By Victor P. Snaith

ISBN-10: 3764367172

ISBN-13: 9783764367176

This monograph provides the state-of-the-art within the concept of algebraic K-groups. it truly is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in similar parts equivalent to quantity thought and algebraic geometry. The recommendations provided listed below are largely algebraic or cohomological. all through quantity concept and arithmetic-algebraic geometry one encounters items endowed with a traditional motion through a Galois workforce. specifically 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. mostly those invariants lie in low-dimensional algebraic K-groups of the essential group-ring of the Galois crew. A relevant topic, predictable from the Lichtenbaum conjecture, is the evaluate of those invariants by way of distinct values of the linked L-function at a destructive integer counting on the algebraic K-theory measurement. additionally, the "Wiles unit conjecture" is brought and proven to steer 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 parts on the topic of algebraic K-theory similar to quantity concept and algebraic geometry. The innovations awarded listed here are mostly algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled while wanted.

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**Example text**

To define f2, f3, f4 we need another element, y E Q/Z(l) [llp], which we shall choose in the same manner as 1341. We begin by considering Q/Z(l)[l/p] as a subgroup of the multiplicative group of the maximal unramified extension of L. If nL is a prime of L such that then we choose y to satisfy yvd-' = &- SO that y is a primitive (vd - 1)r-th root of unity. Chapter 4. 1. R1(L/K,2) in the tame case Next set Then f4(x) E F;, x Z has an image in 1;d(l) of the form (. :,::c( which has second coordinate equal to 4(1) so that g-i)gd-'@l), and so f 4 is surjective.

16. However, as abelian groups there are isomorphisms of the form and w w @ (@f=lz[l/p]). Furthermore the resulting map on the quotients of these direct sums by the submodules given by the first summands has the form by the formula (b E (Q/Z)(r) [llp], m E Z[l/p]) and is given by If u E Z satisfies uv 1 (modulo t) then 9-'agi(O, 1) = aui( 0 , l ) = (ui&, 1) E (QlZ) (r) [llp]6 Z [l/p], writing the first coordinate additively, as usual. 5. Then F is a Z[G(L/K)]-module homomorphism since If (ml,. .

For r 2 1, K2r(L(m)) and K2,+1(L(m)) have no ptorsion [60] and so the results of [148] imply that the tame symbol 72 Chapter 3. 2. The higher K-theory invariants 0, (LIK, 2) and the map induced by the inclusion of the field of constants where each have uniquely divisible kernel and cokernel. 17. Proof. 11. 21, the module ~ n d z [ $ $ { ~ )(pm[ l l p ] 6 z ) is a submod- for all i > 0 and Incidentally, for any one-dimensional local field, L, the indecomposable Kgroup K ~ ( L [80] ) differs fiom Kj(L) only by uniquely divisible groups when j 3 by [15].

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

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