By Raymond Ayoub

ISBN-10: 0821815105

ISBN-13: 9780821815106

ISBN-10: 0821841815

ISBN-13: 9780821841815

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**Example text**

RG-modules that are ﬁnitely generated and projective over R. Note that for a ﬁeld F , every M ∈ M(F G) is an F G-lattice and so M(F G) = PF (F G). 3 Let S be a GSet (G a discrete group). We can associate to S a category S as follows: ob S = elements of S ; morS (s, t) = {(g, s)|g ∈ G, gs = t} . Composition of morphisms is deﬁned by (h, t) ◦ (g, s) = (hg, s), and the identity morphism s → s is (e, s) where e is the identity of G. S is called the translation category of S. • For any category C, let [S, C] be a category of (covariant) functors ζ : S → C, which associates to an element s ∈ S a C-object ζs and to a morphism (g, s) a C-map ζ(g,s) : ζs → ζgs , s ∈ S ζ(e,s) = idζs and ζ(g,hs) ◦ ζ(h,s) = ζ(gh,s) for all g, h ∈ G, s ∈ S.

1. Suppose that Λ satisﬁes the Cartan condition. Then SK0 (Λ) and SG0 (Λ) are ﬁnite. So also is the kernel of the Cartan map K0 (Λ) → G0 (Λ). Moreover, rank (G0 (Λ)) = number of simple factors of Λ. 3 Class groups of Dedekind domains, orders, and grouprings plus some applications For a Dedekind domain R with quotient ﬁeld F , the notion of class groups of R-orders Λ is a natural generalization of the notion of class groups of rings of integers in number ﬁelds as well as class groups of grouprings RG where G is a ﬁnite group.

H Proof. Again, with C1 = T , C2 = G × T , the embedding C1 → C2 H deﬁned above makes C1 a full subcategory of C2 such that any object in C2 is isomorphic to some object in (the image of) C1 . ) (iv) If ϕ : H → G is a group homomorphism, then we have a functor GSet → HSet given by S → S|H . Now, we can associate to any ζ ∈ [S, C] the Hequivariant C-bundle ζ|H over S|H , which has the same ﬁbres as ζ with the H-action deﬁned by restricting the G-action to H via ϕ. We thus get a functor [S, C] → [S|H , C].

### An introduction to the analytic theory of numbers by Raymond Ayoub

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