Gabber rigidity in hermitian K-theory

Abstract

We note that Gabber's rigidity theorem for the algebraic K-theory of henselian pairs also holds true for hermitian K-theory with respect to arbitrary form parameters.

Let $R$ be a commutative ring and $\mathfrak {m} \subseteq R$ an ideal such that $(R,\mathfrak {m})$ is a henselian pair. Standard examples include henselian local rings like valuation rings of complete nonarchimedean fields as well as pairs where $R$ is $\mathfrak {m}$-adically complete or where $\mathfrak {m}$ is locally nilpotent. We write $F = R/\mathfrak {m}$ and let $n$ be a natural number which is invertible in $R$. Then Gabber's rigidity theorem [5] says that the canonical map

\[ K(R)/n \longrightarrow K(F)/n \]

is an equivalence; this result was preceded by work of Suslin [13] who showed this conclusion for henselian valuation rings. See also [4] for an extension of this result, involving topological cyclic homology, to the case where $n$ need not be invertible in $R$ and a general discussion of henselian pairs. The purpose of this short note is to use the results of [2, 3] as well as [6] to show that Gabber's rigidity property also holds true for hermitian K-theory, a.k.a. Grothendieck–Witt theory.

To state the main result, let $\lambda$ be a form parameter over $R$ in the sense of [12, §3], see also [1, definition 4.2.26]. In loc. cit. it is explained that such a form parameter $\lambda$ is equivalently described by a Poincaré structure ${\unicode{x03D9}} ^{\mathrm {g}\lambda }_R$ in the sense of [1] on $\mathscr {D}^{\mathrm {p}}(R)$ which sends projective $R$-modules to discrete spectra. Here, $\mathscr {D}^{\mathrm {p}}(R)$ denotes the stable $\infty$-category of perfect complexes over $R$. We will assume that the $\mathbb {Z}$-module with involution over $R$ underlying the form parameter $\lambda$ is given by $\pm R$, that is, given by the $R$-module $R$ with $C_2$-action either the identity or multiplication by $-1$, viewed as an $R\otimes R$-module via the multiplication map. There is then an induced form parameter on $F$ whose associated Poincaré structure on $\mathscr {D}^{\mathrm {p}}(F)$ we will denote by ${\unicode{x03D9}} ^{\mathrm {g}\lambda }_F$, see remark 4 below for details. The construction is made so that the extension of scalars functor canonically refines to a Poincaré functor $(\mathscr {D}^{\mathrm {p}}(R),{\unicode{x03D9}} ^{\mathrm {g}\lambda }_R) \to (\mathscr {D}^{\mathrm {p}}(F),{\unicode{x03D9}} ^{\mathrm {g}\lambda }_F)$ and therefore a map on Grothendieck–Witt theory. Standard examples of form parameters capture the notion of quadratic, even and symmetric forms (as well as their skew-quadratic, skew-even and skew-symmetric cousins) with associated Poincaré structures ${\unicode{x03D9}} ^{\pm \mathrm {gq}}, {\unicode{x03D9}} ^{\pm \mathrm {ge}}$ and ${\unicode{x03D9}} ^{\pm \mathrm {gs}}$. A further example is provided by the Burnside Poincaré structure ${\unicode{x03D9}} ^\mathrm {b}$ whose L-theory was calculated explicitly for $\mathbb {Z}$ in [3, example 1.3.18] and whose $0$’th Grothendieck–Witt group was studied for commutative rings with 2 invertible in the PhD thesis of Dylan Madden [10]. With this notation fixed, we have the following result.

Theorem 1 Let $(R,\mathfrak {m})$ be a henselian pair, $F=R/\mathfrak {m}$ and let $n$ be a natural number invertible in $R$. Then the canonical map

\[ \mathrm{GW}(R;{\unicode{x03D9}}^{\mathrm{g}\lambda}_R)/n \longrightarrow \mathrm{GW}(F;{\unicode{x03D9}}^{\mathrm{g}\lambda}_F)/n \]

is an equivalence.

