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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).

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
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