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On the Noether exponent. (English) Zbl 1023.13002
The main result of the paper is an very easy proof of an estimation of the Noether exponent of an ideal \(I\subset {\mathbf k}[X]= {\mathbf k}[X_1, \dots,X_n]\) without embedded components, where \({\mathbf k}\) is an algebraically closed field. Let us recall that the Noether exponent \(\mu_I\) of the ideal \(I\) is the smallest number \(\mu\in\mathbb{N}\) such that \((\text{rad} I)^\mu\subset I\). The estimation is given in the following:
Theorem 7. Let \(I=(f_1,\dots, f_k)\) be an ideal generated by polynomials \(f_j\in {\mathbf k}[X]\), where \(\deg f_2 \geq\cdots \geq\deg f_k\geq\deg f_1\). Assume that there is a primary decomposition \(I=\bigcap^m_{i=1} q_i\) without embedded components, where \(q_i\) are \(p_i\)-primary ideals. Set \(r_i:=\text{codim} V(q_i)\), and define \(d_t:=\min \{\deg V(q_i)\mid \text{codim} V(q_i)=t\}\) for \(t\in\{r_1, \dots,r_m\}\). Then \[ \mu_I \leq\max_{t\in\{r_1, \dots,r_m\}} \left\{{\deg f_1\cdot \dots\cdot \deg f_t \over d_t}\right\}. \] The main tools of the proof are “an affine version of the Bézout theorem” (theorem 6) and the reduction to the case of proper intersection (lemma 8). Some technical tricks given in lemma 9 and 10 are also useful.
MSC:
13A10 Radical theory on commutative rings (MSC2000)
14A05 Relevant commutative algebra
13C14 Cohen-Macaulay modules
14R99 Affine geometry
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