International Congress of Mathematicians

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 155Remark Even if T p is not trivial over K^,, the same method allows to twist with amotive whose p-adic realization is trivial over K^. A particular interesting case isthe motive Q(l) if K^ contains the cyclotomic tower. It allows to pass from valuesof the F-function at k to values at k + 1.Strategy This observation allows the following strategy for proving the Main Conjectureand the Bloch-Kato conjecture for all motives:• first prove the equivariant Bloch-Kato conjecture for the motive h°(Q) = Q(0),one fixed k and all finite groups G n . For k = 1 this is an equivariant classnumber formula.• by proposition 3.2.2 this implies the Main Conjecture for the motives Q(k) ®M tnv and all p-adic Lie groups GQO .• for any motive M there is a K^ such that T p becomes trivial. Using corollary3.3.1 it remains to show that ö p (G 00 ,M triv , k) induces 8 p (G n , M, k) for all n.This is a compatibility conjecture for elements in motivic cohomology andallows to reduce to the case of number fields.• the equivariant Bloch-Kato conjecture follows by 3.2.2.4. Relation to classical Iwasawa theory in the criticalcase4.1. Characteristic idealsWe restrict to the case GQO a pro-p-group without p-torsion. In this case theIwasawa algebra is local and Auslander regular ([36]). Its total ring of quotients isa skew field D. Then K 0 (A) ^K 0 (D)^Z, K ± (A) = (A*) ab , and Ki(D) = (F*) abwhere - ab denotes the abelianization of the multiplicative group.Let T be the category of finitely generated A-torsion modules. The localizationsequence for FJ-groups implies an exact sequence(A*) ab -• (F*) ab -• K Q (T) -• 0.If X is a A-torsion module, then we call its class in K 0 (T) the characteristic ideal.By the above sequence it is an element of D* up to [D*,D*] Im A*. If GQO is abelian,K 0 (T) is nothing but the group of fractional ideals that appears in classical Iwasawatheory.The characteristic ideal can also be computed from the theory of determinants.The class of X in K 0 (A) is necessarily 0, hence there exists a generator x £ detA(X).Its image in F®det A (X) = det£>(0) = ID is an element of K\(D). This constructionyields a well-defined element of Ki(D)/lmKi(A) = K 0 (T), in fact the inverse ofthe characteristic ideal of X.Note that a complex is perfect if and only if it is a bounded complex withfinitely generated cohomology. Such complexes also have characteristic ideals if