By Victor P. Snaith

ISBN-10: 3764367172

ISBN-13: 9783764367176

This monograph offers the cutting-edge within the thought of algebraic K-groups. it truly is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in comparable components resembling quantity concept and algebraic geometry. The thoughts offered listed here are mostly algebraic or cohomological. all through quantity concept and arithmetic-algebraic geometry one encounters items endowed with a average motion through a Galois staff. 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. generally those invariants lie in low-dimensional algebraic K-groups of the imperative group-ring of the Galois team. A critical subject, predictable from the Lichtenbaum conjecture, is the evaluate of those invariants when it comes to designated values of the linked L-function at a unfavorable integer reckoning on the algebraic K-theory size. 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 booklet is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in components relating to algebraic K-theory akin to quantity thought and algebraic geometry. The ideas awarded listed here are mostly algebraic or cohomological. must haves on L-functions and algebraic K-theory are recalled whilst wanted.

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Now let us calculate f 4 on (a - l)zl - fz2 and y = (1 a We have + + . . + ar-')z2. as required. 5 Surjectivity of f3((a - l)2l - f 22) f4 Recall that the inclusion homomorphism, 4 : M 4~ n d g ( ~is) given , by g @H g-'m. If y = (1 a . . ar-')z2 E X(2) then $(m) = + + + = (a- + l)(g-' 8 ( y r , l ) 18(yw,o) + g @ (ym2,0) (yrwd-l7 0)) - f (x:~: gi @ (yWi,0)) +gd-2 + gd-3 (yrwd-27 0) (c::: (I, 1) - f gi 8 (yWi,0))) = (9-'av - 9-') 8 ( 1 , l ) - (g-'(1 a . . aV-') - l)(x:~: gi 8 (ywi,0)) = (a - 1)(g-' 8 = g-' s (C,0) - XI;:' = 9-' 8 (er-vrwd-l, However, yr is a primitive (vd - 1)-th root of unity so that f4(y) = yr generates Fit,.

370). Suppose we are in the tame situation. 1. 22, since a E G(L/W) acts triviallyon the roots of unity. The reason for distinguishing Ind(1) and Ind(2) is that they sit in different 2-extensions, which we shall now describe. -P. Serre (see also [I311 Chapter 7). It is a 2extension of Galois modules and (see [132]) is filtered by the usual filtration of the multiplicative group of a local field so as to remain exact at each level. In particular, in the tame case, we may truncate Serre's fundamental class at level one to give a 2-extension of Z[G(L/K)]-modules of the form where ~ ( ga' (7, m)) = g'-l @ (av,m).

Let FIE be a finite Galois extension of two-dimensional local fields of characteristic zero with Galois group, G(F/E), whose residue field extension is a totally ramified Galois extension, L I K , of one-dimensional local fields. There are two cases. Case A: char(K) = 0 In this case ([54] p. 257) F = L((X)) and E = K((Y)), where L I K is a Galois extension of p-adic local fields, so that the elements of L((X)) are formal series, C,",aiX" with ai E L. Case B: char(K) = p In this case ([54] p. 257) we shall only consider the situation1 F = L1{{X)) and E = K1{{Y)), where L I K is a Galois extension of p-adic local fields and the residue fields of K1, L1 are K, L, respectively.

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Algebraic K-groups as Galois modules by Victor P. Snaith

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