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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 \(\mathbb{P}^{n+t-1}\).

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
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