Divisorentheorie einer Halbgruppe

Skula, Ladislav

doi:10.1007/bf01110320

Cited by 31 publications

(25 citation statements)

References 2 publications

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“…This is the divisor theory for the semigroup Γ in the sense of [7,Ch. III] and [20]. Some problems on Cox rings can be reformulated and solved in terms of semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

Cox rings, semigroups and automorphisms of affine algebraic varieties

Arzhantsev¹,

Гайфуллин²

2010

Sb. Math.

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Abstract. We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of a non-degenerate affine toric variety of dimension ≥ 2 has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to Anick's automorphism of the polynomial algebra in four variables.

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“…This is the divisor theory for the semigroup Γ in the sense of [7,Ch. III] and [20]. Some problems on Cox rings can be reformulated and solved in terms of semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

Cox rings, semigroups and automorphisms of affine algebraic varieties

Arzhantsev¹,

Гайфуллин²

2010

Sb. Math.

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“…For po-groups with quasi divisor theory this divisor class group has special importance, since the density condition of h can be substitute by some density condition of ϕ h (see [12,16]). In the following proposition, we show that even for po-groups with GCD this short exact sequence has some density property.…”

Section: Then If G Admits a Gcd Theory The Same Is True For Gmentioning

confidence: 99%

“…After then this axiomatic concept was firstly transformed onto semigroups with cancellation by Skula [16,17] and then transformed and generalized onto po-groups and monoids by several authors (see [2,6,7,8,13,14]). Nevertheless in all these transformations and generalizations the principal structure of this axiomatic concept remains the same and for po-groups it could be expressed as follows.…”

Section: Introductionmentioning

confidence: 99%

“…(1.4) In po-groups and integral domains theory these conditions (1.1), (1.2), (1.3), and (1.4) characterize po-groups with various types of divisor theory or integral domains which are generalizations of Krull domains (see [2,7,13,16]). For example, if Γ is required to be a free abelian group Z (P ) for some set P then the conditions (1.1), (1.2), and (1.3) are equivalent and h is then called the theory of divisors of G. For general l-group Γ the condition (1.1) characterizes po-groups with the so-called theory of quasi divisors (this notion was introduced by Aubert [2], although Jaffard investigated properties of such groups more early [9]), and the condition (1.4) leads to po-groups with the strong theory of quasi divisors (see [14]).…”

Section: Introductionmentioning

confidence: 99%

“…For example, if Γ is required to be a free abelian group Z (P ) for some set P then the conditions (1.1), (1.2), and (1.3) are equivalent and h is then called the theory of divisors of G. For general l-group Γ the condition (1.1) characterizes po-groups with the so-called theory of quasi divisors (this notion was introduced by Aubert [2], although Jaffard investigated properties of such groups more early [9]), and the condition (1.4) leads to po-groups with the strong theory of quasi divisors (see [14]). (This last axiom was originaly formulated by Arnold [1] and later used by Skula [16].) Although the conditions (1.2) and (1.3) were not investigated for general l-group separately, Aubert [2] mistakes (1.1) and (1.3) (and, hence (1.4)) for equivalent.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ordered groups with greatest common divisors theory

Močkoř

2000

International Journal of Mathematics and Mathematical Sciences

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Abstract. An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G / / Γ such that Γ is an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up to o-isomorphisms) and that a po-group admits the complete GCD theory if and only if any v-ideal is v-invertible.

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On v-domains: a survey

Fontana

Zafrullah

2010

Commutative Algebra

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The v-domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of v-domains along with some historical remarks. We also discuss the relationship of v-domains with their various specializations and generalizations, giving suitable examples.

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Divisorentheorie einer Halbgruppe

Cited by 31 publications

References 2 publications

Cox rings, semigroups and automorphisms of affine algebraic varieties

Cox rings, semigroups and automorphisms of affine algebraic varieties

Ordered groups with greatest common divisors theory

On v-domains: a survey

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