Study of blow-up rings

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Let I be an m-primary ideal in a Gorenstein local ring (A, m) with dim A = d and assume that I contains a parameter ideal Q in A as a reduction. Then we say that I is good ideal in A if G = 【symmetry】_n≧_0I^n/I^<n+1> is a Gorenstein ring with a(G) = 1 - d. The associated graded ring G of I is a Gorenstein ring with a(G) = -d if and only if I = Q.Therefore, good ideals in our sense are good ones next to the parameter ideals Q in A.A basic theory of good ideals is developed by this project. We have that I is a good ideal in A if and only if I^2= QI and I = Q : I.Firstly a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have the nonempty sets X_A of good ideals shall be given. Secondly in the case where d=1 we will give a correspondence theorem between the set X_A and the set Y_A of certain overrings of A.A characterization of good ideals of the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato's Rieman-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show the structure of the set X_A of good ideals in A heavily depends on d = dim A.The set X_A may be empty if d ≦ 2, while X_A is necessarily infinite if d ≧ 3. To analyze this phenomenon we shall lastly explore monomial good ideals in the polynomial ring k[X_1, X_2, X_3] in three variables over a field k. Let I be an ideal in a Gorenstein local ring A.Then I is said to be an equimultiple good ideal if I contains a reduction Q = (a_1, a_2, …, a_s) generated by s elements in A and if the associated graded ring G(I)=【symmetry】_n≧_0I^n/I^<n+1> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = ht_AI.The structure of the sets X_<A,s> (s ≧ 0) of equimultiple good ideals I with ht_AI = s. Some of the results in the case where s = dim A are successfully generalized to those of equimultiple case with improvements.