K 1 of a p -adic group ring I. The determinantal image

Chinburg, Ted; Pappas, G.; Taylor, Martin J.

doi:10.1016/j.jalgebra.2010.08.006

Cited by 7 publications

(44 citation statements)

References 13 publications

(3 reference statements)

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“…In [11], we obtain more precise results about SK 1 when we assume that, among other additional hypotheses, our coefficient rings afford a lift of Frobenius. We are going to use the following corollaries of the main result of [11].…”

Section: D3 Suppose Now In Addition Thatmentioning

confidence: 73%

“…For this we need to use results of Oliver on SK 1 -groups of group rings (e.g. [44,45]) and the extension of these results in [11]. In particular, we can see that our notion of elementary structure is appropriately restrictive; for example, our considerations show that any O P 1 [G]-bundle which is trivial along (1 : 1) and has zero degree has an elementary structure.…”

Section: H(r((t))[g]) → Gl ′ (R((t))[g]) →mentioning

confidence: 99%

“…is semi-local we can write y = i y i = i e i δ i d i where the e i and d i lie in the group of elementary matrices E(Z i ), and where δ i is diagonal matrix with all non-leading terms 1; so that the leading diagonal term [13] we have a similar decomposition…”

Section: D3 Suppose Now In Addition Thatmentioning

confidence: 99%

“…We are going to use the following corollaries of the main result of [11]. Here we will assume that W = W (k) is the ring of integers in a finite unramified extension of Q p with residue field k.…”

Section: D3 Suppose Now In Addition Thatmentioning

confidence: 99%

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Higher adeles and non-abelian Riemann–Roch

Chinburg

Pappas

Taylor

2015

Advances in Mathematics

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Abstract. We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P 1 . Our set-up provides an extension of several aspects of the classical Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces. ContentsIntroduction §1. Beilinson-Parshin adeles on a surface §2. Equivariant adelic Chow groups §3. Lattices, determinant functors and determinant theories §4. Pushdown maps and reciprocity laws §5. Transition matrices and the first Chern class §6. Elementary structures and the second Chern class §7. Equivariant Euler characteristics and the Riemann-Roch theorem §8. The proof of the theorem; reduction to the case of P 1 Z §9. The proof of the theorem; bundles over P 1 Z §10. Appendix: Adelic Riemann-Roch for general bundles when G = 1 References

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Section: D3 Suppose Now In Addition Thatmentioning

confidence: 73%

Section: H(r((t))[g]) → Gl ′ (R((t))[g]) →mentioning

confidence: 99%

Section: D3 Suppose Now In Addition Thatmentioning

confidence: 99%

Section: D3 Suppose Now In Addition Thatmentioning

confidence: 99%

See 2 more Smart Citations

Higher adeles and non-abelian Riemann–Roch

Chinburg

Pappas

Taylor

2015

Advances in Mathematics

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“…For any perfect complex of Λ-sheafs F • on the étale site of X we have defined in [16] an L-function L(F • , T ) attached to F • . This is an element in the first K-group K 1 (Λ[ Aside from the result of Emerton and Kisin, a central ingredient for the proof is the recent work of Chinburg, Pappas, and Taylor [2,3] on the first K-group of p-adic group rings. In fact, the main strategy of the proof is to reduce the assertion first to the case Λ = Z p [G] for a finite group G and then use the results of Chinburg, Pappas, and Taylor to reduce it further to the case already treated by Emerton and Kisin.…”

Section: Introductionmentioning

confidence: 99%