The differential operator ring of an affine curve
The differential operator ring of an affine curve
复制标题
仿射曲线的微分算子环
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
Jerry L. Muhasky
中科院分区：
文献类型：
--
作者：
Jerry L. Muhasky
本文的目的是研究特征为零的域k（不一定代数闭）上仿射代数簇X（可能可约）的坐标环上的线性微分算子环D（R）的结构，主要研究dim X < 1的情形.在这种情况下，证明了D（R）是（左和右）Krull维数等于dim X的（左和右）Noether环，任何（左或右）D（R）-模的自同态环是k上有限维的，D（R）有唯一最小理想L本质为左或右理想，D（R）/L是k上有限维的.下面的环理论工具被开发用于导出上述结果。设D是左Noether k-代数E的一个子代数，使得E作为左D-模生成，且所有的单左E-模都有有限维自同态环（在k上），并设D包含E的一个左理想I，使得E/I有有限长.然后证明了D是左Noether模，且任意单左D-模的自同态环是k上有限维的。导论.本文研究交换k-代数R上的k-线性微分算子环D（R），其中k是特征为零的域。特别感兴趣的是R是仿射代数簇X的坐标环的情况。当X是非奇异的时，环D（R）已经被广泛地研究，并享有许多好的性质;例如，D（R）是诺特环。(We将使用术语“诺特”来表示环是左诺特和右诺特的。当X是单数时，D（R）不必是诺特的，如J. N.伯恩斯坦岛M. Gelfand和S. I. Gelfand [3]：如果X是正常的三次锥，即，复三维空间中的曲面由x3 + y3 + Z3 = 0给出，则D（R）既不是左Noether曲面也不是右Noether曲面。因此，一个主要的目标是发现哪些变种X的环D（R）是诺特。本文的主要贡献是证明了当dim X < 1时D（R）是Noether的，并发展了在这种情况下D（R）的一些结构。本文的结构如下。? 1包含了交换环上微分算子的一些基本结果。在哪？2、开发了证明D（R）是诺特的代数工具。这一结果与J.C. Robson和L. W.小[11] ??图3和图4包含了当dim X < 1时D（R）的结构的主要结果。这项工作的动机是计算的I。M. Musson [10].这些结果是由S.史密斯和J. T。斯塔福德[12]的情况下，X是一个不可约曲线上的代数闭域的特征零。? 5包含一个Krull维数为1的非约化k-代数R的例子，使得D（R）是右但不是左诺特。编辑于1986年5月22日收到，修订版于1987年4月13日收到。1980年数学学科分类（1985年修订）。小学16 A33;中学13 B10。? 1988年美国数学学会0002-9947/88 $1.00 + $.25每页
The purpose of this paper is to investigate the structure of the ring D(R) of all linear differential operators on the coordinate ring of an affine algebraic variety X (possibly reducible) over a field k (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim X < 1. In this case, it is proved that D(R) is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim X, that the endomorphism ring of any simple (left or right) D(R)-module is finite dimensional over k, that D(R) has a unique smallest ideal L essential as a left or right ideal, and that D(R)/L is finite dimensional over k. The following ring-theoretic tool is developed for use in deriving the above results. Let D be a subalgebra of a left noetherian k-algebra E such that E is finitely generated as a left Dmodule and all simple left E-modules have finite dimensional endomorphism rings (over k), and assume that D contains a left ideal I of E such that E/I has finite length. Then it is proved that D is left noetherian and that the endomorphism ring of any simple left D-module is finite dimensional over k. Introduction. In this paper, we will study the ring D(R) of k-linear differential operators on a commutative k-algebra R, where k is a field of characteristic zero. Of special interest is the case where R is the coordinate ring of an affine algebraic variety X. When X is nonsingular, the ring D(R) has been extensively studied and enjoys many nice properties; for example, D(R) is noetherian. (We will use the term "noetherian" to indicate that a ring is both left and right noetherian.) When X is singular, D(R) need not be noetherian, as shown by J. N. Bernstein, I. M. Gelfand and S. I. Gelfand [3]: if X is the normal cubic cone, i.e., the surface in complex 3-space given by x3 + y3 + Z3 = 0, then D(R) is neither left nor right noetherian. Thus a major goal is to discover for which varieties X the ring D(R) is noetherian. The main contribution of this paper is to prove that D(R) is noetherian when dim X < 1, and to develop some of the structure of D(R) in this case. The paper is organized as follows. ?1 contains a number of basic results about the differential operators on commutative rings. In ?2, the algebraic tool used in proving D(R) is noetherian is developed. This result overlaps with the independent work of J. C. Robson and L. W. Small [11]. ??3 and 4 contain the main results on the structure of D(R) when dim X < 1. This work was motivated by the calculations of I. M. Musson [10]. These results were independently obtained by S. P. Smith and J. T. Stafford [12] in the case that X is an irreducible curve over an algebraically closed field of characteristic zero. ?5 contains an example of a nonreduced k-algebra R with Krull dimension one, such that D(R) is right but not left noetherian. Received by the editors May 22, 1986 and, in revised form, April 13, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A33; Secondary 13B10. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page