zbMATH — the first resource for mathematics

Lifting monomial ideals. (English) Zbl 1003.13005
Let \(K\) be an infinite field and consider the polynomial rings \(S=K[X_1, \dots,X_n]\) and \(R=K[X_1, \dots,X_n,u_1,\dots,u_t]\). Let \(J=(m_1, \dots, m_r)\) be a monomial ideal in \(S\) and \(N\) an integer \(\geq\) any exponent of an indeterminate in \(m_1,\dots, m_r\). The authors consider, for \(1\leq j\leq n\) and \(1\leq i\leq N\), a linear form \(L_{ji}\in K[X_j,u_1, \dots,u_t]\) such that the coefficient of \(X_j\) in \(L_{ji}\) is non-zero and “lift” any monomial \(m= X_1^{a_1} \dots X_n^{a_n}\in S\) with \(a_i\leq N\), \(i=1,\dots,n\), to \(\widetilde m:= (L_{11} \cdot\dots \cdot L_{1a_1}) \cdot\dots \cdot(L_{n 1} \cdot\dots \cdot L_{na_n})\in R\). Let \(I=(\widetilde m_1,\dots, \widetilde m_r) \subset R\). The authors remark that D. Taylor’s resolution of \(S/J\) over \(S\) lifts naturally to a complex of graded free \(R\)-modules and deduce that \(I\) is a \(t\)-lifting of \(J\), which means that \(u_1,\dots,u_t\) is an \(R/I\)-regular sequence and \((I,u_1,\dots, u_t)/(u_1,\dots, u_t)\simeq J\), and that the graded Betti numbers of \(R/I\) over \(R\) are the same as the graded Betti numbers of \(S/J\) over \(S\). Making general choices of the linear forms \(L_{ji}\), they show that \(I\) is a radical ideal and, in fact, it is the ideal of a union of linear subspaces of \(\mathbb{P}^{n+ t-1}\) which intersect in a “nice way”. When \(S/J\) is Artinian, \(I\) is the ideal of an arithmetically Cohen-Macaulay generalized stick figure of dimension \(t-1\) in \(\mathbb{P}^{n+t-1}\).

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
Full Text: DOI
[1] Ballico E., Comm. Algebra 18 (9) pp 3015– (1990) · Zbl 0711.14031
[2] Bayer D., Source and object code available for Unix and Macintosh computers 18 (9) (1990)
[3] Bolondi, G. and Migliore, J. Configurations of Linear Protective Subvarieties. Algebraic Curves and Projective Geometry, Proceedings. 1988, Trento. Vol. 1389, pp.19–31. Springer-Verlag. Lecture Notes in Mathematics
[4] Bruns W., Cohen-Macaulay Rings (1993)
[5] Buchsbaum, D. and Eisenbud, D. Lifting modules and a theorem on finite free resolutions. Ring theory Proc. Conf. 1971, Park City, Utah. pp.63–74. New York: Academic Press. · Zbl 0248.13011
[6] Buchsbaum D., J. Algebra 25 pp 259– (1973) · Zbl 0264.13007
[7] Carr Ferro G., J. Pure Appl. Algebra 68 pp 279– (1990) · Zbl 0749.13018
[8] Chiantini, L. and Orecchia, F. Plane Sections of Arithmetically Normal Curves in P3. Algebraic Curves and Projective Geometry Proceedings. 1988, Trento. Vol. 1389, pp.32–42. Springer-Verlag. Lecture Notes in Mathematics
[9] Eisenbud D., Commutative Algebra with a View toward Algebraic Geometry 150 (1995) · Zbl 0819.13001
[10] Geramita A.V., J.Pure Appl. Algebra 40 pp 33– (1986) · Zbl 0586.13015
[11] Geramita A.V., Adv. Math 40 (1986)
[12] Geramita A.V., An Alternative to the Hilbert Function for the Ideal of a Finite Set of Points in Pn (1998)
[13] Geramita A.V., Comm. Algebra 26 pp 4285– (1998) · Zbl 0924.13016
[14] Geramita A.V., J. London Math. Soc 28 pp 443– (1983) · Zbl 0535.13012
[15] Geramita A.V., J. Pure Appl. Algebra 122 (1997) · Zbl 0905.13004
[16] Geramita A.V., in J. Algebra 122 (1997)
[17] Harima T., J. Pure Appl. Algebra 103 pp 313– (1995) · Zbl 0847.13003
[18] Hartshorne R., Math. Inst. des Hautes Etudes Sci 29 pp 261– (1966)
[19] Hartshorne R., Mem. Amer. Math. Soc 130 (617) (1997)
[20] Herzog J., J. Math. Kyoto Univ 34 (1) pp 47– (1994)
[21] Migliore J., Introduction to Liaison Theory and Deficiency Modules 165 (1998) · Zbl 0921.14033
[22] Reid, L. and Roberts, L. Intersection points of seminormal configurations of lines. AlgebraicK-theory, commutative algebra, and algebraic geometry. Vol. 151-163, Providence, RI: Santa Margherita Ligure. Contemp. Math., 126, Amer. Math. Soc · Zbl 0783.13017
[23] Schwartau P., Ph.D. thesis (1982)
[24] Stanley R., Adv. Math 28 pp 57– (1978) · Zbl 0384.13012
[25] Taylor D., Ideals generated by monomials in an R-sequence (1960)
[26] Walter C., Comm. Algebra 22 (13) pp 5167– (1994) · Zbl 0814.14034
[27] Walter C., Proc. Amer. Math. Soe 123 (9) pp 2651– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.