Reductive embeddings are Cohen-Macaulay. (English) Zbl 1071.14518
Summary: We prove that in positive characteristic normal embeddings of connected reductive groups are Frobenius split. As a consequence, they have rational singularities and are thus Cohen-Macaulay varieties.
As an application, we study the particular case of reductive monoids, which are affine embeddings of their unit group. In particular, we show that the algebra of regular functions of a normal irreducible reductive monoid \(M\) has a good filtration for the action of the unit group of \(M\).
As an application, we study the particular case of reductive monoids, which are affine embeddings of their unit group. In particular, we show that the algebra of regular functions of a normal irreducible reductive monoid \(M\) has a good filtration for the action of the unit group of \(M\).
MSC:
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14M17 | Homogeneous spaces and generalizations |
| 20M99 | Semigroups |
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