On The Algebraic $K$-Theory of Double Points
 

Recent years have seen a remarkable increase in interest in algebraic $K$-theory. The work of [BGT13] and [Bar16] have described a universal property of $K$-theory. On the more calculational side, the work of Voevodsky and many others has lead to the solution to the Quillen-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 this is 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 homotopy theory, require the scheme to be regular. Once $\mathbb{A}^{1}$-homotopy invariance fails, we lose many of these theorems. Despite this, there have been many successful computations of the $K$-theory of singular schemes using trace methods, particularly in characteristic $p>0$ or rationally, e.g. [HM97], [HM97a], [HN19], [LM08], [Spe19], and [Spe20].

Define for a given ring $R$

and

Then the goal of the present paper is to compute $K_{*}(A_{d},\mathfrak{m})^{\wedge}_{p}$. We manage to do this for $R$ a perfectoid ring. In order to state the main theorem, we will need some terminology from [Spe19]. This is the first of many connections to [Spe19] and [Spe20], which we explore in detail in Section 7.3.

We take from [Spe19] two functions, $t_{ev}$ and $t_{od}$. The function $t_{ev}=t_{ev}(p,r,m^{\prime})$ is the unique positive integer, if it exists, such that $m^{\prime}p^{t_{ev}-1}\leq 2r<m^{\prime}p^{t_{ev}}$. If no such integer exists, $t_{ev}=0$. Similarly, $t_{od}=t_{od}(p,r,m^{\prime})$ is, if it exists, the unique positive such that $m^{\prime}p^{t_{od}-1}\leq 2r+1<m^{\prime}p^{t_{od}}$, and is zero if no such integer exists. We also define the sets $\omega_{s,d}$ as the set of cyclic words, as defined in definition 3.2, of d letters, length $s$, and period exactly $s$. In other words, the elements of $\omega_{s,d}$ 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 calculated the cardinality of this set as

where $\mu(-)$ is the Möbius function. Finally, let $J_{p}=\{m^{\prime}\in\mathbb{Z}_{+}|(m^{\prime},p)=1\}$.

Our main result is now the following.

For the reader unfamiliar with perfectoid rings, the relevant theory is reviewed in Section 5.1. They key example to keep in mind is when $R$ is a perfect $\mathbb{F}_{p}$-algebra, i.e., it is an $\mathbb{F}_{p}$-algebra where the Frobenius is an isomorphism. The main results cited and proven in Section 5.1 are lifts of results known for perfect $\mathbb{F}_{p}$-algebras to the setting of perfectoid rings.

This work originally started as a revisit of [LM08], with the goal of reproducing the calculation using the new methods introduced by Nikolaus and Scholze in [NS18]. We discuss how Theorem 1.1 relates to [LM08] briefly in Section 7.1, where we extend their result from perfect fields of characteristic $p>0$ to perfect $\mathbb{F}_{p}$-algebras. We have also managed to generalize some of the work of Speirs, as discussed in Section 7.3.

The ring $A_{d}$ is particularly well suited for this computation, since on it we may apply the Dundas Goodwillie McCarthy Theorem [DGM13, Theorem 7.0.0.2] (or for a more modern approach and statement, [Ras18, Theorem 1.1.1]) to get a pullback square

where the vertical maps are the quotient by the (nilpotent) ideal $\mathfrak{m}$. As noted in the diagram, the inclusion $R\to A_{d}$ gives a splitting of the above maps, and hence we get an isomorphism

where the last equivalence comes from the above pullback square.

The celebrated computation of Quillen [Qui72] gives the homotopy groups of the first summand in the case of $R$ a finite field. In the more general case of $R$ a perfectoid ring, we will usually not be able to say more in the integral setting.

Our main theorem is then equivalent to the following:

For the remainder of this section we will work in the $p$-complete setting. In this setting we see one of the first benefits of the recent advances in perfectoid rings and homotopy theory.

As noted in [CMM18], their proof of this result specializes nicely to the case of $R$ semiperfectoid. For our purposes we will only need the following corollary.

Hence going from a calculation of the relative $K$-theory $K(A_{d},\mathfrak{m})^{\wedge}_{p}$ to a computation of $K(A_{d})^{\wedge}_{p}$ is a matter of computing $\operatorname{TC}(R)^{\wedge}_{p}$ for these rings. In principal, this is an easier computation. In particular, the recent work of Bhatt and Scholze in [BS19] introduces a promising 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 $\mathbb{F}_{p}$-algebra.

