Group cohomology of the universal ordinary distribution (with an appendix by Greg W. Anderson) (Q2755155)

For any odd squarefree integer \(r\), the universal ordinary distribution \(U_r\) in the sense of \textit{D. Kubert} [Bull. Soc. Math. Fr. 107, 179-202 (1979; Zbl 0409.12021)] is a certain \(\mathbb Z\)-module defined by generators \([a]\), \(a\in \frac 1 r \mathbb Z / \mathbb Z\) and relations \([a]-\sum_{lb=a} [b]\), for all primes \(l\mid r\), \(a\in \frac l r \mathbb Z / \mathbb Z\), endowed with a natural action by \(G_r=\text{ Gal} (\mathbb Q(\mu_r) / \mathbb Q)\). It plays an important role in the theory of cyclotomic fields. For example, \textit{W. Sinnott}'s formula giving the index of the circular units inside the units [Ann. Math. (2) 108, 107-134 (1978; Zbl 0395.12014)] rests mainly on the computation of the \(\{\pm 1 \}\)-cohomology of a certain lattice \(U\subset \mathbb Q[G_r]\) which is a realization of \(U_r\) because in this context the latter is also a lattice. Sinnott's calculations are admittedly not very enlightening. Recently, G. W. Anderson has found a more functorial way to compute the \(\{\pm 1\}\)-cohomology of \(U_r\), using a cochain complex which is a resolution of the universal distribution (see the appendix). NEWLINENEWLINENEWLINEIn the present paper, the author uses Anderson's resolution to construct a double complex related to the \(G_r\)-cohomology of \(U_r\) and he studies the spectral sequence of this double complex. His main result gives the \(G_r\)-cohomology of \(U_r\) in terms of the \(G_r\) cohomology of \(\mathbb Z\). For a fixed integer \(M\) dividing \(l-1\) for all primes \(l\mid\) r, he also studies the \(G_r\)-cohomology of \(U_r/M\). NEWLINENEWLINENEWLINEOne striking application is the ``explanation'' of the construction of Kolyvagin's elements \(\kappa_s\) for \(s\mid r\) in the sense of Rubin [\textit{S. Lang}, Cyclotomic fields. I and II. Combined 2nd ed. Graduate Texts in Mathematics, 121. Springer-Verlag, New York (1990; Zbl 0704.11038)]: these elements come from a certain \(\mathbb Z/M\)-basis of \(H^0(G_r,U_r/M)\) through an evaluation map.