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.
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.
Given a smooth variety X over the field R of real numbers and a line bundle L on X with associated topological line bundle L = L(R), we study the quadratic real cycle class map gamma eRc: fCHc(X, L) , Hc(X (R), Z(L)) from the c-th Chow-Witt group of X to the c-th cohomology group of its real locus X (R) with coefficients in the local system Z(L) associated with L. We focus on the cases c is an element of {0, d-2, d-1, d} where d is the dimension of X, and we formulate a precise conjecture on the image of gamma eR in terms of the exponents of its cokernel that is corroborated by the results obtained in those codimensions.
We study multiplicative structures on the K-theory of the core A of the C*-algebra of a directed graph E. We first study embeddings E -> E & times; E that induce a *-homomorphism A (R) A -> A. Through the K & uuml;nneth formula, any such *-homomorphism induces a ring structure on K*(A). We then give conditions on E for which K*(A) is generated by "noncommutative line bundles" (invertible bimodules). The same conditions guarantee the existence of a homomorphism of abelian groups K0(A) -> 7L[lambda]/(det(lambda 0-1)) (where 0 is the adjacency matrix of E) that is compatible with the tensor product of line bundles. Examples include the C*-algebra C(CPqn-1) of a quantum projective space, the UHF(n infinity) algebra, and the C*-algebra of the space parametrizing Penrose tilings. For the first algebra, we recover as a corollary some identities that classically follow from the ring structure of K0(CPn-1) and that were proved by Arici, Brain and Landi in the quantum case. Incidentally, we observe that the C*-algebra of Penrose tilings is the AF core of the Cuntz algebra O2 if the latter is realized using the appropriate graph.
We define an S center dot-construction for squares categories and introduce a class of squares categories we call proto-Waldhausen which capture the properties required for the S center dot-construction to model the K-theory space. The primary question we investigate is when the S center dot-construction of a squares category produces a 2-Segal space. We show that the answer to this question is affirmative when the squares category satisfies certain "stability" conditions.
Let F be a field of characteristic p>0. We prove that if a symbol A = omega circle times beta(1) circle times ... circle times beta(n) in H-pm(n+1)(F) is of exponent dividing p(m-1), then its symbol length in H-pm-1(n+1)(F) is at most p(n). In the case n=1, we also prove that if A=omega(1)circle times beta(1)+...+omega(r)circle times beta(r) in H-pm(2)(F) satisfies exp(A)|p(m-1), then the symbol length of A in H-pm-1(2)(F) is at most p(r)+r-1. We conclude by looking at the case p=2 and proving that if A is a sum of two symbols in H-2m(n+1)(F) and exp A|2(m-1), then the symbol length of A in H-2m-1(n+1)(F) is at most (2n+1)2(n). Our results use norm conditions in characteristic p in the same manner as Matzri in his 2024 paper "On the symbol length of symbols".
We provide an explicit description of the K-classes of higher Kazhdan projections in degrees greater than 0 for specific free product groups and Cartesian product groups. Employing this description, we obtain new calculations of Lott's delocalised & ell;2-Betti numbers for groups. Notably, we establish the first nonvanishing results for infinite groups.
Let X be a K3 surface over a p-adic field k such that for some abelian surface A isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of k between X and the Kummer surface associated to A. Under some assumptions on the reduction types of the elliptic curve factors of A, we prove that the Chow group A_0(X) of zero-cycles of degree 0 on X is the direct sum of a divisible group and a finite group. This proves a conjecture of Raskind and Spiess and of Colliot-Thélène and it is the first instance for K3 surfaces when this conjecture is proved in full. This class of K3's includes, among others, the diagonal quartic surfaces. In the case of good ordinary reduction we describe many cases when the finite summand of A_0(X) can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Thélene, Sansuc, Kato and Saito which, roughly speaking, predicts that the Brauer-Manin obstruction is the only obstruction to Weak Approximation for zero-cycles. We give examples of Kummer surfaces over a number field F where the ramified places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree 0 and we describe cases when an unconditional local-to-global principle can be proved, giving the first unconditional evidence for this conjecture in the case of K3 surfaces.