Proof.  Since $p=0$ in $A_{d}$, $\mathfrak{m}$ is $p$-power torsion. Applying [LT19, Theorem D] then gives the result.∎

We begin by noting that $A_{d}=R\wedge\Pi$, where $\Pi=\{0,1,x_{1},\ldots,x_{d}\}$ is the pointed monoid with all products $x_{i}x_{j}=0$. Since $\operatorname{THH}$ is symmetric monoidal [NS18, Section IV.2], we then get that

The latter term is equivalent to $\Sigma^{\infty}B^{cy}(\Pi)$. We refer the reader to [Spe19, Sections 3.1 and 3.2] for a comprehensive review of the relevant details about this decomposition. In particular, we have the following lemma.

By naturality, we see that the map $\operatorname{THH}(R)\to\operatorname{THH}(R[\Pi])$ is given by the composition

Consequently, the cofiber of this map is given by $\operatorname{THH}(R)\wedge\operatorname{cofib}(S^{0}\to B^{cy}(\Pi))\cong\operatorname{THH}(R)\wedge\widetilde{B^{cy}(\Pi)}$, where $\widetilde{B^{cy}(\Pi)}$ is the cofiber of the map $S^{0}\to B^{cy}(\Pi)$ in spaces.

In light of the previous section, we get the following decomposition of $\operatorname{THH}(A_{d})$:

where $\omega_{s,d}$ is the set of all cyclic words with cycle length actually equal to $s$ (and not a proper divisor of $s$) in $d$ letters. In addition, the map $\operatorname{THH}(R)\to\operatorname{THH}(A_{d})$ giving the splitting of $\operatorname{TC}$ is the inclusion as the first summand.

Note also that each summand is of the form $\operatorname{THH}(R)\wedge S^{\lambda}\wedge X$ for some representation $\lambda$ and space $X$. The $\mathbb{T}$-action on this can be quite complicated if we allow arbitrary $\mathbb{E}_{1}$-ring spectra $R$. We make the following definition in order simplify these spaces.

We will now spend some time showing that topological hochschield homology of perfectoid rings satisfy this condition. In fact, we will show something stronger:

We will see in Corollary 4.4 that this Proposition applies in our situation, and so we may consider only trivial representation spheres. Before we prove this result, we first need the following lemma.

Proof.  Consider $X$ as an $(\infty,1)$-functor $X:B\mathbb{T}\to E\textrm{-mod}$. It is enough to show that $X$ is contractable as such a map of pointed simplicial sets (pointing $E\textrm{-mod}$ by $X(*)$), since then the identification map $\Sigma^{n}E\simeq X$ will be equivariant.

This map then factors through a map $B\mathbb{T}\to Bhaut_{E}(X(*))$, since the sources is a groupoid. Here $haut_{E}(-)$ is the connected component of the space of $E$-module maps which contain a weak equivalence. Furthermore, since both spaces are Kan complexes, it is then enough to show after geometric realization that this map is contractable. Under the identification of $X(*)\simeq\Sigma^{n}E$, this is then equivalent to a map $B\mathbb{T}\to Bhaut_{E}(\Sigma^{n}E)$. By (de)suspending by trivial representation spheres, we may assume without loss of generality that $n=0$. It remains to classify that maps $B\mathbb{T}\to Bhaut_{E}(E)$.

To this end, note that $haut_{E}(E)\subseteq E\textrm{-mod}(E,E)\simeq\Omega^{\infty}E$¹¹1Warning: This identification is only as a space, and not as an $\mathbb{E}_{\infty}$-space. We go on to take $B$ of the $\mathbb{A}_{\infty}$ structure given by composition and not the lax-monoidal struture of $E\textrm{-mod}(E,-)$, so this distinction matters., and since $haut_{E}(E)$ is a subspace of connected components, $\pi_{i}(haut_{E}(E))\cong\pi_{i}(\Omega^{\infty}E)\cong\pi_{i}(E)$ for $i\geq 1$. In particular, $haut_{E}(E)$ has homotopy groups concentrated in even degree.

Note also that $E\textrm{-mod}$ is a symmetric monoidal $(\infty,1)$-category, and so the monoidal unit will have an $\mathbb{E}_{\infty}$ structure on its homotopy automorphism space, and the standard $\mathbb{A}_{\infty}$ structure will be the one induced by restricting this $\mathbb{E}_{\infty}$ structure. We then have a group-like $\mathbb{E}_{\infty}$-space $haut_{E}(E)$, and so $\pi_{*}(Bhaut_{E}(E))\cong\pi_{*-1}(haut_{E}(E))$ is concentrated in odd degree.

