6,665 papers in this slice of arXiv.
David Chan, Marc Gotliboym, Inbar Klang +1
One of the most effective approaches to computations in algebraic K-theory is trace methods, which compare algebraic K-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology (ETHH
Maxime Ramzi
We prove that the ∞-category of localizing motives in the sense of Blumberg--Gepner--Tabuada is not compactly generated, extending a result of Efimov.
Chirantan Chowdhury
In this article, we study two consequences of abstract six-functor formalisms. Firstly, we show that an abstract six-functor formalism can be extended to specific Ind- and Pro- categories of geometric setups. As an application, we can define the motivic stable homotopy theory for ind-pro-algebraic stacks such as the Hecke stack. Secondly, we also show that Cohomological Purity is functorial using the multisimplicial language developed by Liu-Zheng.
Toni Annala, Piotr Pstrągowski
Using an argument of Jacob Lurie, we prove the existence of a E_2-lax monoidal structure on the Tate filtration coming from the standard cell structure on the infinite complex projective space. We then compare it to the filtration induced by the standard t-structure on spectra. In the last part of the paper, we construct a synthetic variant of the cellular Tate filtration and use it to compare Antieau's, Bhatt-Lurie's and Raksit's HKR filtrations on negative cyclic and periodic homology.
Elvis Torres Pérez
This work revisits the key sufficient conditions on a ring A that guarantee the existence of the exact sequence H3(SM2(A),Z)→H3(SL2(A),Z)→RB(A)→0 and of the isomorphism H3(SL2(A),SM2(A);Z)≃RP1(A). We show that there are rings which satisfy the relations above but do not satisfy the condition that -1 is an square, for example the rings Z[21] and Z[101]. In addition, we study the special case of non-archimedean local fields, obtaining a refined Bloch-Wigner exact sequence when -1 is not an square.
Jebasingh R
Motivated by the work of Ischebeck on the behaviour of algebraic K-theoretic functors under subintegral extensions, we investigate the analogous question in Hermitian K-theory. We prove that Grothendieck-Witt groups in degree zero are invariant under subintegral extensions. We also investigate the cancellation problem for quadratic spaces over geometric subrings of polynomial rings and establish a cancellation theorem for quadratic spaces of sufficiently large Witt index.
Christian Dahlhausen, Can Yaylali, Yicheng Zhou
We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (à la Kerz--Saito--Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is A1-invariant assuming resolutions of singularities, we deduce that it is representable in the A1-homotopy category of rigid spaces (à la Dahlhausen--Yaylali). We identifiy the representing object with both Z×BGL and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is A1-invariant on local Tate pairs (without any regularity assumption).
Niels Feld
Let V be a complete discrete valuation ring of mixed characteristic (0,3) in which 3 is a uniformizer, and put A=V[[x,y]]/(3+x2−y3). We construct a nonzero class a∈K2(A;Z/3) whose restriction to the fraction field of A is zero. Thus Gersten injectivity for algebraic K-theory with Z/3-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of a is zero, while the map K2(A)→K2(F) is injective. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.
Ishan Levy, Jiacheng Liang, Vladimir Sosnilo
We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the AB4∗ and AB6 axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective E1-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category Nuc(R)≥0 of connective nuclear modules (in the sense of Clausen--Scholze) over an adic E∞-ring R via a universal property, identifying it as the additive rigidification of the category of connective complete R-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category Motpst of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic K-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.
Tess Bouis
Using non-A1-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic K-theory of singular rings, such as rings of the form Z/pn, Z[x]/(xe), and C(X;C) for X a compact Hausdorff space. These refinements are made possible by the use of integral p-adic Hodge theory, as a replacement for the standard use of trace methods in K-theory.
Myungsin Cho, Teena Gerhardt, Liam Keenan +2
We develop homological Real trace methods, an approach to understanding the continuous mod two Bredon homology of Real topological periodic homology and Real topological negative cyclic homology, generalizing prior work of Bruner and Rognes. We use this new approach to do several computations, including the continuous mod two Bredon homology of the Real topological negative cyclic homology of Real bordism.
