Beilinson regulator

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

$K_{n}(X)\rightarrow \oplus _{{p\geq 0}}H_{D}^{{2p-n}}(X,{\mathbf Q}(p)).$

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 ${\mathcal O}_{F}$ of a number field F

${\mathcal O}_{F}^{\times }\rightarrow {\mathbf R}^{{r_{1}+r_{2}}},\ \ x\mapsto (\log |\sigma (x)|)_{\sigma }$

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