Lambda-Ring Structure and Adams Operations

The K-theory ring $K_0(X)$ of a scheme carries a rich additional structure: the lambda-ring structure coming from exterior powers. The associated Adams operations provide eigenspace decompositions and connect K-theory to motivic cohomology via the gamma filtration.

Lambda-rings

Definition5.6Lambda-ring

A lambda-ring is a commutative ring $R$ equipped with operations $\lambda^k: R \to R$ for $k \geq 0$ satisfying:

$\lambda^0(x) = 1$ for all $x \in R$ .
$\lambda^1(x) = x$ for all $x \in R$ .
$\lambda^k(x + y) = \sum_{i=0}^k \lambda^i(x) \lambda^{k-i}(y)$ (additivity via exterior power splitting).
$\lambda^k(xy) = P_k(\lambda^1(x), \ldots, \lambda^k(x); \lambda^1(y), \ldots, \lambda^k(y))$ where $P_k$ is a universal polynomial.
$\lambda^k(\lambda^l(x)) = P_{k,l}(\lambda^1(x), \ldots, \lambda^{kl}(x))$ where $P_{k,l}$ is another universal polynomial (the plethysm).

The lambda operations on $K_0(X)$ are defined by $\lambda^k([\mathcal{E}]) = [\Lambda^k \mathcal{E}]$ (the $k$ -th exterior power).

ExampleLambda operations on K₀(ℙⁿ)

On $\mathbb{P}^n$ with $K_0(\mathbb{P}^n) = \mathbb{Z}[h]/(h^{n+1})$ where $h = [\mathcal{O}(1)] - 1$ :

The lambda operations on the generator $[\mathcal{O}(1)]$ are trivial ( $\lambda^k([\mathcal{O}(1)]) = 0$ for $k \geq 2$ since $\mathcal{O}(1)$ has rank 1). For $h$ :

$\lambda_t(h) = \lambda_t([\mathcal{O}(1)] - 1) = \frac{\lambda_t([\mathcal{O}(1)])}{\lambda_t(1)} = \frac{1 + t[\mathcal{O}(1)]}{1 + t} = \frac{1 + t(1+h)}{1+t}$

where $\lambda_t(x) = \sum_{k \geq 0} \lambda^k(x) t^k$ is the generating function. Since $\lambda_t(1) = 1 + t$ (the class $[R]$ has $\lambda^k([R]) = 0$ for $k \geq 2$ ).

Expanding: $\lambda_t(h) = 1 + \frac{th}{1+t} = 1 + th - t^2h + t^3h - \cdots$ , so $\lambda^k(h) = (-1)^{k-1}h$ for $k \geq 1$ .

Adams operations

Definition5.7Adams operations

The Adams operations $\psi^k: K_0(X) \to K_0(X)$ for $k \geq 1$ are ring homomorphisms defined by Newton's identity from the lambda operations:

$\psi^k(x) = (-1)^{k+1} k \lambda^k(x) + \sum_{i=1}^{k-1} (-1)^{i+1} \lambda^i(x) \psi^{k-i}(x).$

Equivalently, they are characterized by the generating function identity:

$\psi_t(x) := \sum_{k \geq 1} \psi^k(x) t^{k-1} = \frac{d}{dt} \log \lambda_{-t}(x)^{-1}$

or the property that for a line bundle $\mathcal{L}$ : $\psi^k([\mathcal{L}]) = [\mathcal{L}^{\otimes k}]$ .

Key properties:

$\psi^k$ is a ring homomorphism: $\psi^k(xy) = \psi^k(x)\psi^k(y)$ .
$\psi^k \circ \psi^l = \psi^{kl}$ .
$\psi^p(x) \equiv x^p \pmod{p}$ for prime $p$ (Frobenius-like).
For a vector bundle of rank $r$ : $\psi^k([\mathcal{E}]) = [\operatorname{Sym}^k(\mathcal{E})] - (\text{lower terms})$ .

ExampleAdams operations on vector bundles

For a vector bundle $\mathcal{E}$ with Chern roots $\alpha_1, \ldots, \alpha_r$ :

$\psi^k([\mathcal{E}]) = \sum_{i=1}^r [\mathcal{L}_i^{\otimes k}]$

where $\mathcal{L}_i$ are the "formal line bundles" with $c_1(\mathcal{L}_i) = \alpha_i$ . Under the Chern character:

$\operatorname{ch}(\psi^k(\mathcal{E})) = \sum_{i=1}^r e^{k\alpha_i}$

So on $CH^p(X) \otimes \mathbb{Q} \subset K_0(X) \otimes \mathbb{Q}$ (via the Chern character), $\psi^k$ acts as multiplication by $k^p$ .

