Milnor-Witt motives

to appear: Memoirs of the American Mathematical Society
(1^st appeared online on the 14-4-2020)

This book contains the construction and study of Milnor-Witt correspondences and motives. They are to Voevodsky's motives what Milnor-Witt K-theory is to Milnor K-theory, Chow-Witt groups to Chow groups and hermitian K-theory to K-theory.

This motivic category is closer to the stable homotopy category of schemes than Voevodsky's motives: in particular it has the same endomorphisms of the point (a field) than the stable homotopy category: the Grothendieck-Witt group.

The construction begins by the replacement of Voevodsky's finite correspondences by Milnor-Witt correspondences, which take into account quadratic forms over the function fields of (irreducible components of) algebraic cycles.