By Martin J. Taylor
This ebook is a self-contained account of the speculation of classgroups of staff earrings. The guiding philosophy has been to explain all of the easy houses of such classgroups when it comes to personality services. This perspective is because of A. Frohlich and it achieves a substantial simplification and readability over earlier strategies. a prime function of the ebook is the creation of the author's crew logarithm, with various examples of its software. the most effects handled are: Ullom's conjecture for Swan modules of p-groups; the self-duality theorem for jewelry of integers of tame extensions; the fixed-point theorem for determinants of team earrings; the lifestyles of Adams operations on classgroups. furthermore, the writer encompasses a variety of calculations of classgroups of particular households of teams akin to generalized dihedral teams, and quaternion and dihedral 2-groups. The paintings contained during this ebook might be effortlessly available to any graduate pupil in natural arithmetic who has taken a path within the illustration conception of finite teams. it is going to even be of curiosity to quantity theorists and algebraic topologists.
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T(~~)) Since = G, as we require. (Er'~) - 1, it is immediate that 45 5. C. Wall on the torsion of determinants of group rings. This result is absolutely vital to our study because it complements the group logarithm introduced in the next chapter. This is because the group logarithm describes determinants of local group rings modulo torsion. The results of this chapter are taken from [Wl}. finite p-group with lrl = pn Let r be a Let K be a finite extension of~ • We p denote the group of p-power roots of unity in K by ~K' and, for the sake of brevity, we shall write 0 in place of OK.
At p, and 1 elsewhere. w) 2 ~ 0 l 19 (being the discriminant of L /K ), it can be seen that p §5. p DUALITY Let K again be a numberfield and r be a finite group. In this section we will describe the effect of duality under the isomorphisms cr,p of §3. The results of this section are all due to Frohlich cf. [Fl], [F2]. Throughout this section we simultaneously use K-linear involution on Kr induced by y virtual character X of r, we write + x for y- 1 to denote the for y E r, and, for a contragredient virtual character.
R2 ) denote the Grothendieck group of Q~ab ~ Qc (resp. Q(~ )oE ~ Qc) modules. n' and we abbreviate HomQq(Ri' 0;) to Hi' and HomQ (R~, 0;) to H. Q Note that for ~. the superscript + on Homnq is superfluous, because ~ has no irreducible symplectic characters. 7) where K is the kernel of the lower right hand map; exact. l at all primes different from p. l [~ }oE)} p p p p t+'p We denote the right hand term by K'. 9 For t "' p Det(7l [ 1;; ] p t Proof ol:*) We are required to show that Det(7l [I;; ] ol:*) (n) p t O* Lt This follows from the following two results: Det(z)(n) (ii) Q(l;; )/L.