Rigidity in equivariant algebraic $K$-theory

TL;DR

This paper establishes a rigidity property for equivariant algebraic K-theory spectra under certain conditions, showing that reduction mod n preserves the K-theory spectrum in the presence of a finite group action.

Contribution

It extends the non-equivariant rigidity results to the equivariant setting by revisiting and adapting recent proofs, under specific algebraic conditions.

Findings

Reduction map of mod-n equivariant K-theory spectra is an equivalence.

Rigidity holds for henselian pairs with finite group actions under certain coprimality conditions.

The proof revisits and adapts recent non-equivariant rigidity proofs to the equivariant context.

Abstract

If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra \[ K^G(R)/n\stackrel{\simeq}{\longrightarrow} K^G(R/I)/n\] is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.

Equations

$$K^G(R)/n\stackrel{\simeq }{⟶}{K}^G(R/I)/n$$

$$K(R)/n\stackrel{\simeq }{⟶}K(R/I)/n$$

$$K^{\mathrm{inv}}(R)/n\stackrel{\simeq }{⟶}{K}^{\mathrm{inv}}(R/I)/n$$

$$K^{\mathrm{inv}}(R\wr G)/n\stackrel{\simeq }{⟶}{K}^{\mathrm{inv}}((R/I)\wr G)/n$$

$$K^G(R)/n\stackrel{\simeq }{⟶}{K}^G(R/I)/n$$

$$f(t)=t(1+t{)}^n+g(t)$$

$$R:{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h}}⟶\mathrm{Sets}$$

$$F_R:\mathrm{Sets}⟶{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h}}$$

$$F_R(n):=F_R(\{1,\dots ,n\})(n\ge 0).$$

$$R^{\prime }:{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h},\mathrm{G}}⟶{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h}}$$

$$F_R^{\prime }:{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h}}⟶{\mathrm{Ring}}_R^{\mathrm{nu},\mathrm{h},\mathrm{G}},$$

$$[N]:F_R^{\prime \prime }(n)⟶F_R^{\prime \prime }(n)$$

$${\mathrm{colim}}_{(N,M)=1}{F}_{\mathbb{Z}\left[\frac{1}{M}\right]}^{\prime \prime }(n)\simeq F_{\mathbb{Q}}^{\prime \prime }(n),$$

$${\tau }_{\ge 1}K((\mathbb{Z}⋉(-))\wr G)⟶K((\mathbb{Z}⋉(-))\wr G)⟶{\tau }_{\le 0}K((\mathbb{Z}⋉(-))\wr G)=K_0((\mathbb{Z}⋉(-))\wr G)$$

$$H\mathbb{Z}\otimes {\Sigma }_{+}^{\infty }{\Omega }^{\infty }{\tau }_{\ge 1}K((\mathbb{Z}⋉(-))\wr G)$$

$${\tau }_{\le n}{C}_{\ast }(BG{L}_{2n+1}((\mathbb{Z}⋉(-))\wr G);\mathbb{Z})⟶{\tau }_{\le n}{C}_{\ast }(BGL((\mathbb{Z}⋉(-))\wr G);\mathbb{Z})$$

$$C_{\ast }(BG{L}_n((\mathbb{Z}⋉(-))\wr G);\mathbb{Z})$$

$$1⟶X(-)⟶G{L}_n((\mathbb{Z}⋉(-))\wr G)\stackrel{\pi }{⟶}G{L}_n(\mathbb{Z}[G])⟶1,$$

$$C_{\ast }(BG{L}_n((\mathbb{Z}⋉(-))\wr G);\mathbb{Z})\simeq {\left(C_{\ast }(BX(-);\mathbb{Z})\right)}_{h(G{L}_n(\mathbb{Z}[G]))}.$$

$$G{L}_n(\mathbb{Z}[G])\subseteq \mathcal{G}(\mathbb{Q})=G{L}_n(\mathbb{Q}[G])$$

$$K^{\mathrm{inv}}((\mathbb{Z}⋉I)\wr G,I\wr G)⟶K^{\mathrm{inv}}((\mathbb{Z}⋉I)\wr G)⟶K^{\mathrm{inv}}(\mathbb{Z}\wr G)=K^{\mathrm{inv}}(\mathbb{Z}[G]).$$

$${\mathrm{Ring}}^{\mathrm{nu},\mathrm{h},\mathrm{G}}⟶\mathrm{Sp},I\mapsto K^{\mathrm{inv}}((\mathbb{Z}⋉I)\wr G,I\wr G)/p$$

$$K^{\mathrm{inv}}({\mathcal{O}}_{X,x}^h\wr G)/p\stackrel{\simeq }{⟶}{K}^{\mathrm{inv}}(k[G])/p$$

$$E([X/G])\stackrel{\simeq }{⟶}{\left(\underset{\sigma \subseteq G\text{mbox}cyclic}{⨁}\tilde{E}\left(X^{\sigma }\times \mathrm{Spec}(k[\sigma ])\right)\right)}^G,$$

$$\bar{x}:[\mathrm{Spec}(k)/G]⟶[X/G]$$

$$\xymatrix{E([X/G])\ar[rr]^{-}{\simeq\,\lx@cref{creftype~refnum}{eq:decomp_map}}\ar[dd]^{E(\bar{x})}&\ar[dd]^{\oplus_{\sigma}\widetilde{E}(x_{\sigma}\times\mathrm{id})}\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(X^{\sigma}\times\mathrm{Spec}(k[\sigma])\right)\right)^{G}\,\\ \\ E([\mathrm{Spec}(k)/G])\ar[rr]^{-}{\simeq}\,\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(\mathrm{Spec}(k[\sigma])\right)\right)^{G},}$$

$$\xymatrix{E([Y/G])\ar[rr]^{-}{\simeq}\ar[dd]^{E(\bar{y})}&\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(Y^{\sigma}\times\mathrm{Spec}(k[\sigma])\right)\right)^{G}\ar[dd]^{\oplus_{\sigma}\widetilde{E}(y_{\sigma}\times\mathrm{id})}\\ \\ E([\mathrm{Spec}(k)/G])\ar[rr]^{-}{\simeq}\,\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(\mathrm{Spec}(k[\sigma])\right)\right)^{G}.}$$

