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Euler characteristics in relative \(K\)-groups. (English) Zbl 1017.19002

Introduction: Suppose that \(M\) is a finite module under the Galois group of a local or global field. Ever since J. Tate’s papers [Proc. Int. Congr. Math. 1962, 288-295 (1963; Zbl 0126.07002); Sém. Bourbaki 1965/66, No. 306 (1966; Zbl 0199.55604)], we have had a simple and explicit formula for the Euler-Poincaré characteristic of the cohomology of \(M\). In this note we are interested in a refinement of this formula when \(M\) also carries an action of some algebra \({\mathcal A}\), commuting with the Galois action. This refinement naturally takes the shape of an identity in a relative \(K\)-group attached to \({\mathcal A}\). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes. This representability may be of independent interest in other contexts.
Our formula for the equivariant Euler characteristic over \({\mathcal A}\) implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the \(L\)-function put forward by D. Burns [“Iwasawa theory and \(p\)-adic Hodge theory over non-commutative algebras”, preprint (1997)], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate’s original paper [loc. cit., 1966]. I am very grateful to J.-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

MSC:

19A99 Grothendieck groups and \(K_0\)
11R34 Galois cohomology
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11R42 Zeta functions and \(L\)-functions of number fields
18G99 Homological algebra in category theory, derived categories and functors
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