In addition, $Bhaut_{E}(E)$ is also and $\mathbb{E}_{\infty}$-space, and so homtopy classes of maps into it can be computed using the Atiyah Hirzebruch spectral sequence:

The $E_{2}$ page is concentrated in odd total degree, and there is no room for differentials so the spectral sequence converges. Thus there are no non-contractable maps of even degree, and in particular no non-contractable maps $B\mathbb{T}\to Bhaut_{E}(E)$.∎

We are now ready to prove Proposition 4.2.

Proof of Proposition 4.2.   Suppose $X$ is any element of $E\textrm{-mod}^{B\mathbb{T}}$, and let $S^{V}$ be any $\mathbb{T}$-representation sphere. Then we have the following chain of Borel equivalences:

By the lemma above, there is a Borel equivalence $E\wedge S^{V}\simeq\Sigma^{\dim(V)}E$ in $E\textrm{-mod}^{B\mathbb{T}}$, where the spectrum on the right has trivial $\mathbb{T}$-action. Hence

in $E\textrm{-mod}^{B\mathbb{T}}$, as desired.∎

In particular, we get the following useful corollary, which in particular applies to $\operatorname{THH}(R)^{\wedge}_{p}$ for $R$ quasiregular semiperfectoid (and hence also perfectoid).

Proof.  Since $X$ is concentrated in even degrees, the homotopy fixed point spectral sequence converging conditionally to the homotopy groups of $X^{h\mathbb{T}}$ has $E_{2}$ page concentrated in even bidegrees. Hence the spectral sequence collapses at the $E_{2}$ page, converges, and has no nonzero entries in odd total degree. Consequentially $X$ and $E=X^{h\mathbb{T}}$ satisfy the hypotheses for Proposition 4.2.∎

For $R$ a perfectoid ring, Corollary 4.4 applies and $\operatorname{THH}(R)^{\wedge}_{p}$ is $\mathbb{T}$-suspension invariant (see Definition 4.1.). Consequentially, we may take the trivial representation spheres in Equation 4.1. Note that since the representations $\lambda_{i}$ have complex dimension $i$ (so real dimension $2i$), these trivial representation spheres are $S^{n}$, where $n$ is $m-1$ or $m-2$, depending on the case. In particular, if we write $\operatorname{THH}(A_{d})=\bigvee_{m\in\mathbb{N}}\operatorname{THH}(m)$ where $\operatorname{THH}(m)$ is the wedge of all summands corresponding to cyclic words of length $m$, then the connectivity of $\operatorname{THH}(m)$ is at least $m-2$ and so

We may further decompose $\operatorname{THH}(A_{d})$ 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 respective products. Therefore, to compute $\operatorname{TC}^{-}(A_{d})$, and the relative $\operatorname{TC}^{-}$, it is enough to compute $(-)^{h\mathbb{T}}$ on each of the the summands in Equation  (4.1), and $(-)^{h\mathbb{T}}$ for the summands where $m\geq 1$, 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, $(-)_{h\mathbb{T}}$ can be computed term by term. In addition, homotopy orbits only increase connectivity, so in the decomposition $\operatorname{THH}(A_{d})_{h\mathbb{T}}\simeq\bigvee_{m\in\mathbb{N}}\operatorname{THH}(m)_{h\mathbb{T}}$, each term $\operatorname{THH}(m)_{h\mathbb{T}}$ is at least $(m-2)$-connected. Consequently, we may express $(\operatorname{THH}(A_{d}))_{h\mathbb{T}}$ as a product in the exact same way we did for $\operatorname{TC}^{-}(A_{d})$, and the canonical map will respect this decomposition. In particular, we get the same decomposition of the topological periodic homology.

Note that each summand in Equation (4.1) with $s=m\mod 2$ is of the form $X\wedge\left(\mathbb{T}/C_{n}\right)_{+}$. We use the following proposition to simplify the computation.

In particular, $\left(X\wedge(\mathbb{T}/C_{n})_{+}\right)^{h\mathbb{T}}\simeq(\Sigma X)^{hC_{n}}$, and similarly for the Tate construction. Furthermore, the Tate construction is initial among functors under the homotopy fixed points vanishing on compact objects by [NS18, Lemma I.1.4(ii)], and so the canonical map must be sent to the canonical map under this equivalence.