$$\xymatrix{\ar[dd]^{E(\iota)}E([\mathrm{Spec}(\mathcal{O}_{X,x}^{h})/G])\ar[rr]^{-}{\simeq}&\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(\mathrm{Spec}(\mathcal{O}^{h}_{X^{\sigma},x})\times\mathrm{Spec}(k[\sigma])\right)\right)^{G}\ar[dd]^{\oplus_{\sigma}\widetilde{E}(\iota_{\sigma}\times\mathrm{id})}\\ \\ E([\mathrm{Spec}(k)/G])\ar[rr]^{-}{\simeq}\,\left(\bigoplus\limits_{\sigma\subseteq G\mbox{ {\small cyclic}}}\widetilde{E}\left(\mathrm{Spec}(k[\sigma])\right)\right)^{G}.}$$

$$\mathrm{Perf}([\mathrm{Spec}({\mathcal{O}}_{X,x}^h)/G])\simeq \mathrm{Perf}({\mathcal{O}}_{X,x}^h\wr G)$$

$${\left(\ker (\pi \wr G)\right)}^n\subseteq (\ker (\pi ){)}^n\wr G.$$

Full text

[TABLE]

is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.

1 Introduction and statement of result

Rigidity is a fundamental feature of algebraic $K$-theory with finite coefficients which was established by Suslin [Sus83] for extensions of algebraically closed fields, and by Gabber and Gillet-Thomason [GT84] for geometric henselian local rings. In [Gab92], inspired by previous results of Suslin [Sus84] for henselian valuation rings of dimension one, Gabber proved a rigidity theorem for algebraic $K$-theory with finite coefficients for general henselian pairs:

Theorem 1** (Gabber).**

If $(R,I)$ is a henselian pair and $n\geq 1$ is an integer such that $n\in R^{*}$, then

[TABLE]

is an equivalence.

In all these results, the coefficients are assumed to be coprime to the characteristic. In [CMM18], the authors established the most comprehensive rigidity statement to date addressing the case of coefficients not necessarily coprime to the characteristic. To formulate it, we denote by $K^{\mathrm{inv}}$ the fiber of the cyclotomic trace $K\longrightarrow TC$. Then their result [CMM18, Theorem A] reads

Theorem 2** (Clausen, Mathew, Morrow).**

If (R,I) is a henselian pair and $n\geq 1$ is an integer, then the reduction map

[TABLE]

is an equivalence.

The purpose of the present note is to generalize this result to an equivariant situation for an action of a finite, abstract group $G$.

Given a commutative ring $R$ with an action of $G$, there is associated the twisted group ring $R\wr G$, see Section 2 for a reminder. Our main theorem is

Theorem 3**.**

If the finite group $G$ acts on the henselian pair $(R,I)$, $|G|\in R^{*}$, and $n\geq 1$ is an integer coprime to $|G|$, then the reduction map

[TABLE]

is an equivalence.

The more traditional invariant in equivariant algebraic $K$-theory is the spectrum $K^{G}(R)$, defined to be the connective $K$-theory of the exact category of finitely generated projective $R$-modules together with a semi-linear $G$-action. We deduce the next result about this.

Corollary 4**.**

Assume in the situation of Theorem 3 that in addition $n\in R^{*}$ holds. Then the reduction map

[TABLE]

is an equivalence.

Proof.

Since $n\in R^{*}$, the $TC$-term in the definition of $K^{\mathrm{inv}}(R\wr G)$ vanishes mod $n$, i.e. $K^{\mathrm{inv}}(R\wr G)/n\simeq K(R\wr G)/n$. Since $|G|\in R^{*}$, a finitely generated projective left $R\wr G$-modules is the same thing as a finitely generated projective $R$-module with a semi-linear $G$-action, hence $K(R\wr G)\simeq K^{G}(R)$; and similarly with $R$ replaced with $R/I$. ∎

Remark 5**.**

The appearance of $R\wr G$ might seem a bit spurious since all our results assume $|G|\in R^{*}$ which forces $K(R\wr G)\simeq K^{G}(R)$ (and similarly for $TC$). Since however several of the intermediate results work without assuming that $|G|\in R^{*}$, we decided to phrase things in terms of $R\wr G$.

Corollary 4 is a generalization of Theorem 1 for equivariant algebraic $K$-theory. Rigidity results for equivariant algebraic $K$-theory have been previously studied for henselian local rings with trivial group actions (but for more general algebraic groups) in [YØ09] and [Kri10] and in [YØ09] and [Tab18] for extensions of algebraically closed fields and extensions of separably closed fields, respectively. In [HRØ18], Corollary 4 was proved in the geometric case, assuming that $G$ is abelian and that $k$ contains $|G|$-th roots of unity.

The proofs of our results are direct generalizations of those of [CMM18]. We made an effort to make this paper reasonably self-contained, which results in repeating some arguments from [CMM18].

We conclude the introduction with an overview of the sections. In Section 2, we establish the equivariant generalization of the key finiteness property, called pseudocoherence, isolated in [CMM18]. This allows to generalize equivariant rigidity from certain nice geometric situations to general henselian pairs. In Section 3 we establish a sufficient supply of equivariant rigidity in nice situations (see Proposition 14) by combining the non-equivariant result with decomposition results of Vistoli and Tabuada-Van den Bergh. Section 4 collects further technical results. The final Section 5 assembles the pieces into a proof of Theorem 3.

2 $G$-projective pseudocoherence

The aim of this section is to establish the equivariant generalizations of the finiteness properties [CMM18, Propositions 4.21, 4.25] of algebraic $K$-theory and of topological cyclic homology with finite coefficients. Fix a finite group $G$ throughout.

