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  We find a place in the minimal free resolution of a module of finite CIdimension after which asymptotically stable behavior develops; in particular, beyond this place the sequence of Betti numbers is either constant or strictly increasing, and gaps between consecutive numbers grow polynomially. 
  CastelnuovoMumford regularity is an important measure of the complexity of the differential in a graded minimal free resolution. 
  Vanishing of the regularity of the residue field defines the class of Koszul algebras, which have received considerable attention due to their extraordinary homological properties and to their appearance in many cases of interest in algebraic geometry, algebraic topology, combinatorics, commutative algebra, and representation theory. 
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