The main theorem of the paper produces a graded minimal free resolution of \(I^s,\;s\in \mathbb{N}\), where \(I\) is the homogeneous ideal of a complete intersection of type \((d_1,\dots,d_r)\) in \(\mathbb{P}^n\). Such a resolution allows: an explicit description of the graded Betti numbers of \(I^s\) in terms of \(s\) and \((d_1,\dots,d_r)\); the computation of the Hilbert function \(H_{R/I^s}\), starting from \(H_{R/I}\). The authors point out that a minimal free resolution of \(I^s\) can also be obtained by means of the Eagon-Northcott complex and that a graded one was produced by Buchsbaum and Eisenbud and used by Srinivasan; the technique of the present paper is more elementary and the description of Betti numbers more explicit.
The last part of the paper consists in some applications. A Herzog-Huneke-Srinivasan conjecture is proved (in this special case) linking the multiplicity of \(R/I^s\) to the shifts of its graded minimal free resolution. A set of fat points, all with the same multiplicity \(m\), having as a support a complete intersection, is an example of scheme having as associated ideal \(I^m\), where \(I\) is a complete intersection; more generally, if \(X\) is a complete intersection that “splits” into smaller complete intersections and \(Y\) is a corresponding set of fat points minus one point, some bounds are produced on the regularity index and on the least degree of a form in its ideal.