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We prove that the degree unramified cohomology of a smooth complex projective variety with small has a filtration of length , whose first piece is the torsion part of the quotient of by its coniveau subgroup, and whose next graded piece is controlled by the Griffiths group when is even and is related to the higher Chow group when is odd. The first piece is a generalization of the Artin–Mumford invariant () and the Colliot-Thélène–Voisin invariant (). We also give an analogous result for the -cohomology , .
We state the Paschke–Higson duality theorem for a transformation groupoid. Our proof relies on an equivariant localized and norm-controlled version of the Pimsner–Popa–Voiculescu theorem. The main consequence is the existence of a Higson–Roe exact sequence, involving the Baum–Connes assembly map for such a groupoid.
We prove that every stably projectionless separable simple amenable -algebra in the UCT class has rationally generalized tracial rank one. Following Elliott’s earlier work, we show that the Elliott invariant of any finite separable simple -algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group (which unifies the unital and nonunital, as well as stably projectionless cases). We also show that, for any given such invariant set, there is a finite separable simple -algebra whose Elliott invariant is the given set, a refinement of the range theorem of Elliott. In the stably projectionless case, modified model -algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features.
We show that the sheaf of -connected components of a Nisnevich sheaf of sets and its universal -invariant quotient (obtained by iterating the -chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of -connected components of any space. Given any natural number , we construct an -connected space on which the iterations of the naive -connected components construction do not stabilize before the -th stage.
We initiate the study of real -algebras associated to higher-rank graphs , with a focus on their -theory. Following Kasparov and Evans, we identify a spectral sequence which computes the -theory of for any involution on , and show that the page of this spectral sequence can be straightforwardly computed from the combinatorial data of the -graph and the involution . We provide a complete description of for several examples of higher-rank graphs with involution.
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