Rings of differential operators on invariant rings of tori
Rings of differential operators on invariant rings of tori
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环面不变环上的微分算子环
DOI：
10.1090/s0002-9947-1987-0902799-2
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发表时间：
1987
影响因子：
1.3
通讯作者：
I. Musson
中科院分区：
文献类型：
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作者：
I. Musson
设k为特征为零的代数闭场，G为对角作用于k′的环面。对于一个子集，8 (s) ={1,2，…， s}, set Up = {{0}, {0}, {0}；然后G作用于正则函数环d（Up）上，研究不变环上所有微分算子的环d（d(Up)G）。更一般地假设/\是s的子集的集合，使得每个不变环d(U,@)G,，B E /\具有相同的商域。我们证明了n:G D（D(Up)G）是Noetherian的有限生成k代数。现在G作用于每个D(D(Up))，并且通过微分算子的限制得到一个自然映射0 n D(D(UB)) n D(D(Up)G) = D（Ya/G）。我们找到了0是合乎的充分必要条件，并描述了#的核。代数n: gdd™>(Up))G和n: gaa™（™>(Up)G）带有由微分算子阶给出的自然过滤。我们证明了相关的分级环是有限生成的交换代数，是Gorensetin环。我们还确定了n:GA D（¢>(Up5)G）和n:GA D（D(U)G）的中心。在本文中，k是一个特征为零的代数闭域。如果K是一个可交换K代数，我们用Do(K)表示K -线性映射的集合K K，如果>1是一个K -线性映射f：K > K属于Dp (K)提供定义的映射[f r] [f r] (s) = f (rs)射频(s) s E K属于Dp_l所有E r K (K)集D (K) = UpaoDp (K)形式的子环Endk (K)称为微分算子的环在K .我们认为K是一个左D (K)模块,r = f (r)为所有f E D (K), r E K的微分算子环有限群的不变环作用于一个多项式环一直在研究(Ka, L1和L2)。我们对环面的不变环进行了类似的研究。我们假设环面G作为一组对角矩阵作用于k。如果jB是s ={1,2，…，s}，则G作用于UA = {u E kSlUJ + O如果j E jB}。G也作用于(p(U)通过（gf）(U) = f（G -1u）对于G E G， f E (G)(U/3), U E U/3。更一般地假设，对于s的子集A，我们有Y = UE UA，其中每个不变环（p(U)c）具有相同的商域f，则每个D(（p(U)6）是A，由编辑于1986年6月18日和1986年12月11日修改。1980 Mclthemcltiss学科分类法（1985修订版）。主要16 a33;二级13N05, 14L30。P ?1987年美国数学学会
Let k be an algebraically closed field of characteristic zero and G a torus acting diagonally on k '. For a subset ,8 of s = {1, 2, . . ., s }, set Up = { u E k ' | u, + O if j E ,8 }. Then G acts on d(Up ), the ring of regular functions on Up, and we study the ring D(d(Up)G) of all differential operators on the invariantvring. More generally suppose that /\ is a set of subsets of s, such that each invariant ring d(U,@)G, ,B E /\, has the same quotient field. We prove thatn:G D(d(Up)G) is Noetherian and finitely generated as a k-algebra. Now G acts on each D(d(Up)) and there is a natural map 0 n D(d(UB)) n D(d(Up)G) = D(Ya/G) obtained by restriction of the differential operators. We find necessary and sufficient conditions for 0 to be suUective and describe the kernel of #. The algebras n: G D ¢>(Up))G and n: G A D(¢>(Up)G) carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of n:GA D(¢>(Up5)G and n:GA D(d(U)G) Introduction. Throughout this paper k will be an algebraically closed field of characteristic zero. If K is a commutative k-algebra we denote by Do(K) the set of k-linear maps K K and if > 1 a k-linear map f: K > K belongs to Dp(K) provided the map [f, r] defined by [f, r](s) = f(rs)-rf(s) for s E K belongs to Dp_l(K) for all r E K. The set D(K) = UpaoDp(K) forms a subring of Endk(K) called the ring of differential operators on K. We consider K as a left D(K)-module where f r = f(r) for all f E D(K), r E K. The ring of differential operators on the invariant ring of a finite group acting on a polynomial ring has been studied in [Ka, L1 and L2]. We carry out a similar study for the invariant ring of a torus. We assume that the torus G acts on kS as a group of diagonal matrices. If jB is a subset of s = {1,2,...,s}, then G acts on UA = {u E kSlUJ + O if j E jB}. Also G acts on (p(U) via (gf)(u) = f(g-1u) for g E G, f E (g)(U/3), u E U/3. More generally suppose that for some set of subsets A of s we have Y = UE UA, where each invariant ring (p(U)c has the same quotient field F. Then each D((p(U)6) is a Received by the editors June 18, 1986 and, in revised form, December 11, 1986. 1980 Mclthemcltiss Subject Clclssificcltion (1985 Revision). Primary 16A33; Secondary 13N05, 14L30. P?1987 American Mathematical S