Graded morphisms of \(G\)-modules. (English) Zbl 0818.13015

Kraft, Hanspeter; Procesi, Claudio

Ann. Inst. Fourier 37, No. 4, 161-166 (1987).

Conjecture (S. Halperin): If \(f_ 1, f_ 2, \dots, f_ n\) is a regular sequence in the polynomial ring in \(n\) indeterminantes \(\mathbb{C} [x_ 1, x_ 2, \dots, x_ n]\), the connected component of the automorphism group of the (finite dimensional) algebra \(\mathbb{C} [x_ 1, \dots, x_ n]/(f_ 1, \dots, f_ n)\) is solvable.
Definition: A morphism \(\varphi : V \to W\) between finite dimensional vector spaces \(V\) and \(W\) is called graded if there is a basis of \(W\) such that the components of \(\varphi\) are all homogeneous polynomials.
Main theorem: Let \(G\) be a connected reductive algebraic group and let \(V\), \(W\) be two \(G\)-modules. Assume that \(V\) and \(W\) do not contain one- dimensional submodules. Then any graded \(G\)-equivariant dominant morphism with finite fibres is a linear isomorphism.
Corollary: Let \(A\) be a finite dimensional local \(\mathbb{C}\)-algebra with maximal ideal \({\mathfrak m}\) and let \(gr_{\mathfrak m} A\) be the associated graded algebra (with respect to the \({\mathfrak m}\)-adic filtration). If \(gr_{\mathfrak m} A\) is a complete intersection then the connected component of the automorphism group of \(A\) is solvable.
Remark: The corollary above implies that the Halperin conjecture (see above) is true in case all \(f_ i\) are homogeneous, i.e., if the algebra \(A = \mathbb{C} [x_ 1, \dots, x_ n]/(f_ 1, \dots, f_ n)\) is finite dimensional and graded with all \(x_ i\) of degree 1.

MSC:

13F20	Polynomial rings and ideals; rings of integer-valued polynomials
20B27	Infinite automorphism groups
13B10	Morphisms of commutative rings

Keywords:

solvable automorphism group; graded morphism; polynomial ring

References:

[1]	[1] , Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik D1, Vieweg-Verlag, 1985. · Zbl 0669.14003