Abstract. We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for ℓ-adic Lie extensions of a separated scheme X of finite type over a finite field of characteristic prime to ℓ.

We extend Deligne's notion of determinant functor to Waldhausen categories and (strongly) triangulated categories. We construct explicit universal determinant functors in each case, whose target is an algebraic model for the 1-type of the corresponding K-theory spectrum. As applications, we answer open questions by Maltsiniotis and Neeman on the K-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional Ktheory of (strongly) triangulated categories and obtain generators and (some) relations for various K 1 -groups. This is achieved via a unified theory of determinant functors which can be applied in further contexts, such as derivators.

Abstract. Let B = A [[t; σ, δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B.happens to be the Iwasawa algebra of a p-adic Lie group G ∼ = H ⋊ Zp, this set S is Venjakob's canonical Ore set. In particular, our result implies thatis split exact for each n ≥ 0. We also prove the corresponding result for the localisation of] with respect to the Ore set S * . Both sequences play a major role in non-commutative Iwasawa theory.

We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated Zp-module. The relationship between this conjecture and Iwasawa's classical µ = 0 conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.

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