# Late-night motives* — MoVid-20: motivic video-conference on 15 April 2020 via Zoom

*The first talk is 1 am Pacific time.

To register, please send an email to denis.nardin@ur.de with the subject “MoVid-20”.

The schedule in Central European Summer Time (aka time in Germany) is as follows.

9:45-10:00 Conference opening
10:00-11:15 Tom Bachmann
11:15-12:00 coffee break
12:00-13:15 Marc Hoyois
13:15-14:30 lunch break
14:30-15:45 Maria Yakerson
15:45-16:30 coffee break
16:30-17:45 Denis Nardin

Here are the titles and abstracts.

Tom Bachmann: Pullbacks for the Rost-Schmid complex
Let F be a “strictly homotopy invariant” Nisnevich sheaf of abelian groups on the site of smooth varieties over a perfect field k. By work of Morel and Colliot-Thélène–Hoobler–Kahn, the cohomology of F may be computed using a fairly explicit “Rost-Schmid” complex. However, given a morphism f : XY of smooth varieties, it is in general (in particular if f is not flat, e.g. a closed immersion) unclear how to compute the pullback map f *: H*(Y,F) → H*(X,F) in terms of the Rost-Schmid complex. I will explain how to compute the pullback of a cycle with support Z such that f-1(Z) has the expected dimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space X, its zeroth P1-stable homotopy sheaf is given by π3P13X)-3.

Marc Hoyois: Milnor excision for motivic spectra
It is a classical result of Weibel that homotopy invariant algebraic K-theory satisfies excision, in the sense that for any ring A and ideal I\subset A, the fiber of KH(A) → KH(A/I) depends only on I as a nonunital ring. In joint work with Elden Elmanto, Ryomei Iwasa, and Shane Kelly, we show that this is true more generally for any cohomology theory represented by a motivic spectrum.

Denis Nardin: A description of the motive of $Hilb(A^\infty)$
The Hilbert scheme of points in infinite affine space is a very complicated algebro-geometric object, whose local structure is extremely rich and hard to describe. In this talk I will show that nevertheless its motive is pure Tate and in fact it coincides with the motive of the Grassmannian. This will allow us to give a simple conceptual description of the motivic algebraic K-theory spectrum. This is joint work with Marc Hoyois, Joachim Jelisiejew, Burt Totaro and Maria Yakerson.

Maria Yakerson: Motivic generalized cohomology theories from framed perspective
All motivic generalized cohomology theories acquire unique structure of so called framed transfers. If one takes framed transfers into account, it turns out that many interesting cohomology theories can be constructed simply as suspension spectra on certain moduli stacks (and their variations). This way important cohomology theories on schemes get new geometric interpretations, and so do canonical maps between different cohomology theories. In the talk we will explain the general formalism of framed transfers and
show how it works for various cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin and Vladimir Sosnilo.

Image: Still from The Third Man (1949, dir. Carol Reed).