The Brauer–Manin pairing, class field theory, and motivic homology

Yamazaki, Takao

doi:10.1017/s0027763000010722

Cited by 1 publication

(2 citation statements)

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“…We note that Wiesend's original definition was not exactly this, but a similar class group for arithmetic schemes of finite type over

Spec Z

. Definition () is a variant due to Yamazaki .…”

Section: Semi‐abelian Varieties and Suslin's Singular Homologymentioning

confidence: 99%

“…We will denote by

W^{0} false(X false)

the subgroup of degree zero cycle classes. When

\bar{C}

is a smooth, complete, geometrically irreducible curve over k and S a finite set of points of

\bar{C}

, then for the smooth curve

C = \bar{C} - S

, the group

W (C)

coincides with the group of classes of divisors on

\bar{C}

prime to S modulo S ‐equivalence, as defined in [, Chapter V]. Notice that when C has a k ‐rational point, the abelian group

W^{0} false(C false)

is isomorphic to the generalized Jacobian of

\bar{C}

corresponding to the modulus

m = \sum_{P \in S} P

. The groups

W (X)

and

H_{0}^{sing}

are covariant functorial for morphisms of varieties

X \to Y

([, Proposition 2.10], [, Lemma 2], [, Lemma 2.3]). Generalized Albanese map: If X is a smooth variety over a perfect field k , there is a generalized albanese map

{prefixalb}_{X} : W^{0} false(X false) \to G_{X} false(k false)

, where

G_{X}

is the generalized Albanese variety of X . For a proof of the fact that the generalized Albanese map is well‐defined we refer to .…”

Section: Semi‐abelian Varieties and Suslin's Singular Homologymentioning

confidence: 99%

See 1 more Smart Citation

Some results about zero‐cycles on abelian and semi‐abelian varieties

Gazaki

2019

Mathematische Nachrichten

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In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of Qp with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration false{Frfalse}r≥0 of Suslin's singular homology group, H0singfalse(Gfalse), such that the successive quotients are isomorphic to a certain Somekawa K‐group.

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“…We note that Wiesend's original definition was not exactly this, but a similar class group for arithmetic schemes of finite type over

Spec Z

. Definition () is a variant due to Yamazaki .…”

Section: Semi‐abelian Varieties and Suslin's Singular Homologymentioning

confidence: 99%

“…We will denote by

W^{0} false(X false)

the subgroup of degree zero cycle classes. When

\bar{C}

is a smooth, complete, geometrically irreducible curve over k and S a finite set of points of

\bar{C}

, then for the smooth curve

C = \bar{C} - S

, the group

W (C)

coincides with the group of classes of divisors on

\bar{C}

prime to S modulo S ‐equivalence, as defined in [, Chapter V]. Notice that when C has a k ‐rational point, the abelian group

W^{0} false(C false)

is isomorphic to the generalized Jacobian of

\bar{C}

corresponding to the modulus

m = \sum_{P \in S} P

. The groups

W (X)

and

H_{0}^{sing}

are covariant functorial for morphisms of varieties

X \to Y

([, Proposition 2.10], [, Lemma 2], [, Lemma 2.3]). Generalized Albanese map: If X is a smooth variety over a perfect field k , there is a generalized albanese map