##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Tom Bachmann

#### LMU Munich

## Cellular motivic invariants of Z[1/2]

##### Abstract:

Report on work in progress, joint with Paul Arne Oestvaer. \\ \\ A cellular motivic invariant is a special type of functor from the category of commutative rings (or the opposite of schemes, say) to spectra. Examples include algebraic K-theory, motivic cohomology, \'e{}tale cohomology and algebraic cobordism. Dwyer-Friedlander observed that for 2-adic \'e{}tale K-theory and certain related invariants, the value on Z[1/2] can be described in terms of a fiber square involving the values on the real numbers, the complex numbers, and the field with three elements. \\ \\ I will explain a generalization of this result to arbitrary 2-adic cellular motivic invariants. As an application, we show that $\pi_0$ of the motivic sphere spectrum over Z[1/2] is given by the Grothendieck-Witt ring of Z[1/2], up to odd torsion.

Host: Zhouli Xu

### January 26, 2021

### 10:30 AM

### Zoom information: Meeting ID: 933 6734 4286 Password: topology

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