Let $R$ be a commutative ring and $I\subset R$ an ideal. Recall that the pair $(R,I)$ is called a henselian pair if for every $f(t)\in R[t]$, $\bar{a}\in R/I$, such that $\bar{a}$ is a simple root of $\bar{f}(t)\in(R/I)[t]$, there exists $a\in R$ such that $a\mapsto\bar{a}$ and $f(a)=0$. By a result of Gabber [Gab92, Corollary 1], the property of $(R,I)$ being a henselian pair depends only on the ideal $I$, regarded as a nonunital ring, and not on $R$. We now briefly recall the definition of nonunital henselian algebras. For a detailed discussion see [CMM18, Section 3].

For a commutative ring $R$, a nonunital $R$-algebra is an $R$-module $I$ endowed with a multiplication $I\otimes_{R}I\to I$ which is associative and commutative. A non-unital $R$-algebra $I$ is said to be henselian if for every $n\geq 0$ and every $g(t)\in I[t]$ of degree at most $n$, the polynomial

[TABLE]

has a (necessarily unique) root in $I$. Let $\mathrm{Ring}^{\mathrm{nu,h}}_{R}$ denote the category of non-unital, henselian $R$-algebras.

Definition 6**.**

We denote by $\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}$ the category of $G$-objects in $\mathrm{Ring}^{\mathrm{nu,h}}_{R}$.

To ease the notation, we will abbreviate $\mathrm{Ring}^{\mathrm{nu,h,G}}:=\mathrm{Ring}^{\mathrm{nu,h,G}}_{\mathbb{Z}}$.

It is observed in [CMM18, Remark 3.10] that the category $\mathrm{Ring}^{\mathrm{nu,h}}_{R}$ is bi-complete, and that the forgetful functor

[TABLE]

to sets is a conservative right adjoint which commutes with sifted colimits.111equivalently, as the categories are discrete, it commutes with filtered colimits and split co-equalizers.

Denoting by

[TABLE]

its left-adjoint, this is remarked to imply that the subcategory $\left(\mathrm{Ring}^{\mathrm{nu,h}}_{R}\right)_{\Sigma}\subseteq\mathrm{Ring}^{\mathrm{nu,h}}_{R}$ of compact projective objects is the idempotent completion of the full subcategory spanned by the free objects

[TABLE]

Moreover, $F_{R}(n)$ is identified in [CMM18, Example 3.9] as the ideal generated by the variables $X_{1},\ldots,X_{n}$ in the $R$-algebra given by the henselization of $R[X_{1},\ldots,X_{n}]$ along the ideal $(X_{1},\ldots,X_{n})$.

This generalizes to the equivariant setting as follows: The category $\mathrm{Ring}^{\mathrm{nu,h,G}}$ is bi-complete and the forgetful functor

[TABLE]

is a conservative right-adjoint which commutes with all colimits. This is clear by thinking of $\mathrm{Ring}^{\mathrm{nu,h,G}}$ as the category of presheaves on $G$ with values in $\mathrm{Ring}^{\mathrm{nu,h}}$. Consequently, denoting the left-adjoint of $R^{\prime}$ by

[TABLE]

and by $F^{\prime\prime}_{R}:=F^{\prime}_{R}\circ F_{R}$, the subcategory $\left(\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}\right)_{\Sigma}\subseteq\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}$ of compact projective objects is the idempotent completion of the full subcategory spanned by the free objects $F^{\prime\prime}_{R}(n):=F^{\prime\prime}_{R}(\{1,\ldots,n\})$ $(n\geq 0)$. These be identified explicitly:

Proposition 7**.**

For every $n\geq 0$, $F^{\prime\prime}_{R}(n)$ is the ideal generated by the variables $X_{\sigma,i}$ $(\sigma\in G,1\leq i\leq n)$ in the $R$-algebra given by the henselization of the polynomial $R$-algebra $R[X_{\sigma,i}\mid\sigma\in G,1\leq i\leq n]$ along the ideal $(X_{\sigma,i})$, and $G$-action determined by $\sigma(x_{\tau,i})=x_{\sigma\tau,i}$.

Said a bit more invariantly, $F^{\prime\prime}_{R}(n)$ is the henselization along the origin of the affine $R$-space afforded by the direct sum of $n$ copies of the regular representation of $G$ over $R$.

Proof of Proposition 7.

Since henselization is a left-adjoint, it suffices to see the analogous statement before henselization. Then using the equivalence between non-unital $R$-algebras and augmented $R$-algebras, the claim follows because the augmented $R$-algebra with $G$-action $R[X_{\sigma,i}\mid\sigma\in G,1\leq i\leq n]$ has the required mapping property. ∎

For every $N\geq 1$, we will denote by

[TABLE]

the “multiplication-by-$N$-map”, namely the unique map in $\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}$ which, under the identification of Proposition 7, maps every $X_{\sigma,i}$ to $NX_{\sigma,i}$.

Proposition 8**.**

For fixed $M\geq 1$ and $n\geq 0$, we have an isomorphism in $\mathrm{Ring}^{\mathrm{nu,h,G}}_{\mathbb{Z}\left[\frac{1}{M}\right]}$

[TABLE]

the (filtered) colimit being taken along the multiplication maps $[N]$ for all $N$ coprime to $M$, partially ordered by divisibility.

Proof.

This is proved exactly as in the special case $M=1,G=\{e\}$ which is due to Gabber (see [CMM18, Corollary 3.20]). We leave the details to the reader. ∎

Recall the twisted group ring (e.g. [CR81, §28]): If $R$ is a commutative ring with a (left) $G$-action, then the twisted group ring $R\wr G$ is the finite free $R$-module on the set $\left\{e_{\sigma}\colon\sigma\in G\right\}$ with multiplication determined by $(re_{\sigma})(r^{\prime}e_{\tau})=r\sigma(r^{\prime})e_{\sigma\tau}$. This construction is functorial in $R$. It is rigged such that the datum of a left $R\wr G$-module is equivalent to the datum of an $R$-module together with a *semi-*linear $G$-action. Observe that when the $G$-action on $R$ is trivial, this construction gives the usual group ring, i.e. $R\wr G=R[G]$ in this case.

