By Victor P. Snaith
This monograph provides the state-of-the-art within the concept of algebraic K-groups. it really is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in similar parts akin to quantity concept and algebraic geometry. The concepts awarded listed here are largely algebraic or cohomological. all through quantity thought and arithmetic-algebraic geometry one encounters gadgets endowed with a common motion by way of 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. normally those invariants lie in low-dimensional algebraic K-groups of the quintessential group-ring of the Galois team. A imperative subject matter, predictable from the Lichtenbaum conjecture, is the assessment of those invariants when it comes to specified values of the linked L-function at a damaging integer looking on the algebraic K-theory measurement. moreover, the "Wiles unit conjecture" is brought and proven to guide either to an assessment 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 thought and algebraic geometry. The concepts provided listed here are mostly algebraic or cohomological. must haves on L-functions and algebraic K-theory are recalled while wanted.
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Extra info for Algebraic K-groups as Galois modules
Kp[G(L/K)]ap 2 We shall abbreviate OK, [G(LQ/Kp)]ap to XQ and set X = OK [G(L/K)]a. This is to be interpreted as meaning that X is the intersection of L with the product of its P-completions, Xp, where both are considered as subgroups of the adkles. Hence X is a locally free OKIG(L/K)]-module whose P-completion is where p is the residue characteristic. There are many to choose from and several are described in ([I341 55), which may be derived (sometimes a little work is needed) from ([ill, [I317 [I617 [461, [511, [551, [661, 1671, [731, [go], [991, [loo], [1411, , ).
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 . These advances made it possible to construct the local fundamental classes associated to the higher K-groups of local fields without any assumptions (for further details, see , ,  and ).
H(ad)). 66 Chapter 3. Higher K-theory of Local Fields Consider the Z [G(L/K)]-submodule, (Q/z) (r) [lM contained in @f=l K2r- (LO). Since L/W is totally ramified, G(L/W) acts trivially on this submodule. 1. 7. Since (Q/Z)(r)[l/p] c U we have ( Q / ~ ) ( r ) [ l / p ]C W. Also G(L/W) acts component-by-component on W so that W (@:=;'K~~-I (Lo)) @U as a Z[G(L/W)]-module. 4, the inclusion induces a cohomology isomorphism of the following form " ) )(qa2, gas,. . ,qad, ql-dal) g(a1,. . ,ad) = (F(a2), .