### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- Character group
- Invariant subring
- UFD

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants.** / Hashimoto, Mitsuyasu.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 43, no. 4, pp. 1524-1562. https://doi.org/10.1080/00927872.2013.867967

}

TY - JOUR

T1 - Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

AU - Hashimoto, Mitsuyasu

PY - 2015/4/3

Y1 - 2015/4/3

N2 - 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.

AB - 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.

KW - Character group

KW - Invariant subring

KW - UFD

UR - http://www.scopus.com/inward/record.url?scp=84923313285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923313285&partnerID=8YFLogxK

U2 - 10.1080/00927872.2013.867967

DO - 10.1080/00927872.2013.867967

M3 - Article

VL - 43

SP - 1524

EP - 1562

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 4

ER -