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We introduce a periodic form of the iterated algebraic -theory of , the (connective) complex -theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory.
A -algebra is said to have the ideal property if each closed two-sided ideal of is generated as a closed two-sided ideal by the projections inside the ideal. -algebras with the ideal property are a generalization and unification of real rank zero -algebras and unital simple -algebras. It was long expected that an invariant that we call , consisting of the scaled ordered total -group (used in the real rank zero case), along with the tracial state spaces for each cut-down algebra , as part of the Elliott invariant of (for each ), with certain compatibility conditions, is the complete invariant for a certain well behaved class of -algebras with the ideal property (e.g., algebras with no dimension growth). In this paper, we construct two nonisomorphic algebras and with the ideal property such that , disproving this conjecture. The invariant to distinguish the two algebras is the collection of Hausdorffified algebraic -groups (for each ), along with certain compatibility conditions. We will prove in a separate article that, after adding this new ingredient, the invariant becomes the complete invariant for algebras (of no dimension growth) with the ideal property.
We use techniques of relative algebraic -theory to develop a common refinement of the theories of metrized and hermitian Galois structures in arithmetic. As a first application of the general approach, we then use it to prove several new results, and to formulate several explicit new conjectures, concerning the detailed arithmetic properties of a natural class of wildly ramified Galois–Gauss sums.
We introduce techniques for uniformly studying the gamma filtration of projective homogeneous varieties. These techniques are utilized in some cases of inner-twisted flag varieties (of type A) to show that functorality known for the Chow rings of these varieties also extends to the associated graded rings for the gamma filtrations of the same varieties. As an application, we show that the associated graded groups for the gamma filtration of these varieties are torsion free in low homological degrees.
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