Spectral sequences of universal distribution and Sinnott's index formula

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.