Refinements to Patching and Applications to Field Invariants

Harbater, David; Hartmann, Julia; Krashen, Daniel

doi:10.1093/imrn/rnu278

Cited by 20 publications

(20 citation statements)

References 38 publications

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“…Since R is a regular local ring of dimension 2, there exists a finite sequence of blow-ups X → Spec R at the closed point of Spec( R) and closed points on the exceptional curves such that the support of µ on X is a union of regular curves with normal crossings [Abh69] or [Lip75]. Since any exceptional curve is the projective line over a finite extension of k, there exists a finite sequence of blow-ups X → Spec(R) such that X × Spec(R) Spec R = X (see [HHK15,prop. 3.6]).…”

Section: The Main Theoremmentioning

confidence: 99%

Hasse principle for hermitian spaces over semi-global fields

2016

Journal of Algebra

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Abstract. In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local-global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves.

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Section: The Main Theoremmentioning

confidence: 99%

Hasse principle for hermitian spaces over semi-global fields

2016

Journal of Algebra

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“…Descent of field extensions. The following two statements generalize Proposition 3.5 of [HHK15b]; by the trivial étale algebra (of degree n) over a field L we mean the direct product of n copies of L. Proposition 2.3. Let X be a normal model of a semi-global field F , let P be a closed point of X , let ℘ be a branch of the closed fiber X at P , and let E ℘ be a finite separable field extension of F ℘ .…”

Section: Descent Resultsmentioning

confidence: 90%

“…But patching holds for finite separable algebras in this context; see Proposition 3.7 and Example 2.7 in [HHK15b]. So there is a finite étale F -algebra E that compatibly induces all the algebras…”

Section: Descent Resultsmentioning

confidence: 99%

Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields

Colliot-Thélène

Harbater

Hartmann

et al. 2019

Trans. Amer. Math. Soc.

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We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. We show that localglobal principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. Many of our results are shown to hold more generally in the henselian case.

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“…Thus Theorem A holds conditionally for regular arithmetic surfaces. Also, recent results of Harbater-Hartmann-Krashen [HHK14] prove condition B for a wide class of local curves over complete discrete valuation rings with finite or algebraically closed residue fields.…”

Section: Surjectivity Of the Total Clifford Invariantmentioning

confidence: 95%

Surjectivity of the total Clifford invariant and Brauer dimension

Auel

2015

Journal of Algebra

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Abstract. A celebrated theorem of Merkurjev-that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms-is in general false when the base is no longer a field. The first counterexamples, when the base is among certain arithmetically subtle hyperelliptic curves over local fields, were constructed by Parimala, Scharlau, and Sridharan. We prove that considering Clifford algebras of all line bundle-valued quadratic forms, such counterexamples disappear and we recover Merkurjev's theorem in these cases: for any smooth curve over a local field or any smooth surface over a finite field, the 2-torsion of the Brauer group is always represented by Clifford algebras of line bundle-valued quadratic forms.

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Refinements to Patching and Applications to Field Invariants

Cited by 20 publications

References 38 publications

Hasse principle for hermitian spaces over semi-global fields

Hasse principle for hermitian spaces over semi-global fields

Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields

Surjectivity of the total Clifford invariant and Brauer dimension

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