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An explicit duality for quasi-homogeneous ideals. (English) Zbl 1171.13006

Let \(m_1,\ldots, m_n\) be some positive integers and \(C=A[x_1,\ldots,x_n]\) a polynomial algebra over a commutative ring \(A\). Consider on \(C\) the grading given by \(\deg\;x_i=m_i\) and \(\deg\;u=0\) for all \(u\in A\). Let \(f_i=u_{i,t} x_1^{t_1}\cdots x_n^{t_n}\), \(u_{i,t}\in A\), \(1\leq i\leq r\) be quasi-homogeneous polynomial of degree \(d_i\), the sum being taken on \(t_1,\ldots t_n\geq 0\) with \(\sum_{i=1}^n t_i m_i=d_i\). Let \(I\) be the ideal generated by \(f_1,\ldots,f_r\) in \(C\), \( J\) the ideal generated by \(x_1,\ldots,x_n\) in \(C\) and \(B=C/I\). Then there exists a graded morphism \(\operatorname{Hom}_A^{gr}(B,A)\rightarrow H_J^0(H_{r-n}(f_1,\ldots,f_r;C))\), which is an isomorphism when \(depth_I\;C=n\). This duality can be seen as an extension of Corollary 3.6.14 of the author’s paper [Electron. J. Comb. 3, No. 2, Research paper R2, 91 p. (1996); printed version J. Comb. 3, No. 2, 35–125 (1996; Zbl 0863.13002)].

MSC:

13B25 Polynomials over commutative rings
13D25 Complexes (MSC2000)
13D45 Local cohomology and commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)

Citations:

Zbl 0863.13002
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References:

[1] Busé, L., 2007. On the equations of the moving curve ideal. Preprint available at http://hal.inria.fr/inria-00198350/en/; Busé, L., 2007. On the equations of the moving curve ideal. Preprint available at http://hal.inria.fr/inria-00198350/en/
[2] Jouanolou, J. P., Idéaux résultants, Adv. Math., 37, 3, 212-238 (1980) · Zbl 0527.13005
[3] Jouanolou, J. P., Aspects invariants de l’élimination, Adv. Math., 114, 1, 1-174 (1995) · Zbl 0882.13007
[4] Jouanolou, J.-P., Résultant anisotrope, compléments et applications, Electron. J. Combin., 3, 2 (1996), Research Paper 2, approx. 91 pp. (electronic), the Foata Festschrift · Zbl 0863.13002
[5] Weibel, C. A., (An Introduction to Homological Algebra. An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (1994), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0797.18001
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