By Victor P. Snaith
This monograph offers the state-of-the-art within the idea of algebraic K-groups. it really is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in similar parts akin to quantity conception and algebraic geometry. The thoughts provided listed here are mostly algebraic or cohomological. all through quantity concept and arithmetic-algebraic geometry one encounters items endowed with a ordinary motion via a Galois staff. particularly 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. in general those invariants lie in low-dimensional algebraic K-groups of the critical group-ring of the Galois staff. A critical subject matter, predictable from the Lichtenbaum conjecture, is the evaluate of those invariants when it comes to specific values of the linked L-function at a adverse integer looking 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 large choice of graduate and postgraduate scholars in addition to researchers in parts on the topic of algebraic K-theory resembling quantity concept and algebraic geometry. The recommendations offered listed below 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
For example, we shall settle for the regulator given by the syntomic Chern classes. More precisely, we have canonical isomorphisms, given by &ale and syntomic Chern classes, of the form Xp = OKp[G(L/K)]ap = Ind G(LQIKp) G(L/K) ( X ~ ) ' In addition, we shall assume henceforth (replacing X by m X for a suitable integer, m E Z, if necessary) that the Q-adic exponential defines an isomorphism for all wild LQ/Kp. This assumption is only needed in the construction of the invariant flo(L/K, 2). Since X is locally free, it is cohomologically trivial and so also is XQ for each Q.
In the case of one-dimensional local fields Cf was K ? 1. 21 associated to K2 and K3 of a local field in characteristic p > 0. 22. For the calculations of this section we shall make things easier for ourselves by considering the tamely ramified case. These calculations are rather long and clumsy, so I shall begin with an outline of contents of this section. 56) homological data is given which is used to calculate the Euler characteristic of the classical (Ko/K1) local fundamental class in the tamely ramified case.
A"-l)z2 (gd-l - I). 12, we wish to determine the class in the class-group of the module Chapter 4. Positive Characteristic 104 Considering the coefficients of zl we see that ai(a - 1) = 0 for each i and therefore ai = Pi(1 a ... a'-') for 0 5 i 5 d - 1 and Pi E Z[g]/(gd - 1). 1. R1(L/K, 2) in the tame case 105 Next we compare coefficients (in the left-factor) of the various powers of g to obtain: Therefore Considering the coefficients of zl we obtain + - 1 ) b y = 0. Hence Po = 0 which and po((vd - l)wZd 1)y = b u d y or (vd I7 implies that Pi = 0 for all 0 5 i 5 d - 1, as required.
Algebraic K-groups as Galois modules by Victor P. Snaith