On the Picard Group: Torsion and the Kernel Induced by a Faithfully Flat Map

Guralnick, Robert M.; Jaffe, David B.; Raskind, Wayne; Wiegand, Roger

doi:10.1006/jabr.1996.0228

Cited by 11 publications

(10 citation statements)

References 32 publications

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“…Therefore L ′ ≃ O Y . It is known (see, e.g., [29], 2.1) that since Y → S is finite and flat, the kernel of the induced map Pic(S) → Pic(Y ) is killed by d. Indeed, choose an invertible sheaf L over S, and consider the scheme X := P(O S ⊕ L), with the associated smooth projective morphism π : X → S. Let C 0 and C ∞ be as in 7.10, and let C := C 0 + C ∞ . Let F := Supp(C).…”

Section: 8mentioning

confidence: 99%

Hypersurfaces in projective schemes and a moving lemma

Gabber¹,

Liu²,

Lorenzini³

2015

Duke Math. J.

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Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if R is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective and flat over S. We also show the existence of a finite surjective S-morphism to the projective space P_S^d for any scheme X projective over S when X/S has all its fibers of a fixed dimension d.Comment: 64 page

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Section: 8mentioning

confidence: 99%

Hypersurfaces in projective schemes and a moving lemma

Gabber¹,

Liu²,

Lorenzini³

2015

Duke Math. J.

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“…By a variant of a result of Roquette (see [7], 1.6) the group B * /k * is finitely generated, so F = {f ∈ B * : f (y 0 ) = 1} is countable. For each f ∈ F , let…”

Section: Functorial Structure Of Units In a Tensor Productmentioning

confidence: 99%

“…, it follows by a version of Roquette's Theorem ( [7], 1.6) that D A (k) is free abelian of finite rank. In fact, D A is represented by the corresponding constant group scheme.…”

Section: Remarks (I) In Part (A) Of the Theorem One Can Choose ( †)mentioning

confidence: 99%

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Functorial structure of units in a tensor product

Jaffe

1996

Trans. Amer. Math. Soc.

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Abstract. The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let k be an algebraically closed field. Let A and B be reduced rings containing k, having connected spectra. Let u ∈ A ⊗ k B be a unit. Then u = a ⊗ b for some units a ∈ A and b ∈ B.Here is a deeper consequence, stated for simplicity in the affine case only. Let k be a field, and let ϕ : R → S be a homomorphism of finitely generated k-algebras such that Spec(ϕ) is dominant. Assume that every irreducible component of Spec(R red ) or Spec(S red ) is geometrically integral and has a rational point. Let B → C be a faithfully flat homomorphism of reduced k-algebras.Then Q satisfies the following sheaf property: the sequenceis exact. This and another result are used to prove (5.2) of [7].

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“…The problem of divisibility in the Picard groups (including Picard groups of arithmetic curves) has been quite vivid in recent years and has drawn attention of numerous authors. We may cite for example [3,4,6,8]. The list is definitely very far from being complete but it already gives the reader some glimpse of the subject.…”

Section: Introductionmentioning

confidence: 99%

Graph of even points on an arithmetic curve

Czogała¹,

Koprowski²

2019

Preprint

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We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the dimension of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.

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On the Picard Group: Torsion and the Kernel Induced by a Faithfully Flat Map

Cited by 11 publications

References 32 publications

Hypersurfaces in projective schemes and a moving lemma

Hypersurfaces in projective schemes and a moving lemma

Functorial structure of units in a tensor product

Graph of even points on an arithmetic curve

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