By Victor P. Snaith
This monograph offers the state-of-the-art within the thought of algebraic K-groups. it really is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in comparable components akin to quantity idea and algebraic geometry. The thoughts provided listed below are largely algebraic or cohomological. all through quantity idea and arithmetic-algebraic geometry one encounters gadgets endowed with a normal motion via a Galois crew. 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. quite often those invariants lie in low-dimensional algebraic K-groups of the indispensable group-ring of the Galois staff. A relevant subject matter, predictable from the Lichtenbaum conjecture, is the review of those invariants when it comes to specified values of the linked L-function at a destructive integer counting on the algebraic K-theory size. moreover, the "Wiles unit conjecture" is brought and proven to guide either to an review of the Galois invariants and to clarification of the Brumer-Coates-Sinnott conjectures. This publication is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in components relating to algebraic K-theory equivalent to quantity idea and algebraic geometry. The ideas offered listed below are mostly algebraic or cohomological. must haves on L-functions and algebraic K-theory are recalled whilst wanted.
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Extra info for Algebraic K-groups as Galois modules
When s = 0 = t these two groups differ by a uniquely divisible group and E:" = EL0. Since G(F/M) acts trivially on K3(Lo), E;" Hs(G(L/K); K3(Lo)) r Z/r 0 if s is odd, if s > 0 is even. Note also that H*(G(L/K); Lz) = 0 when L I K is totally ramified so that the sequence 2't-1 E Ellt if t is odd and s > 0. implies that ) H 2(G(F/E); K F d (Fo)) from Next we calculate H1 (G(F/E); K ? ~ ( F ~ ) and the exact sequence of . The first group is isomorphic to the kernel of the map This homomorphism is an isomorphism on the uniquely divisible part, a surjection with kernel Z l r on the K2-part and the el-th power of the inclusion on the K1part.
2 Now we shall begin the construction of the local fundamental classes by considering the totally ramified case. 17 for finite Galois extensions of padic local fields. Furthermore, for K2r,K2r+1 with r 2 2 in this case we needed to assume that the Lichtenbaum-Quillen Conjecture ([I351 p. 326) was true for the mod p algebraic K-theory of p-adic local fields. When  was written this was known for 2-adic fields by . For p-adic fields when p is odd the conjecture was proved recently . When L I K is a Galois extension of local fields of characteristic p the Lichtenbaum-Quillen Conjecture is now known to be true by , which shows that the K-theory of L has no p-torsion, combined with the results of .
Therefore it remains to verify that G(L/K) - 1) 2 K2r-3(OL/~L) acts on z / ( v ( ~ - ' ) ~ via the quotient to G(W/K) whose generator acts by "multiplication" by vr-l, K ~ ~ - ~ ( Land o ) W. where v = IOK/nKI. Certainly, G(L/E) acts trivially on Chapter 3. Higher K-theory of Local Fields 70 Also the action by a E G(E/W) is trivial on the second coordinates in the direct sums. 1. 6) establishes the passage-toquotients part of the following result and the passage-to-subgroups is relatively straightforward.
Algebraic K-groups as Galois modules by Victor P. Snaith