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RISS 인기검색어
- 검색키워드
- 저자 : Heinzer, William (검색결과 5 건)
- 무료
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- 유료
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Directed unions of local quadratic transforms of a regular local ring
Heinzer, W.,Kim, M.K.,Toeniskoetter, M. Academic Press 2017 Journal of algebra Vol.488 No.-
<P>Let (R, m) be a d-dimensional regular local domain with d >= 2 and let V be a valuation domain birationally dominating R such that the residue field of V is algebraic over R/m. Let v be a valuation associated to V. Associated to R and there exists an infinite directed family {(R-n,m(n))}n >= 0 of d- dimensional regular local rings dominated by V with R = R-0 and Rn+1 the local quadratic transform of R-n along V. Let S := boolean OR(n >= 0) R-n. Abhyankar proves that S = V if d = 2. Shannon oBserves that often S is properly contained in V if d >= 3, and Granja gives necessary and sufficient conditions for S to be equal to V. The directed family {(R-n,m(n))}(n >= 0) and the integral domain S = boolean OR(n >= 0) R-n may be defined without first prescribing a dominating valuation domain V. If {{(R-n,m(n))}(n >= 0) switches strongly infinitely often, then S = V is a rank one valuation domain and for nonzero elements f and g in m, we have v(f)/v(g) = lim(n ->infinity)ordR(n)(f)/ordR(n)(g). If {(R-n,m(n))}(n >= 0) is a family of monomial local quadratic transforms, we give necessary and sufficient conditions for {(R-n,m(n))}(n >= 0) to switch strongly infinitely often. If these conditions hold, then S = V is a rank one valuation domain of rational rank d and v is a monomial valuation. Assume that V is rank one and birationally dominates S. Let s = Sigma(infinity)(i=0) v(m(i)) Granja, Martinez and Rodriguez show that s = infinity implies S = V. We prove that s is finite if V has rational rank at least 2. In the case where V has maximal rational rank, we give a sharp upper bound for s and show that s attains this bound if and only if the sequence switches strongly infinitely often. (C) 2017 Elsevier Inc. All rights reserved.</P>
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Finitely supported @?-simple complete ideals in a regular local ring
Heinzer, W.,Kim, M.K.,Toeniskoetter, M. Academic Press 2014 Journal of algebra Vol.401 No.-
Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). A theorem of Lipman implies that I has a unique factorization as a @?-product of special @?-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if dimR≥3 because of the existence of finitely supported @?-simple ideals that are not special. We consider properties of special @?-simple complete ideals such as their Rees valuations and point basis. Let (R,m) be a d-dimensional equicharacteristic regular local ring with m=(x<SUB>1</SUB>,...,x<SUB>d</SUB>)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a @?-product of monomial ideals to be the @?-product of the complete inverse transforms of the factors. This yields examples of finitely supported @?-simple monomial ideals that are not special. We prove that a finitely supported @?-simple monomial ideal with linearly ordered base points is special @?-simple.
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The leading ideal of a complete intersection of height two, Part II
Goto, Shiro,Heinzer, William,Kim, Mee-Kyoung Elsevier 2007 Journal of algebra Vol.312 No.2
<P><B>Abstract</B></P><P>Let (S,n) be a regular local ring and let I=(f,g) be an ideal in <I>S</I> generated by a regular sequence f,g of length two. Let R=S/I and m=n/I. As in [S. Goto, W. Heinzer, M.-K. Kim, The leading ideal of a complete intersection of height two, J. Algebra 298 (2006) 238–247], we examine the leading form ideal <SUP>I∗</SUP> of <I>I</I> in the associated graded ring G=<SUB>grn</SUB>(S). If <SUB>grm</SUB>(R) is Cohen–Macaulay, we describe precisely the Hilbert series H(<SUB>grm</SUB>(R),λ) in terms of the degrees of homogeneous generators of <SUP>I∗</SUP> and of their successive GCD's. If D=GCD(<SUP>f∗</SUP>,<SUP>g∗</SUP>) is a prime element of <SUB>grn</SUB>(S) that is regular on <SUB>grn</SUB>(S)/(<SUP>f∗</SUP>D,<SUP>g∗</SUP>D), we prove that <SUP>I∗</SUP> is 3-generated and a perfect ideal. If <SUB>ht<SUB>grn</SUB>(S)</SUB>(<SUP>f∗</SUP>,<SUP>g∗</SUP>,<SUP>h∗</SUP>)=2, where h∈I is such that <SUP>h∗</SUP> is of minimal degree in <SUP>I∗</SUP>∖(<SUP>f∗</SUP>,<SUP>g∗</SUP>)<SUB>grn</SUB>(S), we prove <SUP>I∗</SUP> is 3-generated and a perfect ideal of <SUB>grn</SUB>(S), so <SUB>grm</SUB>(R)=<SUB>grn</SUB>(S)/<SUP>I∗</SUP> is a Cohen–Macaulay ring. We give several examples to illustrate our theorems.</P>
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The leading ideal of a complete intersection of height two
Goto, Shiro,Heinzer, William,Kim, Mee-Kyoung Elsevier 2006 Journal of algebra Vol.298 No.1
<P><B>Abstract</B></P><P>Let (S,n) be a Noetherian local ring and let I=(f,g) be an ideal in <I>S</I> generated by a regular sequence f,g of length two. Assume that the associated graded ring <SUB>grn</SUB>(S) of <I>S</I> with respect to n is a UFD. We examine generators of the leading form ideal <SUP>I∗</SUP> of <I>I</I> in <SUB>grn</SUB>(S) and prove that <SUP>I∗</SUP> is a perfect ideal of <SUB>grn</SUB>(S), if <SUP>I∗</SUP> is 3-generated. Thus, in this case, letting R=S/I and m=n/I, if <SUB>grn</SUB>(S) is Cohen–Macaulay, then <SUB>grm</SUB>(R)=<SUB>grn</SUB>(S)/<SUP>I∗</SUP> is Cohen–Macaulay. As an application, we prove that if (R,m) is a one-dimensional Gorenstein local ring of embedding dimension 3, then <SUB>grm</SUB>(R) is Cohen–Macaulay if the reduction number of m is at most 4.</P>
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