Beilinson regulator

Concept in mathematics

In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology:

K n ( X ) → ⊕ p ≥ 0 H D 2 p − n ( X , Q ( p ) ) . {\displaystyle K_{n}(X)\rightarrow \oplus _{p\geq 0}H_{D}^{2p-n}(X,\mathbf {Q} (p)).}

Here, X is a complex smooth projective variety, for example. It is named after Alexander Beilinson. The Beilinson regulator features in Beilinson's conjecture on special values of L-functions.

The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers O F {\displaystyle {\mathcal {O}}_{F}} of a number field F

O F × → R r 1 + r 2 ,     x ↦ ( log ⁡ | σ ( x ) | ) σ {\displaystyle {\mathcal {O}}_{F}^{\times }\rightarrow \mathbf {R} ^{r_{1}+r_{2}},\ \ x\mapsto (\log |\sigma (x)|)_{\sigma }}

is a particular case of the Beilinson regulator. (As usual, σ : F ⊂ C {\displaystyle \sigma :F\subset \mathbf {C} } runs over all complex embeddings of F, where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the Borel regulator.

References

  • M. Rapoport, N. Schappacher and P. Schneider, ed. (1988). Beilinson's conjectures on special values of L-functions. Academic Press. ISBN 0-12-581120-9.