Gabber rigidity in hermitian K-theory

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 m ⊆ R an ideal such that (R, 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 m-adically complete or where m is locally nilpotent.We write F = R/m and let n be a natural number which is invertible in R. Then Gabber's rigidity theorem [Gab92] says that the canonical map K(R)/n −→ K(F )/n is an equivalence; this result was preceded by work of Suslin [Sus84] who showed this conclusion for henselian valuation rings.See also [CMM21] 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 [CDH + 20b, CDH + 20c] as well as [HNS22] 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 λ be a form parameter over R in the sense of [Sch21,§3], see also [CDH + 20a, Definition 4.2.26].In loc.cit. it is explained that such a form parameter λ is equivalently described by a Poincaré structure Ϙ gλ R in the sense of [CDH + 20a] on D p (R) which sends projective R-modules to discrete spectra.Here, D p (R) denotes the stable ∞category of perfect complexes over R. We will assume that the Z-module with involution over R underlying the form parameter λ by is given by ±R, that is, given by the R-module R with C 2 -action either the identity or multiplication by −1, viewed as an R ⊗ R-module via the multiplication map.There is then an induced form parameter on F whose associated Poincaré structure on D p (F ) we will denote by Ϙ gλ F , see Remark 3 below for details.The construction is made so that the extension of scalars functor canonically refines to a Poincaré functor (D p (R), Ϙ gλ R ) → (D p (F ), Ϙ gλ 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 Ϙ ±gq , Ϙ ±ge , and Ϙ ±gs .A further example is provided by the Burnside Poincaré structure Ϙ b whose L-theory was calculated explicitly for Z in [CDH + 20c, 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 [Mad21].With this notation fixed, we have the following result.Theorem A. Let (R, m) be a henselian pair, F = R/m and let n be a natural number invertible in R. Then the canonical map is an equivalence.
Proof.The main result of [CDH + 20b] gives a diagram of horizontal fibre sequences and by Gabber rigidity, the left vertical map becomes an equivalence after tensoring with S/n.Therefore, the statement of the theorem is equivalent to the statement that the map where Ϙ q ±R denotes the homotopy quadratic Poincaré structure associated to the invertible module with involution M which is part of the form parameter λ, and likewise for Ϙ q ±F .We now observe that the formula for relative L-theory obtained in [HNS22] shows that the top and bottom horizontal cofibre are S[ 1 n ]-modules.Indeed, [HNS22] shows that the cofibre of the top horizontal arrow is a filtered colimit of objects of the form ) ⊆ Sp is a full subcategory closed under colimits and limits, both terms in the equaliser, and therefore also the equaliser itself belong to Mod(S[ 1 n ]).Consequently, the horizontal maps in the above diagram become equivalences upon tensoring with 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 work of Wall's [Wal73] Remark 1. Restricting the situation above to form parameters rather than general Poincaré structures on D 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 is a pullback diagram for any Poincaré structure Ϙ on D p (R) whose Z-module with involution over R is given by ±R.The proof presented above therefore shows that for any ring R in which n is invertible, the canonical map GW(R; Ϙ q ±R )/n −→ GW(R; Ϙ)/n is an equivalence and that Gabber rigidity holds for the Poincaré structure Ϙ q ±R .In particular, Gabber rigidity also applies to the homotopy symmetric Poincaré structure Ϙ ±s as well as the Tate Poincaré structure Ϙ t R , see [CDH + 20a, Example 3.2.12].Remark 2. Rigidity in hermitian K-theory has of course been studied in several works before, see for instance [Kar84, Jar83, HY07, Yag22] 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 [CDH + 20b] 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 3. In this remark, we describe how extension of scalars can be used to prolong a form parameter over R along a map R → 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 [CDH + 20a, §3.3], we have shown that the category of Hermitian structures on D p (R) is equivalent to the category Mod N(R) (Sp C 2 ) = Mod(N(R)), that is, the category of modules over the multiplicative norm N(R) in the category Sp C 2 of genuine C 2 -spectra.Moreover, the category Mod(N(R)) is equipped with a canonical t-structure whose heart is equivalent to the category of (possibly degenerate) form parameters over R, see [CDH + 20a, Remark 4.2.27].Objects in Mod(N(R)) are described by triples (M, N, α) where -M is an object of Mod R⊗R (Sp BC 2 ), where R⊗R is an algebra in spectra with C 2 -action where the action flips the two tensor factors, -N is an object of Mod(R), and -α is a map N → M tC 2 of R-modules, see [CDH + 20a]; the Poincaré structures then consist of the above triples where M is invertible in the sense of [CDH + 20a, Def.3.1.3].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.[CDH + 20c, pg.7] for the details.An object (M, N, α) is connective in the canonical t-structure on Mod(N(R)) if and only if M and N are connective.
The Poincaré structure associated to the triple (M, N, α) is denoted by gives rise to a Poincaré structure on D p (R ′ ) for which the extension of scalar functor canonically refines to a Poincaré functor (D p (R), ) was associated to a form parameter, then the same need not be true for the triple (M ′ , N ′ , α ′ ): Indeed, this is the case if and only if Ϙ α ′ M ′ (R ′ ) is a discrete spectrum which in general need not be the case.However, we may consider the composite ) 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 α ′′ : N ′′ → (M ′ ) tC 2 .By construction, the triple (M ′ , N ′′ , α ′′ ) is an object of Mod(N(R ′ )) ♥ and in fact identifies with τ ≤0 (M ′ , N ′ , α ′ ).This object determines a Poincaré structure Ϙ gλ ′ associated to a form parameter λ ′ over R ′ for which the extension of scalars functor refines to a Poincaré functor (D p (R), Ϙ gλ ) −→ (D p (R ′ ), Ϙ gλ ′ ). To ] is an equivalence for all m ∈ 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 Ϙ q ±R → Ϙ ≥m ±R is an equivalence of Poincaré structures, see [CDH + 20c, Remark R.4].
Remark 5. Suppose that R is an associative ring which is m-adically complete for an ideal m ⊂ R. Then the result of Wall, see again [CDH + 20c, Prop.2.3.7], says that the map L ±q (R) → L ±q (R/m) is an equivalence.To the best of our knowledge, it is not known whether also the map K(R)/n → K(R/m)/n is an equivalence.However, if it is, this argument shows that the same is true for Grothendieck-Witt theory and vice versa.

Date:
February 22, 2022.The author was supported by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151).
give an example of this construction, we recall the genuine Poincaré structures Ϙ ≥m ±R which, for m = 0, 1, 2 are the Poincaré structures Ϙ gq ±R , Ϙ ge ±R and Ϙ gs ±R associated to the classical (skew-) quadratic, even, and symmetric form parameter over R, respectively, see [CDH + 20c, Remark R.3 & R.5].In this case, the extension of scalars functor associated to a ring map R → R ′ indeed sends Ϙ ≥m ±R to Ϙ ≥m ±R ′ .