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In upcoming joint work with Matthew Morrow, we extended the theory of motivic cohomology to the setting of arbitrary qcqs schemes (over fields). One quarter of this theory consists of "conventional" motivic homotopy theory, under the guise of a cdh-local version of motivic cohomology. I will explain this picture and how one can use prismatic cohomology to control the values of this theory. As an application, we prove that the zero-th slice of the sphere spectrum is always HZ, as conjectured by Voevodsky. This last part is joint work with Tom Bachmann and Matthew Morrow.

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