For an associative, unital ring $A$, we will denote by $K(A)$ the connective $K$-theory spectrum of the category of finitely generated projective left $A$-modules, cf. [Qui73]. Given any $I\in\mathrm{Ring}^{\mathrm{nu,h,G}}$, we denote by $\mathbb{Z}\ltimes I$ the ring with $G$-action obtained from $I$ by adjoining a unit (necessarily with trivial $G$-action). The augmentation $\mathbb{Z}\ltimes I\longrightarrow\mathbb{Z}$ is $G$-equivariant and thus induces an augmentation $p:(\mathbb{Z}\ltimes I)\wr G\longrightarrow\mathbb{Z}[G]$. We will need the following equivariant generalization of [CMM18, Lemma 4.20].

Proposition 9**.**

Given $I\in\mathrm{Ring}^{\mathrm{nu,h,G}}$, the map $p^{*}:K_{0}\left((\mathbb{Z}\ltimes I)\wr G\right)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}K_{0}(\mathbb{Z}[G])$ is an isomorphism.

Proof.

This is a special case of Proposition 17. ∎

We denote by $\mathrm{Sp}$ the $\infty$-category of spectra. We recall from [CMM18, Definition 4.4] the notions of perfectness and pseudocoherence of spectrum-valued functors on a category relative to a subcategory: Given a small full subcategory $\mathcal{D}$ of a locally small category $\mathcal{C}$, a functor $F:\mathcal{C}\to\mathrm{Sp}$ is called $\mathcal{D}$-perfect if $F$ belongs to the thick subcategory generated by the functors $\{\Sigma^{\infty}_{+}\mathrm{Hom}_{\mathcal{C}}(D,\cdot)\,\mid\,D\in{\mathcal{D}}\}$ in the presentable, stable $\infty$-category $\mathrm{Fun}(\mathcal{C},\mathrm{Sp})$. A functor $F\in\mathrm{Fun}(\mathcal{C},\mathrm{Sp})$ is said to be $\mathcal{D}$-pseudocoherent if for each $n\in\mathbb{Z}$, there exists a $\mathcal{D}$-perfect functor $F_{n}$ and a map $F_{n}\to F$ such that $\tau_{\leq n}F_{n}(C)\to\tau_{\leq n}F(C)$ is an equivalence for all $C\in\mathcal{C}$. In the particular case when $\mathcal{D}=\left(\mathrm{Ring}^{\mathrm{nu,h}}\right)_{\Sigma}\subseteq\mathcal{C}=\mathrm{Ring}^{\mathrm{nu,h}}$, $F$ is called projectively pseudocoherent, see [CMM18, Definition 4.12, (2)]. We pose the immediate equivariant generalization of this as a definition.

Definition 10**.**

A functor $F:\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}\longrightarrow\mathrm{Sp}$ is called $G$-projectively pseudocoherent ($G$-pscoh for short), if it is $\left(\mathrm{Ring}^{\mathrm{nu,h,G}}_{R}\right)_{\Sigma}$-pseudocoherent.

Our first aim then is to establish the following generalization of [CMM18, Proposition 4.21].

Proposition 11**.**

The functor $\mathrm{Ring}^{\mathrm{nu,h,G}}\longrightarrow\mathrm{Sp}$, $I\mapsto K((\mathbb{Z}\ltimes I)\wr G)$ is $G$-pscoh.

Proof.

Using the fiber sequence of functors

[TABLE]

and the fact that $G$-pscoh functors form a thick subcategory [CMM18, Proposition 4.8, (1)], it suffices to see separately the $G$-projective pseudocoherence of $\tau_{\geq 1}K((\mathbb{Z}\ltimes(-))\wr G)$ and of $K_{0}((\mathbb{Z}\ltimes(-))\wr G)$.

For the latter, Proposition 9 yields an isomorphism $K_{0}((\mathbb{Z}\ltimes(-))\wr G)\simeq K_{0}(\mathbb{Z}[G])$ to the constant functor with value the finitely generated abelian group $K_{0}(\mathbb{Z}[G])$, see [Kuk07, Theorem 2.2.1]. This settles the claim for this term.

To see that the other term is $G$-pscoh, we use the criterion [CMM18, Proposition 4.10 and Proposition 4.11] to reduce to seeing that the functor

[TABLE]

is $G$-pscoh. It is well known (see [Wei13, chapter IV, §1]) that this functor is equivalent to $C_{*}(BGL((\mathbb{Z}\ltimes(-))\wr G);\mathbb{Z})$, the complex of integral chains on the classifying space of the infinite general linear group. We now use homology stability as given by [vdK80, Theorem in section 4.11] for the associative ring $A(-):=(\mathbb{Z}\ltimes(-))\wr G$. To do so, we need to see that the stable range of $A(-)$ is bounded independently of the argument $-\in\mathrm{Ring}^{\mathrm{nu,h,G}}$. Firstly, it is easy to see that dividing out a radical ideal does not change the stable range (cf. [Lam99, p. 32] and [Wei13, chapter I, ex. 1.12, (v)]), and at the beginning of the proof of Proposition 17 we will see that $(-)\wr G$ is a radical ideal in $A(-)$ with quotient ring $\mathbb{Z}[G]$. This already gives the independence of the stable range of $A(-)$ of the argument $(-)$, and since $\mathbb{Z}[G]$ is finite over its central subring $\mathbb{Z}$, this is bounded by (in fact, equal to) the stable range of $\mathbb{Z}$ (according to Bass’s stable range theorem [Bas68, Chapter V, Theorem 3.5]). We conclude that for every $n\geq 1$ the obvious map on truncations

[TABLE]

is an equivalence. Renaming indices, this reduces us to seeing that for a fixed $n\geq 1$, the functor

