Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Let X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Archimedean unital groups with finite unit intervals

LetGbe a unital group with a finite unit intervalE, letnbe the number of atoms inE, and letκbe the number of extreme points of the state spaceΩ(G). We introduce canonical order-preserving group homomorphismsξ:ℤn→Gandρ:G→ℤκlinkingGwith the simplicial groupsℤnandℤκ.We show thatξis a surjection andρis an injection if and only ifGis torsion-free. We give an explicit construction of the universal group (unigroup) forEusing the canonical surjectionξ. IfGis torsion-free, then the canonical injectionρis used to show thatGis Archimedean if and only if its positive cone is determined by a finite number of homogeneous linear inequalities with integer coefficients.

Dunford-Pettis and strongly Dunford-Pettis operators on L1(μ)

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.