For $\mathcal{E} = T_{\mathbb{P}^2}$ (tangent bundle of $\mathbb{P}^2$ , rank 2, $c_1 = 3H$ , $c_2 = 3H^2$ ):

$\psi^2([\mathcal{E}]) = 2\operatorname{ch}_0 + 2 \cdot 2 \cdot \operatorname{ch}_1 + 2 \cdot 4 \cdot \operatorname{ch}_2 =$ ... actually the computation uses $\psi^2 = (\lambda^1)^2 - 2\lambda^2 = [\mathcal{E}]^2 - 2[\Lambda^2\mathcal{E}] = [\mathcal{E} \otimes \mathcal{E}] - 2[\det \mathcal{E}]$ .

The gamma filtration

Definition5.8Gamma filtration

The gamma filtration on $K_0(X)$ is defined by the gamma operations $\gamma^k(x) = \lambda^k(x + (k-1) \cdot 1)$ for $x \in \widetilde{K}_0(X) = \ker(\operatorname{rk})$ :

$F^n_\gamma K_0(X) = \text{subgroup generated by products } \gamma^{i_1}(x_1) \cdots \gamma^{i_r}(x_r) \text{ with } i_1 + \cdots + i_r \geq n$

where $x_j \in \widetilde{K}_0(X)$ .

The graded pieces are:

$\operatorname{gr}^n_\gamma K_0(X) = F^n_\gamma / F^{n+1}_\gamma$

The Chern character identifies:

$\operatorname{gr}^n_\gamma K_0(X) \otimes \mathbb{Q} \cong CH^n(X) \otimes \mathbb{Q}$

and the Adams operations act on $\operatorname{gr}^n_\gamma$ by multiplication by $k^n$ .

ExampleGamma filtration and torsion

The gamma filtration can detect torsion phenomena invisible to the Chern character:

For $X = B(\mathbb{Z}/p)$ (the classifying space of a cyclic group, in an algebraic setting via equivariant K-theory):

$K_0(B(\mathbb{Z}/p)) \cong R(\mathbb{Z}/p) = \mathbb{Z}[\zeta_p] / \Phi_p(\zeta_p) \cong \mathbb{Z}[t]/(1 + t + \cdots + t^{p-1})$

The gamma filtration gives $\operatorname{gr}^n_\gamma K_0 \cong \mathbb{Z}/p$ for $1 \leq n \leq p-2$ and $\operatorname{gr}^n = 0$ for $n \geq p-1$ . The successive quotients are all $p$ -torsion, reflecting the $p$ -torsion in the Chow groups/motivic cohomology of $B(\mathbb{Z}/p)$ .

This example illustrates that the gamma filtration captures arithmetic information (torsion primes) that the rational Chern character misses.

Relation to motivic cohomology

RemarkAdams eigenspaces and motivic decomposition

The Adams operations provide a motivic decomposition of K-theory. For a smooth variety $X$ over a field:

$K_n(X) \otimes \mathbb{Q} = \bigoplus_{p \geq 0} K_n(X)^{(p)}$

where $K_n(X)^{(p)} = \{x \in K_n(X) \otimes \mathbb{Q} : \psi^k(x) = k^p x \text{ for all } k\}$ is the weight- $p$ eigenspace.

The Beilinson--Soule conjecture predicts $K_n(X)^{(p)} = 0$ for $p < n/2$ (the "vanishing conjecture"). For $n = 0$ , this says $K_0(X)^{(p)} = 0$ for $p < 0$ , which is trivially true. For $n = 1$ , this says $K_1(X)^{(0)} = 0$ , which is true for connected $X$ .

The eigenspaces are identified with motivic cohomology:

$K_n(X)^{(p)} \cong H^{2p-n}_{\text{mot}}(X, \mathbb{Q}(p))$

This is the rational version of the Atiyah--Hirzebruch spectral sequence in motivic cohomology:

$E_2^{p,q} = H^{p-q}_{\text{mot}}(X, \mathbb{Z}(-q)) \implies K_{-p-q}(X)$

which degenerates rationally by the existence of Adams operations.