### Abstract

Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain property) of (semi-)invariant subrings under the action of affine algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field.

Let F be an affine flat group scheme over a commutative ring R, and S an F-algebra (an R-algebra on which F acts). We define an equivariant analogue Q _{ F}(S) of the total ring of fractions Q(S) of S. It is the largest F-algebra T such that S ⊂ T ⊂ Q(S), and S is an F-subalgebra of T. We study some basic properties.

Original language | English |
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Pages (from-to) | 1524-1562 |

Number of pages | 39 |

Journal | Communications in Algebra |

Volume | 43 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 3 2015 |

Externally published | Yes |

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### Keywords

- Character group
- Invariant subring
- UFD

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*43*(4), 1524-1562. https://doi.org/10.1080/00927872.2013.867967