Stability of the cohomology rings of Hilbert schemes of points on surfaces

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We establish some remarkable properties of the cohomology rings of the Hilbert schemes of n points on a projective surface X, from which one sees to what extent these cohomology rings are (in)dependent of X and n.
v. 2, 19 pages, latex, title modified, streamlined first half, to appear in J. reine Angew. Math

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