[TABLE]

is $G$-pscoh. There is a short exact sequence of groups

[TABLE]

defining $X(-)$. 222To see that $\pi$ is onto, recall that the augmentation $\mathbb{Z}\ltimes(-))\wr G\longrightarrow\mathbb{Z}[G]$ is split surjective. This gives an equivalence

[TABLE]

To conclude the argument exactly as in the proof of [CMM18, Proposition 4.19], it remains to establish that, firstly, the functor $C_{*}(BX(-);\mathbb{Z})$ is $G$-pscoh and that, secondly, there is a finite index normal subgroup $N\subseteq GL_{n}(\mathbb{Z}[G])$ such that its classifying space $BN$ is equivalent to a finite CW-complex. The first claim follows as in loc.cit.‚, because $X(I)\simeq I^{\mid G\mid\cdot n^{2}}$ (as sets), and the second claim follows from work of Borel and Serre, specifically [Ser71, section 2.4, Théorème 4 and section 1.5, Proposition 10], if we can show that $GL_{n}(\mathbb{Z}[G])$ is an arithmetic subgroup of a suitable reductive group ${\mathcal{G}}$ over $\mathbb{Q}$. Indeed, one can take for ${\mathcal{G}}$ the group of units of the $\mathbb{Q}$-algebra $M_{n}(\mathbb{Q}[G])$: It is clear that

[TABLE]

is an arithmetic subgroup, and since $\mathbb{Q}[G]\otimes_{\mathbb{Q}}\mathbb{C}\simeq\mathbb{C}[G]$ is a product of full matrix rings over $\mathbb{C}$, the group ${\mathcal{G}}\otimes_{\mathbb{Q}}\mathbb{C}$ is a finite product of various $GL_{i,\mathbb{C}}$’s, and hence is (connected and) reductive. ∎

The following generalization of [CMM18, Proposition 4.25] is even more immediate.

Proposition 12**.**

For every prime $p$, the functor $\mathrm{Ring}^{\mathrm{nu,h,G}}\longrightarrow\mathrm{Sp}$, $I\mapsto TC((\mathbb{Z}\ltimes I)\wr G)/p$ is $G$-pscoh.

Proof.

This is identical to loc. cit., and we leave the details to the reader. Recall at least that the core part of the argument, namely [CMM18, Proposition 2.19], is a result about $TC(-)/p$ considered on the category of cyclotomic spectra, which applies equally well to the case at hand. ∎

Recall that we write $K^{\mathrm{inv}}$ for the fiber of the cyclotomic trace $K\longrightarrow TC$. We introduce a relative term $K^{\mathrm{inv}}((\mathbb{Z}\ltimes I)\wr G,I\wr G)$ to sit in a fiber sequence

[TABLE]

Combining Propositions 11 and 12 yields the following, which is the finiteness result to be used in the proof of Theorem 3.

Proposition 13**.**

For every prime $p$, the functor

[TABLE]

is $G$-pscoh.

3 A geometric special case

The purpose of this section is to establish a geometric special case of our main result Theorem 3. This equivariant rigidity result will follow from its non-equivariant special case [CMM18, Theorem A] together with decomposition results of Vistoli and Tabuada-Van den Bergh [TVdB18]. To formulate it, fix for the rest of this section a finite group $G$, a field $k$ of characteristic not dividing $|G|$ 333By convention, this condition is satisfied if $k$ is of characteristic zero. and a prime $p$ not dividing $|G|$ (but possibly equal to the characteristic of $k$). Let $X$ be an affine, smooth $k$-algebra with a $G$-action and assume given a rational point $x\in X(k)$ fixed by $G$. Then $G$ acts canonically on the henselization $\mathcal{O}^{h}_{X,x}$ of the local ring $\mathcal{O}_{X,x}$, and the canonical map $\pi\colon\mathcal{O}^{h}_{X,x}\longrightarrow k$ to the residue field is $G$-equivariant (for $k$ endowed with the trivial $G$-action). Hence it induces a map on twisted group rings $\mathcal{O}^{h}_{X,x}\wr G\longrightarrow k\wr G=k[G]$. The result then is the following.

Proposition 14**.**

In the above situation, the map induced by $\pi$

[TABLE]

is an equivalence.

Proof.

We start by setting the stage to apply [TVdB18]. We denote by $E:=\pi_{*}\left(K^{\mathrm{inv}}(-)/p\right)$, and observe that this is an additive invariant taking values in $\mathbb{Z}\left[\frac{1}{|G|}\right]$-modules and commuting with filtered colimits: For algebraic $K$-theory, this is classical and for $TC(-)/p$ it follows from [CMM18, Theorem 2.7]. Now, [TVdB18, Remark 1.3, ii) and iii)] implies that

[TABLE]

where $X^{\sigma}\subseteq X$ is the subscheme fixed by $\sigma$, and $\widetilde{E}$ refers to a certain functorially defined direct summand of $E$ (depending on $\sigma$). Since we will not require knowledge of the exact shape of that summand, we will not review its definition here.

We observe that the $G$-fixed point $x\in X(k)$ determines a map

[TABLE]

such that $E(\bar{x})$ participates in a commutative diagram

[TABLE]

where $x_{\sigma}$ denotes the unique factorization of $x$ through $X^{\sigma}\subseteq X$.

Next we want to pass to henselizations. To do this, we observe that everywhere in the above argument, one can replace $([X/G],x)$ with a pointed étale neighborhood $(Y,y)$ such that $\kappa(x)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\kappa(y)$ is an isomorphism on residue-fields. We obtain a commutative diagram generalizing eq. 3

[TABLE]

Passing to the filtered colimit of all such $(Y,y)$ and recalling that henselization commutes with the closed immersions $X^{\sigma}\subseteq X$ (and more generally with integral extensions, see [Sta18, TAG 0DYE]), we obtain

[TABLE]

Here, $\iota_{\sigma}:\mathrm{Spec}(k)\hookrightarrow\mathrm{Spec}(\mathcal{O}_{X^{\sigma},x}^{h})$ and $\iota:[\mathrm{Spec}(k)/G]\hookrightarrow[\mathrm{Spec}(\mathcal{O}_{X,x}^{h})/G]$ are (induced by) the canonical projection to the residue fields. Since each $\iota_{\sigma}$ is a closed immersion with henselian defining ideal, so is each $\iota_{\sigma}\times\mathrm{id}_{\mathrm{Spec}(k[\sigma])}$, and by [CMM18, Theorem A], every map $\widetilde{E}(\iota_{\sigma}\times\mathrm{id})$ is an isomorphism, and hence so is $E(\iota)$.

