Hecke actions on the \(K\)-theory of commutative rings
\textit{R. Perlis} [J. Number Theory 10, 489-509 (1978; Zbl 0393.12009)] showed that, given a Galois extension \(K\) of \(\mathbb{Q}\) with Galois group \(G\) and any two subgroups \(H\) and \(H'\) of \(G\), there is a functorial way of associating a homomorphism from the class group of \(K^H\) to that of \(K^{H'}\) to each \(\mathbb{Z} G\)-homomorphism from \(\mathbb{Z} [G/H]\) to \(\mathbb{Z} [G/H']\). In particular, the class group of \(K^H\) is a module over the Hecke algebra \(\text{End}_{\mathbb{Z} G} (\mathbb{Z} [G/H])\). Some people extended the results of Perlis to Picard groups of schemes. The author of the present paper extends the results of Perlis to the higher \(K\)-theory of rings of integers. By using the equivalence of Hecke actions and cohomological \(G\)-functors [see \textit{T. Yoshida}, J. Math. Soc. Japan 35, 179-190 (1983; Zbl 0507.20010)], the author proves a quite general result about Hecke actions on the \(K\)-theory of commutative rings, and applies this to the \(K\)-theory of rings of integers. The arguments here are only working for low-dimensional \((n\leq 4)\) and even-dimensional \(K\)-theory. The question for odd-dimensional \(K\)-theory is still open.
scientific article; zbMATH DE number 920028
scientific article