Flach, M. (2000) Euler characteristics in relative Kgroups. Bulletin of the London Mathematical Society, 32 (3). pp. 272284. ISSN 00246093. http://resolver.caltech.edu/CaltechAUTHORS:FLAblms00

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Abstract
Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], 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 [script A], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative Kgroup attached to [script A] (see Section 2). 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 (see Section 3). This representability may be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over [script A] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the Lfunction put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and SwinnertonDyer) was also a motivation for Tate's original paper [18]. I am very grateful to JP. 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.
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Additional Information:  "Reprinted with the permission of Cambridge University Press." Received 15 March 1998; revised 23 April 1999. Published Online 08 September 2000 The author was supported by NSFgrant DMA9624824 and a Sloan fellowship. 
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Alternative URL:  http://dx.doi.org/10.1112/S0024609300006950 
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