This paper lays out the foundations of graded K-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite k-graph Λ without sources, we show that there exists a ℤ[ℤ^k]-module isomorphism between the graded zeroth (integral) homology H_0^gr(𝒢_Λ) of the infinite path groupoid 𝒢_Λ and the graded Grothendieck group K_0^gr(KP_𝗄(Λ)) of the Kumjian-Pask algebra KP_𝗄(Λ), which respects the positive cones (i.e., the talented monoids). We demonstrate that the k-graph moves of in-splitting and sink deletion defined by Eckhardt et al. (Canad. J. Math. 2022) preserve the graded K-theory of associated Kumjian-Pask algebras and produce algebras which are graded Morita equivalent, thus providing evidence that graded K-theory may be an effective invariant for classifying certain Kumjian-Pask algebras. We also determine a natural sufficient condition regarding the fullness of the graded Grothendieck group functor. More precisely, for two row-finite k-graphs Λ and Ω without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving ℤ[ℤ^k]-module homomorphism between K_0^gr(KP_𝗄(Λ)) and K_0^gr(KP_𝗄(Ω)) to a unital graded ring homomorphism between KP_𝗄(Λ) and KP_𝗄(Ω). For this we adopt, in the setting of k-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).
We define and study bivariant equivariant periodic cyclic homology for actions of ample groupoids. In analogy to the group case, we show that the theory satisfies homotopy invariance, stability, and excision in both variables. We also prove an analogue of the Green-Julg theorem for actions of proper groupoids.
Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations π^ℝ_**(kq⊗kq) in the stable motivic homotopy category SH(ℝ), giving a full description in terms of Brown–Gitler comodules. To do this, we decompose the E_2-page of the motivic Adams spectral sequence and show that it must collapse. The description of the E_2-page is accomplished by a series of algebraic Atiyah–Hirzebruch spectral sequences which converge to the summands of the E_2-page. Along the way, we prove a splitting result for the very effective symplectic K-theory ksp over any base field of characteristic not two.
We describe the index pairing between an odd K-theory class and an odd unbounded Kasparov module by a pair of quasi-projections, supported on a submodule obtained from a finite spectral truncation. We achieve this by pairing the K-theory class with an asymptotic morphism determined by the unbounded Kasparov module. We interpret the spectral localiser of Loring and Schulz-Baldes as an instance of such an index pairing.
