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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle2] 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, example 3.2.12].

Remark 3 Rigidity in hermitian K-theory has of course been studied in several works before, see for instance [Reference Hornbostel and Yagunov7–Reference Karoubi9, Reference Yagunov15] 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle2] 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, §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 normFootnote ² $\mathrm {N}(R)$ 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1]; the Poincaré structures then consist of the above triples where $M$ is invertible in the sense of [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, 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. [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle3, 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle3, 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 [Reference Harpaz, Nikolaus and Shah6]. Namely, in [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle3, 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle3, 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 [Reference Calmés, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle3, 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.