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

By Victor P. Snaith

This monograph offers the cutting-edge within the conception of algebraic K-groups. it really is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in comparable components similar to quantity conception and algebraic geometry. The ideas offered listed below 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 team. 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. often those invariants lie in low-dimensional algebraic K-groups of the essential group-ring of the Galois workforce. A important subject matter, predictable from the Lichtenbaum conjecture, is the evaluate of those invariants by way of specific values of the linked L-function at a damaging integer looking on the algebraic K-theory size. additionally, the "Wiles unit conjecture" is brought and proven to steer either to an evaluate of the Galois invariants and to rationalization of the Brumer-Coates-Sinnott conjectures. This publication is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in parts with regards to algebraic K-theory equivalent to quantity thought and algebraic geometry. The thoughts offered listed below are mostly algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled whilst wanted.

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Now we shall use these observations to simplify considerably the construction of the local K-theory fundamental classes in the characteristic p case. First we need a little notation. Let p,[l/p] denote the group of roots of unity of order prime to p. This group is Q/Z[l/p] written multipPcatively. Let p, [llp]6zdenote the Z[G(L/W)]module given by the direct sum of p,[l/p] with the integers where G(L/W) acts trivially on p,[l/p]. 5, then t divides vd - 1 and so is prime to p. We let G(L/W) act on p, [l/p]6Z via the quotient map to G(L/W)/Gl(L/W) 2 Z/t = (a) where a ( 1 , l ) = (St, 1) for some primitive t-th root of unity, St E p, [llp].

Define w by the pushout diagram ( ( a. 13. where G(L/ W) acts trivially on (QIZ) (r) [lip]. Hence we may form the following push-out diagram, which defines the module 62,~ 2 , - 1(LO). I4 is cohomologically trivial. 16 On (Q/Z)(r)[l/p] Fo acts by multiplication by qd where q = vr, v = ~ O K / T KHence, ~. 13 The Z[G(L/K)]-module, ial. 12 is cohomologically trivwhere, on the right, G(L/W) acts trivially on (Q/Z)(r) [llp] and Proof. Since the upper horizontal homomorphism induces isomorphisms in cohomology so does the lower.

Therefore K2(F) -+ K ~ ( F ( s ) ) ~ ( ~ ( ' ) / ~ ) is an isomorphism on the uniquely divisible part and on the K2 part, which is zero, and on the K1 part. Hence we have an exact sequence of the form What is HS(G(F/E);K ~ ~ ( F ~ ) ) ? This time F = L{{X)), E = K{{Y)), M = L{{Y)) and we have an inclusion G(F/M) Z/el We have - since L; is a cohomologically trivial Z[G(L/K)]-module, so d2 must map onto E,2y1,by counting, since E:" is a finite group. If d2 : E:j2 E:" is surjective E :~~ ~ + ~for~all~ s~2 -0,~t 2 1 and, if s > 0 this must then so is d2 : E : ~ ~ H1(G(L/K); pel(Lo)) so that the be an isomorphism, by counting.

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