Karoubi's conjectures on Adams operations on Clifford modules

Karoubi's paper 'Clifford modules and invariants of quadratic forms' (published with an appendix by Serre in the Journal of K-Theory, 2012) constructed a hermitian analogue of the cannibalistic Bott class for quadratic modules over commutative rings, valued in algebraic K-theory, together with Adams operations on the K-theory of Clifford modules. The paper itself states two precise conjectures beyond the reach of its methods, which depend on invertibility of k! or 2k: (1) the multiplicativity of the Adams operation Psi^k on Clifford modules should hold without k! invertible in the base ring; (2) the cyclic-group (Gauss-sum) operation should coincide with Psi^k whenever both are defined. The problem of completing Karoubi's theory by removing these auxiliary hypotheses remains open.

Background

In topological complex K-theory, the Bott class (also called a cannibalistic class) of a complex vector bundle $V$ over a compact space $X$ is the characteristic class $\rho^k(V)$ obtained by applying an Adams operation to the Thom class, where the Thom isomorphism identifies the K-theory of $V$ with that of $X$. It is multiplicative for direct sums and, after inverting the integer $k$, gives a homomorphism from $K(X)$ to the multiplicative group $K(X)[1/k]^{\times}$. Adams operations $\psi^k$ are ring endomorphisms of $K(X)$ built from the lambda-ring structure given by exterior powers; Atiyah gave an alternative description through power maps valued in equivariant K-theory: the $k$-th tensor power of a bundle carries an action of the symmetric group $S_k$, $\psi^k$ is induced by the homomorphism from the integral representation ring $R(S_k)$ to $\mathbb{Z}$ sending a representation to the trace of the $k$-cycle, and the same argument can be run with the cyclic group $\mathbb{Z}/k$, expressing $\psi^k$ as a Gauss sum over eigenmodules weighted by $k$-th roots of unity.

Karoubi, in [1], transposed this circle of ideas to the algebraic K-theory of quadratic forms. Let $R$ be a commutative ring in which $2$ is invertible and let $(V,q)$ be a finitely generated projective $R$-module of finite rank with a nondegenerate quadratic form. Its Clifford algebra $C(V)$ is the quotient of the tensor algebra of $V$ by the relations $v^2 = q(v)$; it is a $\mathbb{Z}/2$-graded $R$-algebra, and all tensor products below are graded ones. One writes $K(C(V))$ for the Grothendieck group of finitely generated graded projective $C(V)$-modules. Graded Azumaya algebras, the graded analogues of algebras that are locally matrix algebras, are classified up to graded Morita equivalence by the Brauer-Wall group $BW(R)$; the class of $C(V)$ in $BW(R)$ encodes the classical primary invariants of $q$ (rank, discriminant, Hasse-Witt invariant). A spinorial module is a pair $(V,E)$ for which $C(V)$ is isomorphic to $\mathrm{End}_R(E)$ as a graded algebra, that is, the Clifford class is trivialized; the corresponding Grothendieck group $K\mathrm{Spin}(R)$ maps to hermitian K-theory $KQ(R)$ (the Grothendieck group of nondegenerate quadratic modules, Karoubi's notation for what is now often called the Grothendieck-Witt group), and the Witt group $W(R)$ is the Grothendieck group of nondegenerate quadratic forms modulo the hyperbolic forms $H(W) = W \oplus W^*$; $\mathrm{Pic}(R)$ denotes the Picard group of invertible $R$-modules.

