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The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of an unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.
We demonstrate that a conjecture of Teh which relates the niveau filtration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that the niveau filtration on reduced Lawson
In this short note, we simply collect some known results about representing algebraic cycles by various kind of nice (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. We
We introduce a notion of Milnor square of stable $infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision theorems in the
In this note, we provide an axiomatic framework that characterizes the stable $infty$-categories that are module categories over a motivic spectrum. This is done by invoking Luries $infty$-categorical version of the Barr--Beck theorem. As an applicat
We define push-forwards for Witt groups of schemes along proper morphisms, using Grothendieck duality theory. This article is an application of results of the authors on tensor-triangulated closed categories to such structures on some derived categor