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We introduce the relative units-Picard complex of an arbitrary morphism ofschemes and apply it to the problem of describing the (cohomological) Brauergroup of a (fiber) product of schemes in terms of the Brauer groups of thefactors. Under certain hypotheses, we obtain a five-term exact sequenceinvolving the preceding groups which enables us to solve the indicated problemin the case of, e.g., (certain types of) ruled varieties over a field ofcharacteristic zero.

The relative units-Picard complex and the Brauer group of a product