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A double complex for computing the sign-cohomology of the universal ordinary distribution. (English) Zbl 0939.11035

Hahn, Sang Geun (ed.) et al., Recent progress in algebra. Proceedings of an international conference, KAIST, Taejon, South Korea, August 11-15, 1997. Providence, RI: American Mathematical Society. Contemp. Math. 224, 1-27 (1999).
Let \(f\) be a positive integer and let \(U(f)\) be the universal level \(f\) distribution on \(\frac 1f\mathbb Z/\mathbb Z\). in the sense of D. Kubert [Bull. Soc. Math. Fr. 107, 179–202 (1979; Zbl 0409.12021)]. The Galois group of the \(f\)-th cyclotomic field acts on \(U(f)\). Let \(G_{\infty}\) be the subgroup generated by complex conjugation. The Tate cohomology of \(G_{\infty}\) with coefficients in \(U(f)\) is called the sign-cohomology of \(U(f)\) and has been studied by Kubert, and by W. Sinnott [Ann. Math. (2) 108, 107–134 (1978; Zbl 0395.12014)] in his work on the indices of the group of cyclotomic units and of the Stickelberger ideal. S. Galovich and M. Rosen [J. Number Theory 14, 156–184 (1982; Zbl 0483.12003)], using Carlitz modules, proved the analogue of Sinnott’s results for function fields. L. S. Yin generalized these results to the case where the Carlitz module is replaced by a general sign-normalized rank one Drinfeld module and computed the analogue of the unit index under the assumption of a certain conjecture on the Galois module structure of the analogues of certain sign-cohomology groups. The present paper proves Yin’s conjecture.
The author develops a new method involving double complexes that keeps track of both distribution relations and the higher syzygies among them. In the classical cyclotomic case, these methods show (when \(f\not\equiv 2\pmod 4\)) that the sign-homology of \(U(f)\) is isomorphic to the Farrell-Tate homology of the multiplicative group generated by \(-1\) and the primes dividing \(f\).
The methods of the paper have been used by P. Das [Trans. Am. Math. Soc. 352, No. 8, 3557–3594 (2000; Zbl 1013.11069)] to study special values of the gamma function.
For the entire collection see [Zbl 0901.00022].

MSC:

11R18 Cyclotomic extensions
11R27 Units and factorization
11R58 Arithmetic theory of algebraic function fields
20J05 Homological methods in group theory
20J06 Cohomology of groups
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