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an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0\n  --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby present the Artin-Rees number as a proper replacement of regularity in the local situation,\n  --> the ring R/I^k is Golod, its Poincer{\\'e}-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree.\n  The first result is an extension of work of Kodiyalam","authors_text":"J\\\"urgen Herzog, Siamak Yassemi, Volkmar Welker","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-08-30T08:04:04Z","title":"Homology of powers of ideals: Artin--Rees numbers of syzygies and the Golod 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