On a theorem of Hazrat and Hoobler

Antieau, Benjamin

doi:10.1090/s0002-9939-2013-11656-4

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…In [19], Klein defined and began the study of relative tensor triangular Chow groups, a family of K-theoretic invariants attached to a compactly generated triangulated category K with an action of a rigidly-compactly generated tensor triangulated category T in the sense of [37]. While in [19], they were used to improve upon and extend results of [20], the initial observation of the present work is that they allow us to enter the realm of noncommutative algebraic geometry: if X is a noetherian scheme and A a (possibly noncommutative) coherent O X -algebra, then the derived category K := D(Qcoh(A)) admits an action by T := D(Qcoh(O X )) which is obtained by deriving the tensor product functor (1) Qcoh(A) × Qcoh(O X ) → Qcoh(A)…”

Section: Introductionmentioning

confidence: 93%

Relative tensor triangular Chow groups for coherent algebras

Belmans

Klein²

2017

Journal of Algebra

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We apply the machinery of relative tensor triangular Chow groups to the action of the derived category of quasi-coherent sheaves on a noetherian scheme $X$ on the derived category of quasi-coherent $\mathcal{A}$-modules, where $\mathcal{A}$ is a (not necessarily commutative) quasi-coherent $\mathcal{O}_X$-algebra. When $\mathcal{A}$ is commutative and coherent, we recover the tensor triangular Chow groups of the relative Spec of $\mathcal{A}$. We also obtain concrete descriptions for integral group algebras and hereditary orders over curves, and we investigate the relation of these invariants to the classical ideal class group of an order. An important tool for these computations is a new description of relative tensor triangular Chow groups as the image of a map in the K-theoretic localization sequence associated to a certain Verdier localization.Comment: 39 pages; all comments welcom

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Section: Introductionmentioning

confidence: 93%

Relative tensor triangular Chow groups for coherent algebras

Belmans

Klein²

2017

Journal of Algebra

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“…Clearly when A is a division algebra and i = 1, we get CK 1 (D) defined in (6.11). The fact that CK i (A) and ZK i (A) are torsion of bounded exponent was studied in several papers (see [8,29]). A consequence of this, is that the K-theory of A coincides with the K-theory of its base ring up to torsions.…”

Section: Corollary 62 Let D Be An F -Central Division Algebra and A B...mentioning

confidence: 99%

Multiplicative groups of division rings

Hazrat¹,

Mahdavi-Hezavehi²,

Motiee³

2014

Mathematical Proceedings of the Royal Irish Academy

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Abstract. Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.

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