Commutative Algebra, Singularities and Computer Algebra: by Ciprian S. Borcea (auth.), Jürgen Herzog, Victor Vuletescu

By Ciprian S. Borcea (auth.), Jürgen Herzog, Victor Vuletescu (eds.)

Commutative Algebra, Singularities and desktop Algebra provides present tendencies in commutative algebra, algebraic combinatorics, singularity thought and machine algebra, and highlights the interplay among those disciplines. Contributions by means of prime overseas mathematicians completely talk about themes in: modules conception, integrally closed beliefs and determinantal beliefs, singularities in projective areas and Castelnuovo-Mumford regularity, Groebner and SAGBI foundation, and using the pc programs Bergman, CoCoA and SINGULAR.

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Extra info for Commutative Algebra, Singularities and Computer Algebra: Proceedings of the NATO Advanced Research Workshop on Commutative Algebra, Singularities and Computer Algebra Sinaia, Romania 17–22 September 2002

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Then J /1 is the graded canonical module (up to the same shift). Proof For the sake of simplicity and since it is sufficient for our applications, we restrict ourselves to the graded case in (b) and 1 = O. Since A is a CohenMacaulay positively graded K -algebra which is a domain, to prove that J is the canonical module of A it suffices to show that J is a maximal Cohen-Macaulay module whose Hilbert series satisfies the relation HJ( t) = (-1 )dtkHA(I-I) for some integer k and d = dimA [13, Thm. 5, Cor.

9]. It remains to consider normality. We show that AI has the Serre properties (R I) and (S2) if these hold for AI /(l) . Let P be a prime ideal of A' with heightp ::; 1. If t E p, then p = p/(t) is a minimal prime ideal of AI/(l}, and the regularity of (N/(t)}jj = A~ /(t) implies that of AI. If t ft p, we choose a minimal prime overideal q of p + (t). Since AI is an integral domain and Ii localization of an 30 affine K*algebra, we must have height q = height I' + 1. Moreover, heightq /(t) = height q - 1 = heightI' .

C) Let b be a positive weight veclOr, and suppose that A and I are b-graded. Then A/ I and in(A)/in(l) have the same Hilbertfunction. (d) If, in addition lO the hypothesis o/(c), in(A}/ in(l) is Koszul with respect to e-normalized b-degree/or some e, then so is A/I. 15, we may represent them as initial objects with respect to a suitable positive integral weight vector. 9 there exist a polynomial ring, say B, an ideal H , and a positive weight c such that B/H ~ A/ I and B/ inc(H } 2:': in(A) / in(l).

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