double Points and TP r ( + ) ∶ = t ev − ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω ∈ ω s,d W ν − v p ( s ) ( R ) . We then have the following result.
Lemma 5.1.2.
The composite map t ev − ∏ ν = ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s )+ ( R ) ↪ TC − r ( + ) can − ϕ p ÐÐÐÐ→ TP r ( + ) is an isomorphism. Proof.
In this range, can − ϕ p = can − ϕ p , the difference only comes from the ν = t ev − factor in TC − r ( + ) , where the canonical map is unaffected, but the Frobenius map is thezero map. In addition, ϕ p is an isomorphism on each component of the source. We will nowtransfer the proof of Lemma 5.1.1 to this setting. Let x ∈ ∏ t ev − ν = ∏ s ∣ m ′ p ν s even ∏ ω s,d W ν − v p ( s )+ ( R ) be nonzero. Define the degree of x as the largest n such that p n ( x ) ≠ , p i the projectionsof the outermost product. Then ϕ p ( x ) has degree n + , and can ( x ) has degree at most n .Consequentially, the degree of ( can − ϕ p )( x ) is n + , and in particular ( can − ϕ p )( x ) ≠ .To see that the map is also surjective, we first show that ϕ p is surjective, i.e. it hits allthe non-zero factors in the product. Since ω s,d depends only on s and d , and ϕ p preservesboth, it follows that ϕ p will hit all factors in this product if it hits one. For the product over s , the Frobenius will not change s , so we are getting the factors with s such that s ∣ m ′ p ν indegree ν + . In other words, the Frobenius is only missing the factors with v p ( s ) = v p ( m ) .These terms on the right, however, are exactly the W -terms, and so are all zero. Finally,in the outermost product we only miss the factor ν = which is also only W -terms.Thus, ϕ p is surjective in this range. To see that ϕ p − can is surjective, take an ele-ment y ∈ ∏ ν = t ev − ν = ∏ s ∣ m ′ p ν s even ∏ ω ∈ ω s,d W ν − v p ( s ) ( R ) . Inductively define a pre-image z by z t ev − = ϕ − p ( y t ev − ) , and z t ev − k = ϕ − p ( y t ev − k − can ( z t ev − k + )) . (cid:3) Thus the cokernel of the map in question is zero. Furthermore, this also shows that thekernel is, in addition to , comprised of elements X such that x t ev − ≠ . For any such x ,for it to be in the kernel, x t ev − = − ϕ − p ( can ( x t ev )) . Inductively, x i is determined by x t ev − .Conversely, every element in ∏ s ∣ m ′ p tev − s even ∏ ω s,d W t ev − v p ( s ) ( R ) determines an element in thekernel, so p t ev − ∣ ker ( can − ϕ p ) is an isomorphism. Corollary 5.1.1.
Let X be as above, p an odd prime. Then there are isomorphisms TC r − ( X ) ∧ p ≅ and TC r ( X ) ∧ p ≅ ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev − s even ∏ ω s,d W t ev − v p ( s ) ( R ) . Proof.
By the decomposition explained above, it is enough to show that this holds when m ′ is fixed. This follows from Lemma 5.1.2 and the discussion after it. (cid:3) Case Two: m odd. The case when m is odd is essentially identical in proof to CaseOne. For this reason, we will only record the result . RIGGENBACH On The Algebraic K -Theory of Double Points Lemma 5.2.1.
Let X = ∏ m ∈ Z + ∖ Z + ∏ s ∣ m ∏ ω s,d ( Σ m − THH ( R ) ∧ ( T / C ms ) + ) , p an odd prime.Then there are isomorphisms TC r ( X ) ∧ p = and TC r + ( X ) ∧ p ≅ ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod − ∏ ω s,d W t od − v p ( s ) ( R ) Theorem 2.0.1, and therefore Theorem 1.0.1, now follows.6.
Examples
We conclude this paper with some calculations which can be derived from our results. Thefirst example is a direct corollary, but one which tends to be the main class of computationsdone for K -theory. The second is not an example of a perfectiod ring, but is sufficiently closethat we may still apply our results. Finally, the last example recovers recent computationsdone by Speirs.6.1. The case of R a perfect F p -algebra. As stated above, this is a direct application ofthe main theorem, since all perfect F p -algebras are perfectiod. Corollary 6.1.1.
Let p be an odd prime, and k a perfect F p -algebra, i.e., an F p -algebrawhere the Frobenius is an isomorphism. Then there is an isomorphism π ∗ ( K ( A d , m )) ≅ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev − s even ∏ ω s,d W t ev ( p,r,m, )− v p ( s ) ( k ) if ∗ = r ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod − ∏ ω s,d W t od ( p,r,m ′ )− v p ( s ) ( k ) if ∗ = r + This recovers and slightly extends the calculation Lindenstrauss and McCarthy [LM08]did for perfect fields of positive characteristic. Note that we are not p -completing relative K -theory in the above corollary. This is because of Proposition 2.0.1.In particular, we get the following result. Corollary 6.1.2.
