INTRODUCTION
7
This is illustrated by the following example. Let
R = C[xux2, x3]/(x\+xl + xl) = 0{X)
be the coordinate ring of the cubic cone. Then a = Z/3Z acts on R by the rule
a : (xi,X2,x3) —» ( a ? 2 ^ , ^ ^ , ^ ^ ) , where a; is the cube root of unity. By [Wa, §4.6],
is a 2-dimensional, normal ring with an isolated rational singularity at the origin.
Moreover, (#1,22, £3) is the unique prime ideal of R that is fixed by Z/3Z . Using the
techniques of, for example [Lei], it follows easily that V(R)Z/3Z £ V(RZ^Z). Thus,
combining [BGG] with standard results on fixed rings (see, in particular [Mo, Corollaries
1.12 and 2.6]), one obtains that V(RZ'3Z) is neither Noetherian nor simple. Further
details may be found in [Le3].
This raises the following question:
0.13.1. Suppose that Z has rational singularities. Then what other hypotheses are
required to ensure that the algebra T(Z) is finitely generated or simple or
Noetherian?
Some related questions are given in [CS]. As a particular special case of (0.13.1) and as
a generalisation of the commutative theory:
0.13.2. Let G be a reductive, algebraic group acting linearly on C n and set 0(Cn)G
O(Z). Then does T(Z) satisfy the properties listed in (0.13.1)?
Finally:
0.13.3. When is O(Z) a simple V(Z) -module? For example, this is the case if Z
satisfies the hypotheses of (0.13.2) (see I, Proposition 3.5).
It is easy to see that the simplicity of V{Z) forces v(Z)@(Z) to be simple. However,
the converse is false. An example is given by the subring
O(Z) = C + xC[x, y] 4- 2/2C[x, y]
of the polynomial ring C[z, y]. This example was found jointly with M. Chamarie. The
details are left to the reader.
0.14. Much of this research was conducted while the second author was visiting the
University of Paris VI in November 1986, and he would like to thank the department there
for its hospitality and financial support. This work was first reported at the conference
in honour of I.N. Herstein in Chicago in March 1987.
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