The smallness is proved of étale fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the HermiteMinkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain finiteness results of representations of the fundamental groups over algebraically closed fields.

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We introduce a Milnor type K-group associated to commutative algebraic groups over a perfect field. It is an additive variant of Somekawa's K-group. We show that the K-group associated to the additive group and q multiplicative groups of a field is isomorphic to the space of absolute Kähler differentials of degree q of the field, thus giving us a geometric interpretation of the space of absolute Kähler differentials. We also show that the K-group associated to the additive group and Jacobian variety of a curve is isomorphic to the homology group of a certain complex.

We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito's class field theory for projective curves (Saito in J Number Theory 21: 1985).

In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i : Y ֒→ X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O X -Modules supported on Y . The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh nonconnected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting nonconnected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes. * Supported by the JSPS Fellowships for Young Scientists. † This research is supported by JSPS core-to-core program 18005 pseudo-coherent O X -Modules K S (Wt(X on Y )) (resp. HC(Wt(X on Y ))). That is, we have isomorphismsfor each q ∈ Z. For the connected K-theory this result is nothing other than Exercise 5.7 in [TT90]. For Grothendieck groups (q = 0), there is a detailed proof if X is the spectrum of a Cohen-Macaulay local ring and Y is the closed point of X ([RS03], Prop. 2). For K-theory, as mentioned in Exercise 5.7, this problem is related with the works [Ger74], [Gra76] and [Lev88]. Namely the problem about describing the homotopy fiber of K B (X) → K B (X Y ) (or rather than K Q (X) → K Q (X Y )) by using the K-theory of a certain exact category. As described in [Ger74], there is an example due to Deligne which suggests difficulty of the problem for a general closed immersion. Conversely, the example indicate that for an appropriate scheme X, there is a good class of pseudo-coherent O X -Modules. That is, Modules of pure weight. This concept is intimately related to Weibel's K-dimensional conjecture [Wei80] (see Conj. 6.4), Gersten's conjecture [Ger73] and its consequences. These subjects will be treated in [HM08], [Moc08]. Notice that there are different notions of pure weight by Grayson [Gra95] and Walker [Wal00] and these two notions are compatible in a particular situation [Wal96]. In a future work, the authors hope to compare the Walker weight with the Thomason-Trobaugh one by utilizing the (equidimensional ) bivariant algebraic K-theory [GW00]. Now we explain the structure of the paper. In §2, we describe to our motivational picture. After reviewing the fundamental facts in §3, we will define the notion of weight and state the main theorem in §4. The proof of the main theorem will be given in §5. Finally we will give applications of the main theorem in §6.Convention. Throughout this paper, we use the letter X to denote a scheme. A complex means a chain complex whose boundary morphism is increase level of te...