The paper's central device is the rescaled module $V^{(k)} = (V, kq)$ (same underlying module, form multiplied by $k$), for which the diagonal $V \to V^k$ is an isometry. The graded $k$-th tensor power of a $C(V)$-module is a module over a crossed-product algebra involving $S_k$ (the algebra generated by the graded tensor power of the Clifford algebra together with the permutations of its factors), and restriction of scalars yields a power map $K(C(V)) \to K_{S_k}(C(V^{(k)}))$ into equivariant K-theory, the Grothendieck group of modules carrying an action of the group $S_k$. Applying the cycle character (the homomorphism given by the trace of the $k$-cycle) gives an Adams operation $\Psi^k : K(C(V)) \to K(C(V^{(k)}))$; Theorem 2.5 proves its multiplicativity $\Psi^k(x \cdot y) = \Psi^k(x) \cdot \Psi^k(y)$ under the hypothesis that $k!$ is invertible in $R$, a hypothesis used to decompose $R(S_k)$-modules via semisimplicity. Conjugating by the Morita equivalence $C(V) \cong \mathrm{End}(E)$ defines the hermitian Bott class $\rho_k(V) \in K(R)[1/k]^{\times}$ of a spinorial module; it is multiplicative, and using a square-root lemma due to Serre (communicated in a letter of 2 July 2007 and published as the appendix of the paper: for $k$ odd and $V_0$ of even rank and self-dual, the classical Bott class $\rho^k(V_0)$ is canonically a square in $K(R)$) one corrects it to a homomorphism from the spinorial Witt group $W^{\mathrm{Spin}}(R)$ (the Witt group of spinorial quadratic modules) to $(K(R)[1/k])^{\times}/\mathrm{Pic}(R)^{(k-1)/2}$ with image in the $2$-torsion — but only for $k$ odd; Remark 3.7 states that the case of even $k$ does not fit this strategy for the corrected untwisted class, although the topological examples of Section 4 show that the $k = 2$ analogue detects nontrivial Witt classes. Without a spinorial structure the class is defined by twisting: with $W = V(-1)$ (so $C(W)$ is the graded opposite algebra of $C(V)$), and $k$ odd with $2k$ invertible and the $k$-th roots of unity present in $R$, the twisted hermitian Bott class takes values in $K(C(V^{(k)}) \otimes C(W^k)) \otimes_{\mathbb{Z}} \Omega_k$, where $\Omega_k = \mathbb{Z}[\omega]$ is the ring of integers in the $k$-th cyclotomic extension of $\mathbb{Q}$ (Theorem 7.1); a corrected version descends to the Witt group $W(R)$ with values in the corresponding quotient by $\pm\mathrm{Pic}(R)^{(k-1)/2}$. In parallel, Section 6 defines an alternative cyclic Adams operation $\overline{\Psi}^k(E) = \sum_j F_{\omega^j} \cdot \omega^j$, a Gauss sum over the eigenmodules $F_{\omega^j}$ of a generator of the cyclic group, valid when $2k$ is invertible and $R$ contains the $k$-th roots of unity, and extends such operations to oriented $\mathbb{Z}/2$-graded Azumaya algebras, where an orientation is the choice of an order-two unit trivializing the permutation of the two factors of $A^{\otimes 2}$ and the $S_k$-action is lifted to units through the Goldman element (a canonical unit of $A^{\otimes 2}$, constructed in the theory of Azumaya algebras, whose inner automorphism implements that permutation); the resulting operations depend on that lifting, which over an integral domain is unique only up to the sign representation — changing the sign of the lift replaces exterior powers by symmetric powers — although in that case the paper obtains a canonical choice of the lifting by extending scalars to the algebraic closure of the quotient field and matching the canonical lifting of $S_k$ on the resulting matrix algebra.

Problem Statement

Complete Karoubi's theory of Adams operations on Clifford modules and hermitian Bott classes of quadratic forms by removing the auxiliary hypotheses under which it is currently proved. Throughout, $R$ is a commutative ring with $2$ invertible where stated, $(V,q)$ is a nondegenerate quadratic $R$-module, $C(V)$ is its $\mathbb{Z}/2$-graded Clifford algebra with graded tensor products understood, $K(C(V))$ is the Grothendieck group of finitely generated graded projective $C(V)$-modules, $V^{(k)} = (V, kq)$ is the rescaled module, $\Omega_k = \mathbb{Z}[\omega]$ is the ring of $k$-th cyclotomic integers, and all notions are as in Karoubi's 'Clifford modules and invariants of quadratic forms' (J. K-Theory 9 (2012) 1-44). The problem has two linked components with one common objective:

1. (Conjecture, Remark 2.6) Prove or refute that the multiplicativity of the Adams operation $\Psi^k$ on Clifford modules (Theorem 2.5: for classes $x, y \in K(C(V))$, whose product $x \cdot y$ is induced by the graded tensor product of the underlying modules with their Clifford actions, one has $\Psi^k(x \cdot y) = \Psi^k(x) \cdot \Psi^k(y)$ in $K(C(V, 2kq))$) holds without the hypothesis that $k!$ is invertible in $R$. (Footnote 4 of the published version records that the conjecture holds, for instance, when the module is hyperbolic or of rank one.)

