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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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