Proceedings of the 44th Symposium on Ring Theory and ...

Proposition 4 (Paper III, Proposition 6.7 in [1], Proposition 4.14 in [6]). For any m ≥ 1,
D (m) (I) is a free left S-module with a basis
{ε m , P 1 δ m 1 , . . . , P m δ m m} if m < r − 1,
{P 1 δ m 1 , . . . , P r δ m r } if m = r − 1,
{P 1 δ m 1 , . . . , P r δ m r , Qη (m)
r+1, . . . , Qη (m)
m+1} if m > r − 1,
where <strong>the</strong> set {δ m 1 , . . . , δ m r , η (m)
r+1, . . . , η (m)
m+1} forms a K-basis for ∑ |α|=m K∂α if m > r − 1.
By Proposition 1, we have
( ⊕r−2
( ) )
D(I) = S ⊕ Sεm ⊕ SP 1 δ1 m ⊕ · · · ⊕ SP m δm
m
m=1
( ⊕ (
⊕ SP1 δ1 m ⊕ · · · ⊕ SP r δr m ⊕ SQη (m)
r+1 ⊕ · · · ⊕ SQη m+1) ) (m) .
m≥r−1
For i = 1, . . . , r, we define an additive group
L i := D(I) ∩ (p 1 · · · p i )D(S).
Proposition 5. For i = 1, . . . , r, <strong>the</strong> additive group L i is a two-sided ideal <strong>of</strong> D(I).
We consider a sequence
(3.1)
ID(S) = L r ⊆ L r−1 ⊆ · · · ⊆ L 1 ⊆ L 0 = D(I)
<strong>of</strong> two-sided ideals <strong>of</strong> D(I). If L i−1 /L i is a right Noe<strong>the</strong>rian D(I)-module for any i, <strong>the</strong>n
D(I)/ID(S) is a right Noe<strong>the</strong>rian ring. By proving that L i−1 /L i is right Noe<strong>the</strong>rian for
all i, we obtain <strong>the</strong> following main <strong>the</strong>orem.
Theorem 6. The ring D(S/I) ≃ D(I)/ID(S) <strong>of</strong> differential operators <strong>of</strong> <strong>the</strong> coordinate
ring <strong>of</strong> a central 2-arrangement is Noe<strong>the</strong>rian (i.e., D(S/I) is right Noe<strong>the</strong>rian and left
Noe<strong>the</strong>rian).
In contrast, Gr D(S/I) is not Noe<strong>the</strong>rian when r ≥ 2.
Remark 7. The graded ring Gr D(S/I) associated to <strong>the</strong> order filtration is a commutative
ring. We consider <strong>the</strong> ideal M := 〈P 1 δ1
m | m ≥ 1〉 <strong>of</strong> Gr D(S/I).
Assume that M is finitely generated with generators η 1 , . . . , η l . Then <strong>the</strong>re exists a
positive integer m such that
Since P 1 δ m 1
(3.2)
∈ M, we can write
M = 〈η 1 , . . . , η l 〉 ⊆ 〈P 1 δ 1 , . . . , P 1 δ m−1
1 〉.
P 1 δ m 1
= P 1 δ 1 · θ 1 + · · · + P 1 δ m−1
1 · θ m−1
for some θ 1 , . . . , θ m−1 ∈ D(I).
If θ ∈ D(I) with ord(θ) ≤ 1, <strong>the</strong>n <strong>the</strong> polynomial degree <strong>of</strong> θ is greater than or equal
to 1 by Proposition 4. Since <strong>the</strong> order <strong>of</strong> <strong>the</strong> LHS <strong>of</strong> (3.2) is m, <strong>the</strong>re exists at least one
θ j such that <strong>the</strong> order <strong>of</strong> θ j is greater than or equal to 1. Thus <strong>the</strong> polynomial degree <strong>of</strong>
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