The authors study Weyl symmetry from a cohomological point of view in theories including gravity in order to determine 0- and 1-cocycles (Weyl invariants and Weyl anomalies). For pedagogical reasons they rederive known results in four dimensions. Then they solve the same problem in six dimensions. The relation of Weyl anomalies to cocycles of the diffeomorphisms are determined: they can be obtained from each other by subtracting a suitable counterterm from the action. Finally it is shown by an example that such a subtraction corresponds to changing the regularisation scheme.

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We prove that in 10-D SUGRA+SYM the only essential T-constraint is [Formula: see text]; the other conditions on the torsion follow from suitable field redefinitions. We discuss also the H-constraints. We argue that there can be a choice of constraints such that the corresponding models can accommodate the effective actions coming from the superstring theories.