By Victor P. Snaith
This monograph offers the state-of-the-art within the idea of algebraic K-groups. it really is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in similar parts corresponding to quantity concept and algebraic geometry. The ideas offered listed here are largely algebraic or cohomological. all through quantity conception and arithmetic-algebraic geometry one encounters items endowed with a ordinary motion via a Galois staff. particularly this is applicable to algebraic K-groups and ?tale cohomology teams. This quantity is anxious with the development of algebraic invariants from such Galois activities. more often than not those invariants lie in low-dimensional algebraic K-groups of the indispensable group-ring of the Galois crew. A vital subject, predictable from the Lichtenbaum conjecture, is the evaluate of those invariants by way of certain values of the linked L-function at a adverse integer reckoning on the algebraic K-theory size. moreover, the "Wiles unit conjecture" is brought and proven to guide either to an evaluate of the Galois invariants and to clarification of the Brumer-Coates-Sinnott conjectures. This ebook is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in parts on the topic of algebraic K-theory corresponding to quantity idea and algebraic geometry. The innovations provided listed below are largely algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled while wanted.
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Extra info for Algebraic K-groups as Galois modules
Generators of the Z[G(L/K)]-module, X(2), are given by the elements t = (a - l)zl - f z2, y = (1 a . . a'-')z2 and x = (1 g-l . . + gl-d)zl + (a-l . . 5. 5. 7. 7 associated to the lower 2-extension in the diagram is equal to - [Ker(g4)1E C w w ( L I K ) I ) . Proof. If g4 is surjective then, since g4 induces isomorphisms on all Tate cohomology groups, its kernel is a finitely generated, cohomologically trivial, torsion-free Z[G(L/K)]-module. Hence it is projective. Therefore so that, for 1 5 i 5 d, + is a perfect complex which is quasi-isomorphic to + + + + + Note that, since 6 is a primitive r(v2d - 1)-th root of unity, (6 )V i + d is a primitive (v2d - 1)-th root of unity and hence generates K3(Fvd).
Ar-'). Hence we can list the generators for W f~(X(2) 8 Qo) by setting each of al,d, a2,1, a2,2,. . ,a2,d-1, equal to one and the rest to zero in the previous immense expression. The result is as follows: These elements have been chosen so that all except y 8 g are in ker(g4 : X(2) 8 Qo --t K3(Fvd) F : , ) . Fkom the exact sequence of zlg]/(gd - 1)-modules " 102 Chapter 4. 1. 13 shows that the interesection in the denominator is / (1)g(y@ ~ (v - g +rad)). the direct sum of a free module with the submodule ~ [ ~ ] Note that X(y @ (v - g Tad)) = y @ (v - g rad).
By ,the map on the prime-to-p torsion may be identified with the injective map on K2r-l of the residue fields. For the torsion-free part it suffices to show that ,K ::: (L(s)) @ QP -+ K ~ ; ? ~ ( L ( S @ ~ )Qp ) is injective and this follows from . Therefore we may define -+ resulting in the associated J-invariantlcoinvariant 2-extension in which the Z[G/J]-modules, B J and C J , are cohomologically trivial. 20). 2 Now we shall begin the construction of the local fundamental classes by considering the totally ramified case.
Algebraic K-groups as Galois modules by Victor P. Snaith