Let $k$ be a field of characteristic $0$ endowed with a fixed field embedding $\sigma: k \hookrightarrow \mathbb{C}$. In this paper we complete the construction of the six functor formalism on perverse Nori motives over quasi-projective $k$-varieties initiated by F. Ivorra and S. Morel. Our main contribution is the construction of a canonical closed unitary symmetric monoidal structure on the bounded derived categories of perverse Nori motives compatible with the analogous structure on the underlying constructible derived categories; along the way, we extend some of Nori's original results to perverse Nori motives. As a consequence, we obtain well-behaved Tannakian categories of motivic local systems over smooth, geometrically connected $k$-varieties. Our constructions do not depend on the chosen complex embedding of $k$ and, in fact, our results generalize to arbitrary base fields of characteristic $0$.
In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of exact and subtractive categories to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves.
Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$.
We study the assembly map on K-theory and on topological cyclic homology, written as TC. Specifically, we study the maps S1 +boolean AND K(S) -> K(S[x +/- 1]) and S1 +boolean AND TC(S) -> TC(S[x +/- 1]). In the second case we are able to describe geometrically what the cofiber is. Using this we show that the Whitehead space of the circle has a countable sum of Q infinity coker(j)boolean AND p as a summand. This paper was part of the author's thesis.
Let R be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring GWs(R) of R as an abelian group when R has residue field 2. This completes the work of Rogers and Schlichting (Math. Z. 307:2 (2024), art. id. 41), where an explicit presentation of GWs(R) is given when the residue field is different from 2. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings 7L/2n and 2Lx]/(xn).
Our aim in this paper is to prove in the setting of Kato-Milne cohomology in characteristic 2 an exact sequence which is analogue to the Milnor-Scharlau sequence [8, Theorem 6.2]. This is an extension of the Milnor exact sequence proved in [6].
We show that the continuous & eacute;tale cohomology groups H-cont(n)(X, Z(l)(n)) of smooth varieties X over a finite field k are spanned as Z(l)-modules by the n-th Milnor K-sheaf locally for the Zariski topology for all n >= 0. Here l is a prime invertible in k. This is the first general unconditional result towards the conjectures of Kahn (1998) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective k-varieties.
Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category.
Using the Evans spectral sequence and its counterpart for real K-theory, we compute both the real and complex K-theory of several infinite families of C*-algebras based on higher-rank graphs of rank 3 and 4. The higher-rank graphs we consider arise from double-covers of cube complexes. By considering the real and complex K-theory together, we are able to carry these computations much further than might be possible considering complex K-theory alone. As these algebras are classified by K-theory, we are able to characterize the isomorphism classes of the graph algebras in terms of the combinatorial and number-theoretic properties of the construction ingredients.
We prove a comparison result between two duality statements - Takai duality, which is implemented by the crossed product functor-x G : KKG -> KKGb on equivariant Kasparov categories, and Treumann duality, which asserts the existence of an exotic equivalence of stable infinity-categories
In the setting of a proper, cocompact action by a locally compact, unimodular group $G$ on a Riemannian manifold, we construct equivariant spectral flow of paths of Dirac-type operators. This takes values in the $K$-theory of the group $C^*$-algebra of $G$. In the case where $G$ is the fundamental group of a compact manifold, the summation map maps equivariant spectral flow on the universal cover to classical spectral flow on the base manifold. We obtain "index equals spectral flow" results. In the setting of a smooth path of $G$-invariant Riemannian metrics on a $G$-spin manifold, we show that the equivariant spectral flow of the corresponding path of spin Dirac operators relates delocalised $\eta$-invariants and $\rho$-invariants for different positive scalar curvature metrics to each other.
