The canonical module of a ring of invariants. (English) Zbl 0693.14005
Invariant theory, Proc. AMS Spec. Sess., Denton/Tex. 1986, Contemp. Math. 88, 43-83 (1989).
[For the entire collection see Zbl 0665.00004.]
An interesting overview is given of what is known about the canonical module of the ring \(A=R^ G\) of invariants of a linearly reductive group G acting linearly on the polynomial ring \(R=K[X_ 1,...,X_ n]\), K an algebraically closed field. The extensive first section on background material makes the paper almost self-contained. The central theme is the \(conjecture\quad G\) which says that, with the notations above and assuming that A is Cohen-Macaulay, A is Gorenstein if the action of G factors through SL(V), where V denotes the space \(RX_ 1+...+RX_ n\), or, in other words, if the determinant of any \(g\in G\) acting on V is the trivial character. Cases where the conjecture is known to be true include finite G such that the order of G is invertible in K, or G an algebraic torus, or a connected semi-simple group over a field of characteristic zero. Results of K. Watanabe and of R. Stanley, based on Molien’s formula (for a finite group G) for the Hilbert-Poincaré series are discussed. The condition for A to be Gorenstein can be translated in terms of the canonical module \(\omega_ A\) and this fact implies a functional equation for the Hilbert-Poincaré series of A if and only if A is Gorenstein.
In the third section the canonical module \(\omega_ A\), \(A=R^ G\), is calculated under the assumption that \(conjecture\quad G\) holds. A useful, auxiliary action of G on R[z], z a new indeterminate, is introduced by letting \(g\in G\) act on z by multiplication with the inverse of its determinant and \(\omega_ A\) turns out to be the module of semi- invariants for the inverse of the determinant for many good actions of G, e.g. when for the auxiliary action of G on R[z], \(R[z]^ G\) is Gorenstein. Explicit descriptions of \(\omega_ A\) are obtained for finite G, in which case one recovers Watanabe’s result, and for G an algebraic torus.
In the fourth section a special situation where the Gorenstein property of \(A=R^ G\) holds is discussed. The notion of excellent action of the connected linear algebraic (not necessarily reductive) group G on the ring R is defined (in terms of characters) and the main result which says that for an excellent action and under an additional condition on the normalizer of G in GL(V), \(A=R^ G\) (assumed to be Cohen-Macaulay) is Gorenstein.
The last section treats several examples in some detail, in particular, certain determinantal rings are studied.
In a footnote on the first page the author remarks that he recently had become aware of a result of F. Knop that says that \(conjecture\quad G\) is not true in general.
An interesting overview is given of what is known about the canonical module of the ring \(A=R^ G\) of invariants of a linearly reductive group G acting linearly on the polynomial ring \(R=K[X_ 1,...,X_ n]\), K an algebraically closed field. The extensive first section on background material makes the paper almost self-contained. The central theme is the \(conjecture\quad G\) which says that, with the notations above and assuming that A is Cohen-Macaulay, A is Gorenstein if the action of G factors through SL(V), where V denotes the space \(RX_ 1+...+RX_ n\), or, in other words, if the determinant of any \(g\in G\) acting on V is the trivial character. Cases where the conjecture is known to be true include finite G such that the order of G is invertible in K, or G an algebraic torus, or a connected semi-simple group over a field of characteristic zero. Results of K. Watanabe and of R. Stanley, based on Molien’s formula (for a finite group G) for the Hilbert-Poincaré series are discussed. The condition for A to be Gorenstein can be translated in terms of the canonical module \(\omega_ A\) and this fact implies a functional equation for the Hilbert-Poincaré series of A if and only if A is Gorenstein.
In the third section the canonical module \(\omega_ A\), \(A=R^ G\), is calculated under the assumption that \(conjecture\quad G\) holds. A useful, auxiliary action of G on R[z], z a new indeterminate, is introduced by letting \(g\in G\) act on z by multiplication with the inverse of its determinant and \(\omega_ A\) turns out to be the module of semi- invariants for the inverse of the determinant for many good actions of G, e.g. when for the auxiliary action of G on R[z], \(R[z]^ G\) is Gorenstein. Explicit descriptions of \(\omega_ A\) are obtained for finite G, in which case one recovers Watanabe’s result, and for G an algebraic torus.
In the fourth section a special situation where the Gorenstein property of \(A=R^ G\) holds is discussed. The notion of excellent action of the connected linear algebraic (not necessarily reductive) group G on the ring R is defined (in terms of characters) and the main result which says that for an excellent action and under an additional condition on the normalizer of G in GL(V), \(A=R^ G\) (assumed to be Cohen-Macaulay) is Gorenstein.
The last section treats several examples in some detail, in particular, certain determinantal rings are studied.
In a footnote on the first page the author remarks that he recently had become aware of a result of F. Knop that says that \(conjecture\quad G\) is not true in general.
Reviewer: W.W.J.Hulsbergen
MSC:
14L24 | Geometric invariant theory |
13D99 | Homological methods in commutative ring theory |
14L30 | Group actions on varieties or schemes (quotients) |