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Let I = (F1, . . . , Fr) be a homogeneous ideal of the ring R = k(x0, . . . , xn) generated by a regular sequence of type (d1, . . . , dr). We give an elementary proof for an explicit description of the graded Betti numbers of I s for any s  1. These numbers depend only upon the type and s. We then use this description to: (1) write HR/Is, the Hilbert function of R/Is, in terms of HR/I; (2) verify that the k- algebra R/I s satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in P n whose support is a complete intersection or a complete

Powers of complete intersections: graded Betti numbers and applications