Beilinson regulator

Short description: Concept in mathematics

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

$K_n(X)\to {\oplus }_{p\ge 0}{H}_D^{2p-n}(X,\mathbf{𝐐}(p)).$

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 ${{𝒪}}_F$ of a number field F

${{𝒪}}_F^{\times }\to {\mathbf{𝐑}}^{r_1+r_2},x\mapsto (\log |\sigma (x)|{)}_{\sigma }$

is a particular case of the Beilinson regulator. (As usual, $\sigma :F\subset \mathbf{𝐂}$ 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.