Let p be an odd prime, and let R = F q . Then there is a isomorphism π ∗ ( K ( A d )) ≅ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ Z if ∗ = ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev − s even ∏ ω s,d O F / p t ev ( p,r,m ′ )− v p ( S ) if ∗ = r, ∗ ≠ Z /( q r − − ) × ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod − ∏ ω s,d O F / p t od ( p,r,m ′ )− v p ( s ) if ∗ = r + where F = Q p ( ζ q − ) is the unique unramified extension of Q p of degree log p ( q ) . The case of R = Z p [ ζ p ∞ ] . For this example, note that as stated above Z p [ ζ p ∞ ] is notperfectiod. In particular, every perfectoid ring is p -complete and this ring is not. This,however, is the only problem: the ring Z cyclp = Z p [ ζ p ∞ ] ∧ p is perfectoid. In addition, since Z p [ ζ p ∞ ] has bounded p ∞ -torsion (because it is torsion free), this is also the derived p -completion. In particular, [CMM18, Lemma 5.2] applies and THH ( Z p [ ζ p ∞ ]) ∧ p ≃ THH ( Z cyclp ) ∧ p . Since this map comes from the natural map of rings Z p [ ζ p ∞ ] → Z cyclp , the map on THH must be a map of cyclotmic spectra, and we may safely replace
THH ( Z p [ ζ p ∞ ]) ∧ p with THH ( Z cyclp ) ∧ p in all of the above arguments. Hence we get the following: . RIGGENBACH On The Algebraic K -Theory of Double Points Corollary 6.2.1.
Let p be an odd prime, and let R = Z p [ ζ p ∞ ] . Then there is an isomorphism π ∗ ( K ( A d , m ) ∧ p ) ≅ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∏ m ′ ∈ J p ∏ s ∣ m ′ p tev − s even ∏ ω s,d W t ev ( p,r,m, )− v p ( s ) ( Z cyclp ) if ∗ = r ∏ m ′ ∈ J p ∖ J p ∏ s ∣ m ′ p tod − ∏ ω s,d W t od ( p,r,m ′ )− v p ( s ) ( Z cyclp ) if ∗ = r + This argument will work for any ring O F , where F / Q p is an algebraic extension such that Q p ( ζ p ∞ ) ⊆ F . In fact, this will work for any ring R such that R ∧ p is perfectoid and R hasbounded p ∞ -torsion.6.3. Relationship to the work of Speirs.
There are many connections between thispaper and the papers [Spe19] and [Spe20] of Speirs. Many of the arguments found here wereinspired from those found in these papers. That being said, we can actually recover manyof the results in both of these papers from our result, either as a direct application or as aconsequence of the proof.The first connection is to [Spe20], which itself is a revisit of [HM97a]. Directly applyingour result when d = , we see that π ∗ ( K ( R [ x ]/ x , x ) ∧ p ) ≅ ∏ m ′ ∈ J p W h ( R ) where h = ⎧⎪⎪⎨⎪⎪⎩ t od if ∗ is odd otherwise . This is because when d = , ω s,d = unless s = . Thus theinner product is only non-zero when s = , and hence only when m ′ is odd. Note that thisagrees with the h function in [Spe20], and so we have as a consequence π r − ( K ( k [ x ]/ x , x ) ∧ p ) ≅ W r ( k )/ V W r ( k ) by [Spe20, Lemma 2], where k a perfect field of characteristic p > , and W the big Wittvectors.What is perhaps more surprising is that we can also recover and extend calculations from[Spe19] as well. To see this, let Π S be the pointed monoid { , , x , . . . , x d , x , x , . . . } , inother words, the pointed monoid considered in [Spe19] to study THH of the coordinateaxes. We then get a map B cy ( Π S ) → B cy ( Π ) , which breaks up along the cyclic worddecomposition on both sides. For a given cyclic word ω without cyclic repetitions, the map B cy ( Π S , ω ) → B cy ( Π , ω ) is a T -equivariant homeomorphism. On the other hand, if ω hascyclic repetitions, then by [Spe19, Lemma 9(2)] B cy ( Π S , ω ) is T -equivariantly contractable.In summation, the map B cy ( Π S ) → B cy ( Π ) is exactly the inclusion of the summand ⋁ ω No cyclic repetitions B cy ( Π , ω ) ↪ ⋁ ω B cy ( Π , ω ) Following this summand through the argument above then gives the following.
Corollary 6.3.1.
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