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Relative $$K_0$$, annihilators, Fitting ideals and Stickelberger phenomena. (English) Zbl 1108.19001
For a finite group $$G$$ and a prime number $$\ell,$$ let $$R = {\mathbb Z}_\ell [G]$$ and $$K_0 T(R) = K_0 ({\mathbb Z}_\ell[G], {\mathbb Q}_\ell [G])$$ the relative $$K_0$$-group appearing in the localisation exact sequence of $$K$$-theory in dimensions 0, 1. If $$G$$ is abelian, $$K_0 T(R) \simeq {\mathbb Q}_\ell [G]^\times/{\mathbb Z}_\ell [G]^\times$$ by a determinant map. In this case, the author’s main algebraic result is the construction of a determinant living in $${\mathbb Q}_\ell [G]^\times /{\mathbb Z}_\ell [G]^\times,$$ starting from any bounded perfect complex $$F$$ of $$R$$-modules having all its homology groups finite. More precisely, for such a complex $$F,$$ the author defines a certain natural $${\mathbb Q}_\ell[G]$$-isomorphism $$X$$ : $$F_{even} \otimes {\mathbb Q}_\ell {\buildrel\sim\over\rightarrow} F_{odd} \otimes {\mathbb Q}_\ell,$$ and det$$(X)$$ is the element of $${\mathbb Q}_\ell [G]^\times/{\mathbb Z}_\ell [G]^\times$$ corresponding to the relative class $$[F_{even}, X, F_{odd}].$$ The additional hypothesis that $$H_i (F_\ast) = 0$$ for $$i \not= 0,1$$ gives rise to interesting annihilation results.
Let us only cite: if the $$R$$-module $$\text{Hom}(H_1 (F_\ast), {\mathbb Q}_\ell/{\mathbb Z}_\ell)$$ is monogeneous, then $$\text{det}(X)^{-1} \ldotp \text{ann}_R (H_1(F_\ast)) \subseteq \text{ann}_R (H_0(F_\ast)).$$ In examples coming from number theory or arithmetic geometry, this yields Stickelberger-type relations. For instance, taking $$K = {\mathbb Q} (\xi_{m\ell^{s+1}}), s \geq 1, \ell |\!\!/ m, G =\text{Gal}(K/{\mathbb Q})$$ and $$r \in {\mathbb Z},$$ the author constructs a bounded perfect cochain complex $$P(r)^\ast$$ of $$R$$-modules such that $$H^i (P(r)^\ast) = 0$$ for $$i \not= 1,2,$$ $$H^2(P(r)^\ast) \simeq H^2_{\text{ét}} ({\mathcal O}_K [1/m\ell], {\mathbb Z}_\ell (1-r)),$$ $$H^1(P(r)^\ast)$$ is the quotient of $$H^1_{\text{ét}} ({\mathcal O}_K [1/m\ell],$$ $${\mathbb Z}_\ell(1-r))$$ by a certain $$R$$-monogeneous submodule of “cyclotomic elements”, and $$\text{det}(X)^{-1} = \Bigl( {1 \over 2} (1+(-1)^r c)\Bigl) + \Bigl( {1 \over 2} (1 + (-1)^{r-1} c)\Bigl) g_m^r,$$ where $$c$$ denotes complex conjugation and $$g_m^r$$ is a certain Stickelberger element. Essentially, this is done by “descending” (not a trivial task !) a certain perfect complex of $${\mathbb Z}_\ell [[G({\mathbb Q} (\xi_{m\ell^\infty})/{\mathbb Q})]]$$-modules construted by D. Burns and C. Greither in their proof of the equivariant Tamagawa number conjecture for Tate motives over absolute abelian fields [Invent. Math. 153, 303–359 (2003; Zbl 1142.11076)].
For $$r < 0,$$ odd, the annihilation result obtained by the author implies the ($$\ell$$-adic version of) the Coates-Sinnott conjecture for absolute abelian fields. But note that a sharper theorem has been proved in the relative abelian case by D. Burns and C. Greither [Doc. Math., J. DMV Extra Vol., 157–185 (2003; Zbl 1142.11371)], and also by the reviewer [T. Nguyen Quang Do, J. Théor. Nombres Bordx. 17, 643–668 (2005; Zbl 1098.11054)] (whose formulation is even a global, not merely $$\ell$$-adic, one). For $$r < 0,$$ even, the Stickelberger higher ideal is zero and the Coates-Sinnott conjecture becomes trivial, but the author’s result reveals a new phenomenon, namely the existence of a fractional ideal $${\mathcal J}_r$$ of $${\mathbb Q} [G(K^+/{\mathbb Q})]$$ such that $${\mathcal J}_r \cap {\mathbb Z}_\ell [G(K^+/{\mathbb Q})]$$ annihilates $$H^2_{\text{ét}} \Bigl( {\mathcal O}_{K^+} [1/m\ell], {\mathbb Z}_\ell (1-r)\Bigl).$$ This annihilator is non trivial, since for $$m = 1, s = 0,$$ it contains $$\xi^\ast (r)/R_r({\mathbb Q}),$$ the order, up to powers of 2, of the group $$H^2_{\text{ét}} ({\mathbb Z} [1/\ell], {\mathbb Z}_\ell (1-r))$$ given by the Lichtenbaum conjecture (now a theorem for absolute abelian fields).

##### MSC:
 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 11G55 Polylogarithms and relations with $$K$$-theory 11R34 Galois cohomology 11R42 Zeta functions and $$L$$-functions of number fields
##### Keywords:
annihilators; complexes
##### Citations:
Zbl 1142.11076; Zbl 1142.11371; Zbl 1098.11054
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