We compute the cohomology ring H * (X, Z/nZ) for X the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. Contents 1. Introduction 1 1.1. Organization 2 2. Background on the étale cohomology of a number field 3 2.1. The Artin-Verdier site of a number field 3 2.2. Artin-Verdier duality for general number fields 8 2.3. Galois coverings in Xét 10 3. The cohomology ring of a number field 12 4. A non-vanishing criterion for Kim's invariant 23 References 25
We compute some arithmetic path integrals for BF$BF$‐theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to double-struckGm${\mathbb {G}}_{m}$ and abelian varieties.
We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.
We compute the cohomology ring H * (U, Z/nZ) for U = X \ S where X is the spectrum of the ring of integers of a number field K and S is a finite set of finite primes. Contents 1. Introduction 1 2. The cohomology of a punctured arithmetic curve 3 3. The cup product 9 4. Some computations of cup products 19 Appendix A. Computations 21 Appendix B. Cohomology of idèles 22 References 23 * G(F ) ν is a resolution of a ν * F ν into acyclics. As explained in [GS18, §2], there is a resolution D • (a ν * F ν ) of a ν * F ν which is pointwise acyclic, and splicing D • (a ν * F ν ) and a ν * G(F ) ν together, one gets a complete (pointwise) acyclic resolution Î(F ν ) of a ν * F ν . The resolution Î(F ν ) computes the Tate-hypercohomology of a ν * F ν and there is a canonical map awhere C(−) denotes the mapping cone. We will also at times view RΓ c (U fppf , F ) as an object of D(Ab).Definition 2.1. For a bounded complex F of abelian sheaves on U fppf , we define
We compute the cohomology ring $$H^*(X,{{\mathbb {Z}}}/n{{\mathbb {Z}}})$$
H
∗
(
X
,
Z
/
n
Z
)
for X the spectrum of the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.