By Victor P. Snaith
This monograph provides the cutting-edge within the concept of algebraic K-groups. it's of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in comparable components equivalent to quantity thought and algebraic geometry. The recommendations awarded listed here are largely algebraic or cohomological. all through quantity concept and arithmetic-algebraic geometry one encounters items endowed with a typical motion through a Galois team. specifically this is applicable to algebraic K-groups and ?tale cohomology teams. This quantity is worried with the development of algebraic invariants from such Galois activities. regularly those invariants lie in low-dimensional algebraic K-groups of the critical group-ring of the Galois crew. A important subject matter, predictable from the Lichtenbaum conjecture, is the overview of those invariants when it comes to precise values of the linked L-function at a unfavorable integer reckoning on the algebraic K-theory measurement. additionally, 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 large choice of graduate and postgraduate scholars in addition to researchers in components regarding algebraic K-theory equivalent to quantity idea and algebraic geometry. The recommendations provided listed below are largely algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled whilst wanted.
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Additional resources for Algebraic K-groups as Galois modules
1 Let L / K be a Galois extension of number fields. The second Chinburg invariant (Qo(L/K,2) in our notation) is constructed from a projective Z[G(L/K)]module, X, and the classical fundamental classes of local class field theory (see [I311 Chapter 7). 17 in place of the classical one to construct Galois module structure invariants %-1(LIK, 2) E CC(Z[G(L/K)]) lying in the class-group of Z[G(L/K)]. These invariants measure the Galois module structure of algebraic K-groups in dimensions 2r - 1 and 2r - 2 and are expected to be connected with the values of the L-functions at s = 1 - r (or, equivalently, s = r).
Md) E Ker(X) then ml = vr-lmd,m2 = vr-lml,. . and m d ( l v('-')~) = 0 SO that X is injective. Also we can solve the equation X(ml, . . ,md) = (xl, . . ,xd) by choosing xd, X I , . . ,xd-2 successively but the equations are consistent if and only if - Therefore, if 7r2 is defined to depend only on the coordinates in the second summand and to be given by while a(fi(g% (0, I))) = a(gG1 8 (0, vr-l)) (0, Vr-1) 8 (vr-lui-l St, vr-l) = 92-1 8 a"' = gi-l and these are equal since vr-lui-l = quui-l r qui (modulo t).
Xd) = (FO(xd),xl,.. ,xd-1). Since K-theory is additive there is an induced isomorphism > for each s 0. This isomorphism induces a G(L/K)-action on the right-hand group, induced by the Galois action on L. Since the F'robenius acts on the other factor, KO,in the tensor product we may define a Z[G(L/K)]-homomorphism by the same formula as before. 2). 64 Chapter 3. 6 Suppose that g E G(L/K) satisfies (gl W) = (FjI W) for some 0 5 j 5 d - 1. K ~ (LO) ~ is-given ~ by Then the action of g on (ul, u2, .
Algebraic K-groups as Galois modules by Victor P. Snaith