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Critical cones of characteristic varieties. (English) Zbl 1298.13015

Let \(W\) be the \(n\)th Weyl algebra over a field \(K\) of characteristic \(0\) and let \(\Omega=\{\omega\in {\mathbb N}^{2n}_0\mid \omega_i+\omega_{i+n}>0\) for \(1\leqslant i\leqslant n\}\). Let \(M\) be a left \(W\)-module. For each \(\omega\in \Omega\) consider the \(\omega\)-degree filtration \(F^\omega W=(F^\omega_iW)_{i\in\mathbb Z}\) of \(W\) and any good \(F^\omega W\)-filtration \(F^\omega M= (F^\omega_i M)_{i\in \mathbb Z}\) of \(M\). Put \(G^\omega W=\bigoplus_{i\in \mathbb Z}F^\omega_iW/F^\omega_{i-1}W\) and \(G^\omega M=\bigoplus_{i\in \mathbb Z}F^\omega_iM/F^\omega_{i-1}M\). Then \(G^\omega W\) is a ring canonically isomorphic to the commutative polynomial ring \(K[X, Y]\) in the indeterminates \(X_1,\ldots, X_n\) and \(Y_1,\ldots, Y_n\), and \(G^\omega M\) is a finitely generated \(K[X, Y]\)-module. The main result of this paper is that for each \(\nu\in\mathbb N_0^{2n}\) there exists \(s_0\in \mathbb N_0\) such that for all \(s\in\mathbb N\) with \(s>s_0\) and all \(\omega\in \Omega\) in \(K[X, Y]\) it holds \(\surd(0: G^\nu G^\omega M)=\surd (0:G^{\nu+s\omega}M)\).

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13N10 Commutative rings of differential operators and their modules
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
16P90 Growth rate, Gelfand-Kirillov dimension
16W70 Filtered associative rings; filtrational and graded techniques
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References:

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