To equate $E(\iota)$ with $\pi_{*}(\lx@cref{creftype~refnum}{eq:claim})$, and thus to conclude the proof, it remains to recall that $E(-)=\pi_{*}(K^{\mathrm{inv}}(-)/p)$ and that since the order $|G|$ is invertible, a finitely generated projective module with a semi-linear $G$-action is the same thing as a finitely generated projective left module over the twisted group ring, so that we have an equivalence of $\infty$-categories of perfect modules

[TABLE]

and similarly with $\mathcal{O}^{h}_{X,x}$ replaced by $k$. ∎

4 Nil-invariance, excision and exactness

4.1 Nil-invariance

Proposition 15**.**

Let $G$ be a finite group and $\pi:R\longrightarrow R^{\prime}$ a surjective homomorphism of commutative rings with a $G$-action such that $\mathrm{ker}(\pi)$ is nilpotent. Then $K^{\mathrm{inv}}(R\wr G)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}K^{\mathrm{inv}}(R^{\prime}\wr G)$ is an equivalence.

Proof.

This will follow from [DGM13, Chapter VII, Theorem 0.0.2] if we can show that the kernel of (the obviously surjective) ring homomorphism $\pi\wr G:R\wr G\longrightarrow R^{\prime}\wr G$ is nilpotent. However, an immediate computation shows that for every $n\geq 0$ we have

[TABLE]

∎

4.2 Excision

Assume that

[TABLE]

is a Milnor square of commutative rings, i.e. a pull-back diagram of rings with $g$ surjective, see [Bas68, Chapter IX, §5] for an early account and [LT18] for a current development. If, in addition, a finite group $G$ acts on eq. 6, then the induced square of twisted group rings

[TABLE]

is clearly still a Milnor square. Denoting by $\mathbb{K}$ non-connective algebraic $K$-theory and by $\mathbb{K}^{\mathrm{inv}}$ the fiber of the cyclotomic trace $\mathbb{K}\longrightarrow TC$, we then deduce the following from [LT18, Theorem 3.3].

Proposition 16**.**

In the above situation,

[TABLE]

is a pull-back square.

To pass to connective $K$-theory here, we need the equivariant generalization of [CMM18, Corollary 4.34], namely Proposition 17 below.

For an associative, unital ring $A$, we denote by

[TABLE]

the set of isomorphism classes of finitely generated projective left $A$-modules.

We start by establishing the equivariant generalization of [CMM18, Lemma 4.20]:

Proposition 17**.**

Let $(R,I)$ be a henselian pair and $G$ a finite group acting on $(R,I)$. Then the obvious homomorphism

[TABLE]

is an isomorphism.

Proof.

To see that the map is injective, according to [Bas68, Chapter IX, Proposition 1.3] it suffices to check that the kernel of the projection $R\wr G\to(R/I)\wr G$, namely

[TABLE]

is contained in the radical of $R\wr G$. Otherwise, $I\wr G$ was not contained in some maximal left ideal $\mathfrak{n}\subseteq R\wr G$. Then the subset

[TABLE]

was a left-ideal properly containing $\mathfrak{n}$, and hence

[TABLE]

We consider $R=Re_{e}\subseteq R\wr G$ as a (non-central !) subring. Then eq. 9 holds as an equality of $R$-modules, and since $I\wr G=I(R\wr G)$ and $R\wr G$ is a finite (and free) $R$-module, Nakayama’s lemma444recall that $I$ is contained in the radical of $R$. implies that $\mathfrak{n}=R\wr G$, a contradiction which completes the proof of injectivity.

To see the surjectivity, we will establish the stronger claim that the reduction map

[TABLE]

is surjective. Write $\bar{R}:=R/I$ and fix some $\bar{M}\in\mathbb{P}\mathrm{roj}(\bar{R}\wr G)$. We first descend everything to a situation of finite type over the integers. The ring with $G$-action $R=\cup_{\alpha}R_{\alpha}$ is the union of its finitely generated, $G$-stable subrings $R_{\alpha}\subseteq R$. Accordingly, we also have $\bar{R}=\cup_{\alpha}(R_{\alpha}/I_{\alpha})=:\cup_{\alpha}\bar{R}_{\alpha}$ for $I_{\alpha}:=R_{\alpha}\cap I$. Since then also $\bar{R}\wr G=\cup_{\alpha}\left(\bar{R}_{\alpha}\wr G\right)$, the given $\bar{M}$ descends to some $\bar{M}_{\alpha}\in\mathbb{P}\mathrm{roj}(\bar{R}_{\alpha}\wr G)$ for suitably large indices $\alpha$.

Write $(R_{\alpha},I_{\alpha})^{h}$ for the henselization of $R_{\alpha}$ along $I_{\alpha}\subseteq R_{\alpha}$, and note that $G$ naturally acts on $(R_{\alpha},I_{\alpha})^{h}$, because henselization is functorial. It will suffice to lift the given $\bar{M}_{\alpha}\in\mathbb{P}\mathrm{roj}(\bar{R}_{\alpha}\wr G)$ to some element of $\mathbb{P}\mathrm{roj}((R_{\alpha},I_{\alpha})^{h}\wr G)$, because the inclusion $R_{\alpha}\subseteq R$ factors through $(R_{\alpha},I_{\alpha})^{h}$ $G$-equivariantly.

