Cohomology of \(\lambda\)-rings
A \(\lambda\)-ring is a commutative ring with identity endowed with operations \(\lambda^i:R\to R\), for \(i\geq 0\), which satisfy identities encoding the properties of exterior power operations. Key examples of \(\lambda\)-rings are the complex \(K\)-theory \(K(X)\) of a topological space \(X\) and the representation ring \(R(G)\) of a group \(G\). The author introduces a cohomology theory \(H_{\lambda}^*(-)\) for \(\lambda\)-rings. This is defined via a cochain complex whose differential involves the Adams operations of the \(\lambda\)-ring. Writing \(T\) for the commutative monoid of the positive integers under multiplication, it is shown that \(H_{\lambda}^*(R)\) coincides with the Hochschild cohomology of the monoid-ring \(\mathbb{Z}[T]\) with coefficients in \(\text{End}(R)\). The cohomology theory \(H_{\lambda}^*(-)\) is used to study the deformation theory of \(\lambda\)-rings. A rigidity theorem is proved: If \(H_{\lambda}^1(R)=0\), then every deformation of \(R\) is equivalent to the trivial deformation. The question of extending a \(1\)-cocycle to a deformation is also explored. The obstruction is a sequence of \(2\)-cocycles.
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