Abstract. The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . . , xn] and a finitely generated graded S-module, the Hilbert coefficients ei(M/I k M ) are polynomial functions. Given two families of graded ideals (I k ) k≥0 and (J k ) k≥0 with J k ⊂ I k for all k with the property that J k J ℓ ⊂ J k+ℓ and I k I ℓ ⊂ I k+ℓ for all k and ℓ, and such that the algebras A = L k≥0 J k and B = L k≥0 I k are finitely generated, we show the function k →0 (I k /J k ) is of quasipolynomial type, say given by the polynomials P0, . . . , Pg−1.we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that lim k→∞ ℓ(Γm (S/I k ))/k n ∈ Q. We also study analogous statements in the local case.

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Let R be a commutative Noetherian local ring with residue field k. We show that if a finite direct sum of syzygy modules of k surjects onto 'a semidualizing module' or 'a non-zero maximal Cohen-Macaulay module of finite injective dimension', then R is regular. We also prove that R is regular if and only if some syzygy module of k has a non-zero direct summand of finite injective dimension.2010 Mathematics Subject Classification. Primary 13D02; Secondary 13D05, 13H05.

Abstract. Let (R, m) be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI * , G, G * or CM. We prove that, if for a finite R-module M of positive depth, H-dim R (m i M ) is finite for some i ≥ reg(M ), then the ring R has property H.

Abstract. For a finitely generated, non-free module M over a CM local ring (R, m, k), it is proved that for n 0 the length of Tor R 1 (M, R/m n+1 ) is given by a polynomial of degree dim R − 1. The vanishing of Tor R i (M, N/m n+1 N ) is studied, with a view towards answering the question: If there exists a finitely generated R-module N with dim N ≥ 1 such that the projective dimension or the injective dimension of N/m n+1 N is finite, then is R regular? Upper bounds are provided for n beyond which the question has an affirmative answer.