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By G. Shilov

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To construct the free pro-C product G one proceeds as follows: let Gabs = α∈A Gα ∗ be the free product of the Gα as abstract groups. Consider the pro-C topology on Gabs determined by the collection of normal subgroups N of finite index in Gabs such that Gabs /N ∈ C, N ∩ Gα is open in Gα , for each α ∈ A, and N ≥ Gα , for all but finitely many α. Put G = lim G/N. ←− N Then G together with the maps ια : Gα −→ G is the free pro-C product rα∈A Gα . If the set A is finite, the ‘convergence’ property of the homomorphisms ια is automatic; in that case, instead of r , we use the symbol .

10 Luis Ribes Cohomological dimension Let G be a profinite group and let p be a prime number. , the subgroup of X consisting of its elements whose order is a power of p. The cohomological p-dimension cdp (G) of G is the smallest non-negative integer n such that H k (G, A)p = 0 for all k > n and A ∈ DMod([[ZG]]), if such an n exists. Otherwise we say that cdp (G) = ∞. Similarly, the strict cohomological p-dimension scdp (G) of G is the smallest non-negative number n such that H k (G, A)p = 0 for all k > n and A ∈ DMod(G).

An abelian group homomorphism for which ϕ(ga) = gϕ(a), for all g ∈ G, a ∈ M. The class of G-modules and G-morphisms constitutes an abelian category which we denote by Mod(G). The profinite G-modules form an abelian subcategory of Mod(G), denoted PMod(G), while the discrete G-modules form an abelian subcategory denoted DMod(G). In turn, the discrete torsion G-modules form a subcategory of DMod(G). 2 The complete group algebra ˆ and a profinite Consider a commutative profinite ring R (for example R = Z) group H.

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Calculus of Rational Functions by G. Shilov


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