2. (Conjecture, Remark 2.6 and Section 6) Assume $2k$ is invertible in $R$ and that $R$ contains the $k$-th roots of unity, so that the cyclic-group operation $\overline{\Psi}^k(E) = \sum_{j=0}^{k-1} F_{\omega^j} \cdot \omega^j \in K(C(V^{(k)})) \otimes_{\mathbb{Z}} \Omega_k$ is defined (here $F_{\omega^j}$ are the eigenmodules of a generator of the cyclic group acting on the graded tensor power). Prove or refute that $\overline{\Psi}^k = \Psi^k$ whenever both operations are defined (this is known when $k!$ is invertible in $R$), and likewise for the corresponding operations on oriented $\mathbb{Z}/2$-graded Azumaya algebras restated in Section 6.

The common objective is that the resulting theory of Adams operations and hermitian Bott classes for quadratic modules, Clifford algebras, and graded Azumaya algebras should hold in the same generality as the classical topological theory (which needs no invertibility of $k!$).

References

1. Max Karoubi and Jean-Pierre Serre, 'Clifford modules and invariants of quadratic forms', Journal of K-Theory 9 (2012), no. 1, 1-44. DOI: 10.1017/is011003006jkt145, https://doi.org/10.1017/is011003006jkt145
2. Max Karoubi, 'Clifford modules and invariants of quadratic forms', preprint arXiv:1012.3924 (2010), https://arxiv.org/abs/1012.3924
3. Michael F. Atiyah, 'Power operations in K-theory', Quarterly Journal of Mathematics 17 (1966), 165-193. DOI: 10.1093/qmath/17.1.165, https://doi.org/10.1093/qmath/17.1.165
4. Michael F. Atiyah, Raoul Bott and Arnold Shapiro, 'Clifford modules', Topology 3 (1964), 3-38. DOI: 10.1016/0040-9383(64)90003-5, https://doi.org/10.1016/0040-9383(64)90003-5
5. Jean Fasel and Olivier Haution, 'The stable Adams operations on Hermitian K-theory', Geometry and Topology 29 (2025), 127-169. DOI: 10.2140/gt.2025.29.127, https://doi.org/10.2140/gt.2025.29.127; preprint arXiv:2005.08871, https://arxiv.org/abs/2005.08871
6. Michael K. Brown, Claudia Miller, Peder Thompson and Mark E. Walker, 'Cyclic Adams operations', Journal of Pure and Applied Algebra 221 (2017), 1589-1613. DOI: 10.1016/j.jpaa.2016.12.018, https://doi.org/10.1016/j.jpaa.2016.12.018; preprint arXiv:1601.05072, https://arxiv.org/abs/1601.05072
7. Max Karoubi, 'Twisted bundles and twisted K-theory', preprint arXiv:1012.2512 (2010), https://arxiv.org/abs/1012.2512
8. Tom Bachmann and Michael J. Hopkins, 'eta-periodic motivic stable homotopy theory over fields', preprint arXiv:2005.06778 (2020), https://arxiv.org/abs/2005.06778
9. Max Karoubi and Charles Weibel, 'The Real graded Brauer group', preprint arXiv:1901.04393 (2019), https://arxiv.org/abs/1901.04393; published in The Quarterly Journal of Mathematics (online publication 18 November 2019; no volume or pages assigned), DOI: 10.1093/qmathj/haz037, https://doi.org/10.1093/qmathj/haz037
10. Gerd Laures and Martin Olbermann, 'Cannibalistic Classes of String Bundles', preprint arXiv:1701.01287 (2017), https://arxiv.org/abs/1701.01287