Mathematics

We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in infinity categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.

Let R be a strong n -coherent ring such that each finitely n -presented R -module has finite projective dimension. We consider FP n (R) the full subcategory of R -Mod of finitely n -presented modules. We prove that FP n (R) is an exact category, K i (R)= K i ( FP n (R)) for every i≥0 and obtain an expression of Nil i (R) .

We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms models. Along the way we also prove some results about the algebraic K-theory of Tate rings.

We give a short proof of Kaledin's theorem on the degeneration of the noncommutative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschild homology and the theory of cyclotomic spectra. As a consequence, we also obtain relative versions of the degeneration theorem, both in characteristic zero and for regular bases in characteristic p .

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A _1 and A _2 , vanishes in any (co)homological degree p>2 . Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees p and 2−p , and we prove a generalised version of the 2-Calabi-Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi-Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi-Yau property and we say that they are Koszul complex Calabi-Yau (Kc-Calabi-Yau) of dimension 2 . For Kc-Calabi-Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.

In this paper, we study the graded ring gr(X) defined by the K-theory of twisted flag variety X. In particular, Kunneth map from gr(R')(\otimes)gr(R') to gr(R) is studed explicitly for an original Rost motive R' and a generalized Rost motive R. Using this, we give an example that T(X)^2 is nonzero for the torsion ideal T(X) in the Chow ring CH(X).

We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is d -Calabi-Yau.

In this paper, we verify the L p coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for p∈[1,∞) . We also show that the K -theory of L p Roe algebras are independent of p∈(1,∞) for spaces with finite asymptotic dimension.

Let T be a circle group, and LT be its loop group. We hope to establish an index theory for infinite-dimensional manifolds which LT acts on, including Hamiltonian LT -spaces, from the viewpoint of KK -theory. We have already constructed several objects in the previous paper \cite{T}, including a Hilbert space H consisting of " L 2 -sections of a Spinor bundle on the infinite-dimensional manifold", an " LT -equivariant Dirac operator D " acting on H , a "twisted crossed product of the function algebra by LT ", and the "twisted group C ∗ -algebra of LT ", without the measure on the manifolds, the measure on LT or the function algebra itself. However, we need more sophisticated constructions. In this paper, we study the index problem in terms of KK -theory. Concretely, we focus on the infinite-dimensional version of the latter half of the assembly map defined by Kasparov. Generally speaking, for a Γ -equivariant K -homology class x , the assembly map is defined by μ Γ (x):=[c]⊗ j Γ (x) , where j Γ is a KK -theoretical homomorphism, [c] is a K -theory class coming from a cut-off function, and ⊗ denotes the Kasparov product with respect to Γ⋉ C 0 (X) . We will define neither the LT -equivariant K -homology nor the cut-off function, but we will indeed define the KK -cycles j LT τ (x) and [c] directly, for a virtual K -homology class x=(H,D) which is mentioned above. As a result, we will get the KK -theoretical index μ LT τ (x)∈KK(C,LT ⋉ τ C) . We will also compare μ LT τ (x) with the analytic index ind LT ⋉ τ C (x) which will be introduced.

In these lectures, we provide a toolkit to work with Chow-Witt groups, and more generally with the homology and cohomology of the Rost-Schmid complex associated to Milnor-Witt K -theory.