Milnor-Witt motivic cohomology of complements of hyperplane arrangements

In this paper, we compute the (total) Milnor-Witt motivic cohomology of the complement of a hyperplane arrangement in an affine space.

Low dimensional Milnor-Witt stems over R

This article computes some motivic stable homotopy groups over R. For 0 <= p - q <= 3, we describe the motivic stable homotopy groups of a completion of the motivic sphere spectrum. These are the first four Milnor-Witt stems. We start with the known Ext groups over C and apply the rho-Bockstein spectral sequence to obtain Ext groups over R. This is the input to an Adams spectral sequence, which collapses in our low dimensional range.

Duality Theorem for Motives

Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodskys duality to the case of arbitrary base field characteristic.

Witt ring localized

The rings of $p$-typical Witt vectors are interpreted as spaces of vanishing cycles for some perverse sheaves over a disc. This allows to localize an isomorphism emerging in Drinfelds theory of prismatization [Dr], Prop. 3.5.1, namely to express it as an integral of a standard exact triangle on the disc.

Milnor excision for motivic spectra

We prove that the $infty$-category of motivic spectra satisfies Milnor excision: if $Ato B$ is a morphism of commutative rings sending an ideal $Isubset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/IB$. Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoubs etale motives over schemes of finite virtual cohomological dimension.