Proof. The main result of [2] gives a diagram of horizontal fibre sequences

and by Gabber rigidity, the left vertical map becomes an equivalence after tensoring with $\mathbb {S}/n$. Therefore, the statement of the theorem is equivalent to the statement that the map

\[ \mathrm{L}(R;{\unicode{x03D9}}^{\mathrm{g}\lambda}_R)/n \longrightarrow \mathrm{L}(F;{\unicode{x03D9}}^{\mathrm{g}\lambda}_R)/n \]

is an equivalence. We then consider the diagram

where ${\unicode{x03D9}} ^{\mathrm {q}}_{\pm R}$ denotes the homotopy quadratic Poincaré structure associated with the invertible module with involution $\pm R$ which is part of the form parameter $\lambda$, and likewise for ${\unicode{x03D9}} ^{\mathrm {q}}_{\pm F}$. We now observe that the formula for relative L-theory obtained in [6] shows that the top and bottom horizontal cofibres are $\mathbb {S}[\tfrac {1}{n}]$-modules.

Indeed, [6] shows that the cofibre of the top horizontal arrow is a filtered colimit of objects of the form

\[ \mathrm{Eq} \left( \mathrm{map}_{R}(T\otimes_R T,R) \rightrightarrows (\Sigma^{1-\sigma}\mathrm{map}_{R}(T\otimes_R T,R))_{hC_2} \right) \]

for $T \in \mathscr {D}^{\mathrm {p}}(R)$, the bottom horizontal cofibre is described similarly¹. Since $\mathrm {map}_{R}(T\otimes _R T,R)$ is canonically an $R$-module and $n$ is invertible in $R$, it is also an $\mathbb {S}\left [\tfrac {1}{n}\right ]$-module. Moreover, since $\mathrm {Mod}\left (\mathbb {S}\left [\tfrac {1}{n}\right ]\right ) \subseteq \mathrm {Sp}$ is a full subcategory closed under colimits and limits, both terms in the equalizer, and therefore also the equalizer itself belong to $\mathrm {Mod}\left (\mathbb {S}\left [\tfrac {1}{n}\right ]\right )$. Consequently, the horizontal maps in the above diagram become equivalences upon tensoring with $\mathbb {S}/n$. The statement of the main theorem is therefore equivalent to the statement that the left vertical map in the above commutative square is an equivalence. This is a consequence of the work of Wall's [14] as explained in [3, prop. 2.3.7 and remark 2.3.8].

Remark 2 Restricting the situation above to form parameters rather than general Poincaré structures on $\mathscr {D}^{\mathrm {p}}(R)$ was merely a cosmetic choice to obtain a result about classical Grothendieck–Witt theory: Indeed, it is again a consequence of the main theorem of [2] that the diagram

is a pullback diagram for any Poincaré structure ${\unicode{x03D9}}$ on $\mathscr {D}^{\mathrm {p}}(R)$ whose $\mathbb {Z}$-module with involution over $R$ is given by $\pm R$. The proof presented above therefore shows that for any ring $R$ in which $n$ is invertible, the canonical map

\[ \mathrm{GW}(R;{\unicode{x03D9}}_{{\pm} R}^\mathrm{q})/n \longrightarrow \mathrm{GW}(R;{\unicode{x03D9}})/n \]

is an equivalence so that Gabber rigidity also holds for the Poincaré structure ${\unicode{x03D9}}$.

In particular, Gabber rigidity also applies to the homotopy symmetric Poincaré structure ${\unicode{x03D9}} ^{\pm \mathrm {s}}$ as well as the Tate Poincaré structure ${\unicode{x03D9}} ^{\mathrm {t}}_R$, see [1, example 3.2.12].

