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An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

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Abstract

Given an Azumaya algebra with involution \((A,\sigma )\) over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of \((A,\sigma )\) and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of \({\mathbf {GL}}_n\) and \({\mathbf {Sp}}_{2n}\), provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map \(W(R)\rightarrow W(S)\) when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

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Notes

  1. The term \(W_{-\varepsilon }(B,\tau _1)\) on the bottom row of (0.1) and the maps adjacent to it differ from their counterparts in [30, p. 980]. However, the octagons become the same once identifying the term \(W_{-\varepsilon }(B,\tau _1)\) on the bottom row of (0.1) with the corresponding term \(W_{\varepsilon }(B,\tau _1)\) in op. cit. via \(\lambda \)-conjugation (“scaling by \(\lambda \)”) in the sense of 2.7 below.

  2. If A is a finite projective R-algebra of rank \(n\in \mathbb {N}\), then the trace and norm maps \({{\,\mathrm{Tr}\,}}_{A/R},{{\,\mathrm{Nr}\,}}_{A/R}:A\rightarrow R\) take \(a\in A\) to \(-c_1(a)\) and \((-1)^n c_n(a)\), respectively, where \(X^n+c_1(a)X^{n-1}+\dots +c_n(a)X^0\) is the characteristic polynomial of \([x\mapsto ax]\in {{\,\mathrm{End}\,}}_R(A)\) in the sense of [27, Example 5.3.3].

  3. This should be understood as “Azumaya algebra-with-involution” rather than “Azumaya-algebra with involution”.

  4. Some texts use “regular” or “nondegenerate”.

  5. We do not write \(\rho (Q,g)\) as \((P_A,f_A)\) because we reserve the subscript notation for base change relative to the base ring R.

  6. One can show that \({{\,\mathrm{Nrd}\,}}_{E/R}\) maps \(U(E,\theta )\) to \(\mu _2(R)\), and similarly after base-changing to S, as follows: By [39, III.§8.5] or [26, Theorems 5.17 & 5.37, Examples 7.3 & 7.4], there exists a faithfully flat R-ring \(R'\) such that \((E_{R'},\theta _{R'})\cong (\mathrm {M}_{n}(R'),\mathrm {t})\), where \(\mathrm {t}\) is the transpose involution. Now, for all \(x\in U(\mathrm {M}_{n}(R'),\mathrm {t})\), we have \({{\,\mathrm{Nrd}\,}}(x)^2=\det (x)^2=\det (x^{\mathrm {t}}x)=1\), so \({{\,\mathrm{Nrd}\,}}(x)\in \mu _2(R')\). As a result, \({{\,\mathrm{Nrd}\,}}_{E/R}\) maps \(U(E,\theta )\) to \(R\cap \mu _2(R')=\mu _2(R)\).

  7. With the appropriate definitions, this functor also extends to a sheaf on the site of all R-schemes with the fpqc topology.

  8. When R is a field, our definition of D(f) does not agree with the definition given in [40, §10]. However, both definitions give the same Brauer class by [40, Corollary 10.35].

  9. Note that in contrast with Sect. 3.1, we do no require \(\lambda ^\sigma =-\lambda \) and \(\mu ^\sigma =-\mu \). Indeed, this cannot be guaranteed in general. The situations considered in cases (i) and (ii) of Lemma 4.3 below are such examples, the reason being that \(\sigma |_B={{\,\mathrm{id}\,}}_B\) or \(\sigma |_E={{\,\mathrm{id}\,}}_E\).

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Acknowledgements

We are grateful to Eva Bayer-Fluckiger for suggesting us the project at hand. We further thank Eva Bayer-Fluckiger and Raman Parimala for many useful conversations and suggestions. The research was partially conducted at the department of mathematics at University of British Columbia, where the author was supported by a post-doctoral fellowship. We thank Ori Parzanchevski for encouragement and motivation.

We are also grateful to the anonymous referees for many useful suggestions which have improved the exposition.

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    Uriya A. First

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First, U.A. An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution. manuscripta math. 170, 313–407 (2023). https://doi.org/10.1007/s00229-021-01352-0

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