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

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.


13F20 Polynomial rings and ideals; rings of integer-valued polynomials
20B27 Infinite automorphism groups
13B10 Morphisms of commutative rings
Full Text: DOI Numdam EuDML


[1] [1] , Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik D1, Vieweg-Verlag, 1985. · Zbl 0669.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.