Remark 3 Rigidity in hermitian K-theory has of course been studied in several works before, see for instance [7–9, 15] for the case of rings with involution. The main purpose here is to show how to use the formalism of Poincaré categories and the main result of [2] to reduce rigidity in hermitian K-theory to rigidity in algebraic K-theory and L-theory in a way that allows to treat general form parameters.

Remark 4 In this remark, we describe how extension of scalars can be used to prolong a form parameter over $R$ along a map $R \to R'$ of rings. It is here that the assumption on the underlying module with involution is used. Indeed, we will describe a general construction on Hermitian structures, and the assumption is used to ensure that the given Poincaré structure is sent to a Poincaré structure rather than merely a Hermitian structure.

Namely, in [1, §3.3], we have shown that the category of Hermitian structures on $\mathscr {D}^{\mathrm {p}}(R)$ is equivalent to the category $\mathrm {Mod}_{\mathrm {N}(R)}(\mathrm {Sp}^{C_2}) = \mathrm {Mod}(\mathrm {N}(R))$, that is, the category of modules over the multiplicative norm² $\mathrm {N}(R)$ in the category $\mathrm {Sp}^{C_2}$ of genuine $C_2$-spectra. Moreover, the category $\mathrm {Mod}(\mathrm {N}(R))$ is equipped with a canonical $t$-structure whose heart is equivalent to the category of (possibly degenerate) form parameters over $R$, see [1, remark 4.2.27]. Objects in $\mathrm {Mod}(\mathrm {N}(R))$ are described by triples $(M,N, \alpha )$ where

1. – $M$ is an object of $\mathrm {Mod}_{R\otimes R}(\mathrm {Sp}^{BC_2})$, where $R\otimes R$ is an algebra in spectra with $C_2$-action where the action flips the two tensor factors,

2. – $N$ is an object of $\mathrm {Mod}(R)$ and

3. – $\alpha$ is a map $N \to M^{tC_2}$ of $R$-modules,

see [1]; the Poincaré structures then consist of the above triples where $M$ is invertible in the sense of [1, def. 3.1.4]. We warn the reader that caution has to be taken in regards to how $M^{tC_2}$ is to be viewed as an $R$-module, see e.g. [3, p. 7] for the details. An object $(M,N, \alpha )$ is connective in the canonical $t$-structure on $\mathrm {Mod}(\mathrm {N}(R))$ if and only if $M$ and $N$ are connective.

The Poincaré structure associated with the triple $(M,N,\alpha )$ is denoted by ${\unicode{x03D9}} ^\alpha _M$. Assuming that $M$ is in the image of the canonical functor $\mathrm {Fun}(BC_2,\mathrm {Mod}(R)) \to \mathrm {Mod}_{R\otimes R}(\mathrm {Fun}(BC_2,\mathrm {Sp}))$, the triple

\[ (M',N', \alpha') = (R'\otimes_R M, R'\otimes_R N, R'\otimes_R N \to R'\otimes_R M^{tC_2} \to (R'\otimes_R M)^{tC_2}) \]

gives rise to a Poincaré structure on $\mathscr {D}^{\mathrm {p}}(R')$ for which the extension of scalar functor canonically refines to a Poincaré functor $(\mathscr {D}^{\mathrm {p}}(R),{\unicode{x03D9}} ^\alpha _M) \to (\mathscr {D}^{\mathrm {p}}(R'),{\unicode{x03D9}} ^{\alpha '}_{M'})$, see [1, lemma 3.4.3]. Now, if $(M,N,\alpha )$ was associated with a form parameter, then the same need not be true for the triple $(M',N',\alpha ')$: Indeed, this is the case if and only if ${\unicode{x03D9}} ^{\alpha '}_{M'}(R')$ is a discrete spectrum which in general need not be the case (but by construction it is always a connective spectrum). However, we may consider the composite

