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Divided powers in the Witt ring of symmetric bilinear forms

Burt Totaro

Vol. 8 (2023), No. 2, 275–284
Abstract

The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi, Merkurjev, and Serre’s work on cohomological invariants that all operations on the Witt ring are 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.

Keywords
Witt ring, symmetric bilinear forms, quadratic forms, divided powers
Mathematical Subject Classification
Primary: 11E81
Secondary: 11B68, 19D45
Milestones
Received: 21 September 2022
Revised: 13 April 2023
Accepted: 14 April 2023
Published: 14 June 2023
Authors
Burt Totaro
Mathematics Department
University of California
Los Angeles, CA
United States