Divided powers in the Witt ring of symmetric bilinear forms
arXiv:2209.08634 · doi:10.2140/akt.2023.8.275
Abstract
The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are essentially linear combinations of exterior powers. We find the explicit formula for the divided powers as a linear combination of exterior powers. The coefficients involve the ``tangent numbers'', related to Bernoulli numbers. The divided powers on the Witt ring give another construction of the divided powers on Milnor K-theory modulo 2.
10 pages; v2: proof of Lemma 2.2 improved after publication, thanks to J.-P. Serre