  We prove an abstract index formula about Sinnott's symbol between two
different lattices. We also develop the theory of the universal distribution
and predistribution in a double complex point of view. The theory of spectral
sequence is used to interpret the index formula and to analyze the cohomology
of the universal distribution. Combing these results, we successfully prove
Sinnott's index formula about the Stickelberger ideal. In addition, the {1,
-1}-cohomology groups of the universal distribution and the universal
predistribution are obtained.