\[ M'_{hC_2} \longrightarrow {\unicode{x03D9}}^{\alpha'}_{M'}(R') \longrightarrow \tau_{{\leq} 0}{\unicode{x03D9}}^{\alpha'}_{M'}(R') \]

and denote its cofibre by $N''$. The pushout diagram of spectra

and the fact that $(M')^{hC_2}$ is coconnective shows that there is a canonical map $\alpha ''\colon N'' \to (M')^{tC_2}$. By construction, the triple $(M', N'', \alpha '')$ is an object of $\mathrm {Mod}(\mathrm {N}(R'))^{\heartsuit }$ and in fact identifies with $\tau _{\leq 0}(M',N',\alpha ')$. This object determines a Poincaré structure ${\unicode{x03D9}} ^{\mathrm {g}\lambda '}$ associated with a form parameter $\lambda '$ over $R'$ for which the extension of scalars functor refines to a Poincaré functor

\[ (\mathscr{D}^{\mathrm{p}}(R),{\unicode{x03D9}}^{\mathrm{g}\lambda}) \longrightarrow (\mathscr{D}^{\mathrm{p}}(R'),{\unicode{x03D9}}^{\mathrm{g}\lambda'}). \]

To give an example of this construction, we recall the genuine Poincaré structures ${\unicode{x03D9}} ^{\geq m}_{\pm R}$ which, for $m=0,1,2$ are the Poincaré structures ${\unicode{x03D9}} ^{\mathrm {gq}}_{\pm R}$, ${\unicode{x03D9}} ^{\mathrm {ge}}_{\pm R}$ and ${\unicode{x03D9}} ^{\mathrm {gs}}_{\pm R}$ associated with the classical (skew-) quadratic, even and symmetric form parameter over $R$, respectively, see [3, remark R.3 and R.5]. In this case, the extension of scalars functor associated with a ring map $R \to R'$ indeed sends ${\unicode{x03D9}} ^{\geq m}_{\pm R}$ to ${\unicode{x03D9}} ^{\geq m}_{\pm R'}$.

Remark 5 For the Poincaré structures ${\unicode{x03D9}} ^{\geq m}_{\pm R}$ one can give the following argument that the map

\[ \mathrm{GW}(R;{\unicode{x03D9}}^{{\geq} m}_{{\pm} R})/n \longrightarrow \mathrm{GW}(F;{\unicode{x03D9}}^{{\geq} m}_{{\pm} R})/n \]

is an equivalence without appealing to the general formula for relative L-theory of [6]. Namely, in [3, prop. 3.1.14] we have shown that the map

\[ \mathrm{L}(R;{\unicode{x03D9}}^{\mathrm{q}}_{{\pm} R})\left[\tfrac{1}{2}\right] \longrightarrow \mathrm{L}(R;{\unicode{x03D9}}^{{\geq} m}_{{\pm} R})\left[\tfrac{1}{2}\right] \]

is an equivalence for all $m \in \mathbb {Z}$. Therefore, the proof of the theorem applies in the case where 2 does not divide $n$. In the case where $2$ divides $n$, we deduce that $2$ is invertible in $R$ in which case already the map ${\unicode{x03D9}} ^{\mathrm {q}}_{\pm R} \to {\unicode{x03D9}} ^{\geq m}_{\pm R}$ is an equivalence of Poincaré structures, see [3, remark R.4].

Remark 6 Suppose that $R$ is an associative ring which is $\mathfrak {m}$-adically complete for an ideal $\mathfrak {m} \subset R$. Then the result of Wall, see again [3, prop. 2.3.7], says that the map $\mathrm {L}^{\pm \mathrm {q}}(R) \to \mathrm {L}^{\pm \mathrm {q}}(R/\mathfrak {m})$ is an equivalence. To the best of our knowledge, it is not known whether also the map $K(R)/n \to K(R/\mathfrak {m})/n$ is an equivalence. However, if it is, this argument shows that the same is true for Grothendieck–Witt theory and vice versa.