On the Hodge decomposition of differential graded bi-algebras

We give a natural decomposition of a connected commutative differential graded bi-algebra over a commutative algebra in the case of characteristic zero. This gives the ordinary Hodge decomposition of the Hochschild homology when we apply this natural decomposition to the cyclic bar complex of a commutative algebra. In the case of characteristic p>0, we show that, in the spectral sequence induced by the augmentation ideal filtration of the cyclic bar complex of a commutative algebra, the only possible non-trivial differentials are dk(p−1) for k≥1. Also we show that the spectral sequence which converges to the Hochschild cohomology is multiplicative with respect to the Gerstenhaber brackets and the cup products.