We next claim an inclusion of ($G$-invariant) ideals for all sufficiently large $M>>0$, namely

[TABLE]

Indeed, the second inclusion is obvious, and the first one follows from the fact that $I_{\alpha}$ is finitely generated together with the relation, valid for every $x\in I_{\alpha}$:

[TABLE]

which implies that $x^{|G|}\in I_{\alpha}^{G}\cdot R_{\alpha}$.

By eq. 11, the kernel of the projection

[TABLE]

is nilpotent, and an easy calculation then shows that so is the kernel of the projection

[TABLE]

(see the proof of Proposition 15).

By [Bas68, ch. III, Corollary 2.4 and Proposition 2.12] then, we can lift the given $\bar{M}_{\alpha}\in\mathbb{P}\mathrm{roj}(\bar{R}_{\alpha}\wr G)$ to some $\bar{M}^{\prime}_{\alpha}\in\mathbb{P}\mathrm{roj}((R_{\alpha}/I_{\alpha}^{G}\cdot R_{\alpha})\wr G)$.

As a final piece of preparation, we need to see what happens to the $G$-invariants under henselization. Since $R_{\alpha}^{G}\subseteq R_{\alpha}$ is integral, and eq. 11 shows that $\sqrt{I_{\alpha}^{G}\cdot R_{\alpha}}=\sqrt{I_{\alpha}}$, [Sta18, TAG 0DYE] implies that the canonical map

[TABLE]

is an isomorphism.

We are now in a position to lift the given $\bar{M}^{\prime}_{\alpha}\in\mathbb{P}\mathrm{roj}((R_{\alpha}/I_{\alpha}^{G}\cdot R_{\alpha})\wr G)$ using [Gre69, Theorem 4.1], as follows555The application of this Theorem here is a bit involved because in general neither is $R\wr G$ an $R$-algebra in any obvious way (but only an $R^{G}$-algebra), nor is $R^{G}\subseteq R$ finite.:

As our henselian pair, we take $(R_{\alpha}^{G},I_{\alpha}^{G})^{h}$, and as our algebra we take $A:=(R_{\alpha},I_{\alpha})^{h}\wr G$: The algebra $A$ is a finite $(R^{G}_{\alpha})^{h}$-module, because it is clearly finite over $R_{\alpha}^{h}$, and eq. 12 shows that $R_{\alpha}^{h}$ is finite over $(R_{\alpha}^{G})^{h}$, because $R_{\alpha}^{G}\subseteq R_{\alpha}$ is finite, being both integral and of finite type.

We then compute the reduction of $A=(R_{\alpha},I_{\alpha})^{h}\wr G$ to be

[TABLE]

Now, [Gre69, Theorem 4.1] shows that the given $\bar{M}_{\alpha}^{\prime}\in\mathbb{P}\mathrm{roj}((R_{\alpha}/I_{\alpha}^{G}\cdot R_{\alpha})\wr G)=\mathbb{P}\mathrm{roj}(\bar{A})$ lifts to some element of $\mathbb{P}\mathrm{roj}(A)=\mathbb{P}\mathrm{roj}((R_{\alpha},I_{\alpha})^{h}\wr G)$, as desired. ∎

We can now start to work on the version of Proposition 16 for connective algebraic $K$-theory, at least for those diagrams eq. 6 coming from suitable maps of henselian pairs: In the following, fix henselian pairs $(R,I)$ and $(S,J)$ with an action of the finite group $G$, and assume that $(R,I)\longrightarrow(S,J)$ is a map of pairs which respects the $G$-action and maps $I$ isomorphically to $J$. Then

[TABLE]

is a diagram as in eq. 6, i.e. a Milnor-square with a $G$-action.

We then define $K(R\wr G,I\wr G)$ by the fiber sequence

[TABLE]

and analogously for $K(S\wr G,J\wr G)$ and with $K$ replaced by $\mathbb{K}$. The map of pairs $(R,I)\to(S,J)$ induces a map

[TABLE]

and similarly for $\mathbb{K}$. Recall that there is a canonical transformation $K\longrightarrow\mathbb{K}$.

Proposition 18**.**

In the above situation, the diagrams

• i)

[TABLE]

and

• ii)

[TABLE]

are pull-back squares.

Proof.

To prove part $i)$, let $F\longrightarrow{\mathbb{F}}$ denote the map induced by eq. 14 on horizontal fibres. The claim is that this map is an equivalence. Since $K\longrightarrow\mathbb{K}$ induces an isomorphism on $\pi_{k}$ for $k\geq 0$, we have $\pi_{k}({\mathbb{F}})\simeq\pi_{k}(F)$ for $k\geq 0$. The excision theorem of Milnor-Bass-Murthy [Bas68, Chapter XII, Theorem 8.3] applied to the diagram obtained from eq. 13 by passing to twisted group rings shows that $\pi_{k}({\mathbb{F}})=0$ for all $k\leq-1$. It remains to see that $\pi_{k}(F)=0$ in this range, too. Since $K_{0}(R\wr G)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}K_{0}((R/I)\wr G)$ is an isomorphism (Proposition 17) and the maps

[TABLE]

[TABLE]

are surjections (this is true more generally for any surjective ring homomorphism with kernel contained in the radical, see [Bas68, ch. IX, Proposition 1.3, (1)]), we conclude that the fibers $K(R\wr G,I\wr G)$ and $K(S\wr G,J\wr G)$ are concentrated in degrees $\geq 1$, and thus $F$ is concentrated in degrees $\geq 0$, as claimed.

Part $ii)$ follows from part $i)$ by passage to fibers over $TC$, because the canonical transformation $K\to TC$ factors as $K\to\mathbb{K}\to TC$. ∎

Corollary 19**.**

If $(R,I)$ is a henselian pair with a $G$-action and $R\to S$ is a map of commutative rings with $G$-action mapping $I$ isomorphically to an ideal $J\subseteq S$, then

[TABLE]

is an equivalence.

Proof.

The diagram

[TABLE]

is a Milnor-square with $G$-action. Note that the pair $(S,J)$ is also henselian by [CMM18, Lemma 3.18]. Therefore an application of Proposition 18, ii) reduces our claim to the analogous statement with $K^{\mathrm{inv}}$ replaced with $\mathbb{K}^{\mathrm{inv}}$. This is then a special case of Proposition 16. ∎

These results will be used to reduce rigidity of arbitrary pairs to rigidity of those pairs of the form $(\mathbb{Z}\ltimes I,I)$ already encountered in Section 2.

Corollary 20**.**

For a fixed finite group $G$, there is an equivalence of spectra, functorial in the henselian pair $(R,I)$ with $G$-action

[TABLE]

4.3 Exactness

We call a sequence $I^{\prime}\to I\to\bar{I}$ in $\mathrm{Ring}^{\mathrm{nu,h,G}}$ short exact if it is so when considered non-equivariantly, i.e. in $\mathrm{Ring}^{\mathrm{nu,h}}$, i.e. if the underlying sequence of abelian groups is short exact (see [CMM18, Definition 3.4]). We consider the functor

[TABLE]

and claim that it is exact:

Proposition 21**.**

Given a short exact sequence $I^{\prime}\to I\to\bar{I}$ in $\mathrm{Ring}^{\mathrm{nu,h,G}}$, then $F(I^{\prime})\to F(I)\to F(\bar{I})$ is a fiber sequence.

Proof.

We consider the following commutative diagram

[TABLE]

The top row is the one we want to recognize as a fiber sequence. The two right colomns are the fiber sequences defining $F(I)$ and $F(\bar{I})$. The indicated equality implies that the upper right square is a pullback. Hence the top row is a fiber sequence if and only if so is the second row. We verify this by observing that using Corollary 19 for the obvious map of pairs with $G$-action $(\mathbb{Z}\ltimes I^{\prime},I^{\prime})\to(\mathbb{Z}\ltimes I,I^{\prime})$ gives

[TABLE]

[TABLE]

Since $(\mathbb{Z}\ltimes I)/I^{\prime}\simeq\mathbb{Z}\ltimes\bar{I}$, this concludes the proof. ∎

5 The proof of the main result

In this section, we give the proof of our main result, Theorem 3, which we restate for convenience.

Theorem 22**.**

If the finite group $G$ acts on the henselian pair $(R,I)$, $|G|\in R^{*}$, and $n\geq 1$ is an integer coprime to $|G|$, then the reduction map

[TABLE]

is an equivalence.

Proof.

We can assume that $n=p$ is a prime (not dividing $|G|$). Since $K^{\mathrm{inv}}(R\wr G,I\wr G)/p\simeq K^{\mathrm{inv}}((\mathbb{Z}\ltimes I)\wr G,I\wr G)/p$ (see Corollary 20), our claim is that the functor

[TABLE]

is trivial. We start by collecting properties of $F$ that were established previously: By Proposition 13,

[TABLE]

By Proposition 21,

[TABLE]

and by Proposition 15,

[TABLE]

For every prime field $\Omega$ of characteristic not dividing $|G|$666by convention, this is fulfilled for characteristic zero. recall the compact projective generators $F^{\prime\prime}_{\Omega}(n)\in\mathrm{Ring}^{\mathrm{nu,h,G}}_{\Omega}$ $(n\geq 0)$ from Proposition 7. We deduce from Proposition 14 777This is the step which forces us to assume that $p$ does not divide $|G|$, and that the characteristic of $\Omega$ does not divide $|G|$. that $F(F^{\prime\prime}_{\Omega}(n))=0$ for all $n\geq 0$. Since by eq. 16, the restriction of $F$ to $\mathrm{Ring}^{\mathrm{nu,h,G}}_{\Omega}$ is $G$-pscoh, and in particular left Kan extended from its subcategory of compact projective objects (see [CMM18, Lemma 4.6]), which in turn is the idempotent completion of all the $F^{\prime\prime}_{\Omega}(n)$, we see that for every prime field $\Omega$ of characteristic not dividing $|G|$, we have

[TABLE]

We now boot-strap to see that for every $N\geq 1$, $I\in\mathrm{Ring}^{\mathrm{nu,h,G}}_{\mathbb{Z}\left[\frac{1}{|G|}\right]}$:

[TABLE]

Since $F$ preserves finite products, we can assume that $N=q^{r}$ is a prime-power (with the prime $q$ not dividing $|G|$) and then consideration of the short exact sequence $qI\to I\to I/qI$ together with eq. 17, eq. 18 and eq. 19 (for $\Omega=\mathbb{F}_{q}$) proves eq. 20.

Since $F$ is bounded below, there is an integer $d\in\mathbb{Z}$ such that

[TABLE]

We will be done if we can show that the functor to abelian groups $F_{0}:=\pi_{d}F:\mathrm{Ring}^{\mathrm{nu,h,G}}_{\mathbb{Z}\left[\frac{1}{|G|}\right]}\longrightarrow\mathrm{Ab}$ vanishes, because $d$ being arbitrary, this will imply that $F=0$. To see this, we will establish the following:

[TABLE]

[TABLE]

Given this, using eq. 17 and eq. 21, we obtain an exact sequence

[TABLE]

where $F_{0}(I/NI)=0$ by eq. 20, hence $F_{0}(I)=0$.

To prove eq. 22, we recall (Proposition 8) that we have a relation between free objects

[TABLE]

Since $F_{0}(F^{\prime\prime}_{\mathbb{Q}}(n))=0$ by eq. 19 for $\Omega=\mathbb{Q}$ and $F_{0}$ commutes with filtered colimits, we deduce that for every $x\in F_{0}({F^{\prime\prime}_{\mathbb{Z}\left[\frac{1}{|G|}\right]}(n)})$ there is some $N$ coprime to $|G|$ (depending on $x$ and $n$) such that $[N](x)=0$. To deduce from this the more uniform statement eq. 22 one uses that $F_{0}$ is finitely generated and takes the product of all $N$’s for the generators. Since the details of this step are literally the same as in the proof of [CMM18, Lemma 4.16], we omit them here. ∎