Sergio Estrada, Alina Iacob, Yucheng Wang
Let R be a ring and let Ch(R) be the category of complexes of left R-modules. We consider the C-E (Cartan-Eilenberg) exact structure on Ch(R) and show that it is an efficient exact category. We prove that if R is a left virtually Gorenstein ring, the pair (C-E(GProj),C-E(GInj)) is a C-E-admissible balanced pair. Under suitable additional hypotheses, the converse holds. We also consider the C-E version of Tate cohomology and establish C-E analogues of the Avramov-Martsinkovsky exact sequences and balance results for both Ext and Tor.
Xinan Dai, Kuok Fai Chao
Let G be a connected linear real semisimple group with finite centre and discrete series, and let Gc be a fixed compact inner form. We isolate an integral structure behind the discrete-series character identity. There are natural homomorphisms CG:K0(Cr∗(G))⟶R(Gc) and JG:R(Gc)⟶K0(Cr∗(G)), characterised, respectively, by stable and ordinary elliptic orbital integrals. The first sends a noncompact Dolbeault--Dirac index to its compact counterpart; the second sends an irreducible representation of Gc to the signed sum of the K-theory classes in the corresponding discrete-series L-packet. We prove CGJG=[WG:WK]idR(Gc). Thus the packet cardinality is the precise integral cost of splitting stable orbital averaging. Writing SG=imJG and UG=kerCG, we obtain the exact obstruction sequence 0⟶SG⊕UG⟶K0(Cr∗(G))⟶R(Gc)/[WG:WK]R(Gc)⟶0. After inverting the packet cardinality, this yields a canonical stable projector and functorial transfers between the stable K-theory lattices of real inner forms. For SL(2,R) the obstruction is (Z/2Z)[z+z−1]. For the inner forms of type Cn, the relevant multiplier is 2n for Sp(2n,R) and (pn) for Sp(p,n−p).
Shay Ben-Moshe
We give a new proof that algebraic K-theory increases chromatic height by at most one, originally established by Clausen-Mathew-Naumann-Noel. Our argument proceeds by descent from the Lubin-Tate spectrum, for which the required vanishing follows from Hahn-Wilson's Quillen-Lichtenbaum result.
Steven Rayan
Topological K-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary K-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface X, we prove nonfactorization in three settings: fibrewise K0(X), occupied-state K0(B), and operator-algebraic K-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani--Sunada bottom-band Hessian is the Hodge inner product on H1(X;R). Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichmüller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as O(∣Gn∣−η) for some η>0, and Cs observables as O((log∣Gn∣)−s), while established coherent large-rank limits recover bulk density-of-states moments.
Saad El Boukhari
Let K/k be a finite abelian extension of number fields with Galois group G and let n≥2. We prove, assuming the relevant p-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture. We construct a rank-one leading term from the first derivatives at s=1−n of the S-truncated Artin L-functions and show that it satisfies an integral annihilation property. We then attach to this leading term a fractional ideal of Qp[G] and prove that, up to the natural torsion factor coming from K2n−1(OK), this ideal annihilates the even K-group K2n−2(OK,S). The proof uses determinant methods, Σ-modified étale complexes, and a cancellation argument which removes the auxiliary Euler factors.
Satoshi Naito, Daisuke Sagaki
We give an explicit description, in terms of the quantum Bruhat graph, of the (torus-equivariant) 3-point, genus 0, K-theoretic Gromov-Witten invariants ⟨O(−λ),Ow,Ou⟩d for the (full) flag manifold X=G/B, where O(−λ) denotes the class in the (torus-equivariant) K-theory ring KT(X) of X of the line bundle OX(−λ)=G×BCλ over X=G/B associated to a weight λ∈Wϖi lying in the Weyl group orbit of a minuscule fundamental weight ϖi, and Ou, Ow are the Schubert and opposite Schubert classes in KT(X) for u,w∈W. This result can be thought of as a partial generalization of the quantum K-theoretic divisor axiom, which we obtained in our previous work; our proof utilizes a generalization of the Chevalley formula in the (torus-equivariant) quantum K-theory ring QKT(X) of X, which computes the quantum product with the line bundle class O(−λ) associated to the weight λ above.
Roozbeh Hazrat, Promit Mukherjee, Sujit Kumar Sardar
Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian--Pask algebra of the associated 2-graph. The main objective of this paper is to extend textile systems to dimension 3 so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for 3-graphs extending the well-known connection between 2-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and 3-graphs. In particular, our investigation shows that the conditions required to form a 3-graph from a 3-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated 3-graphs, thus establishing that our construction is homologically consistent with 3-graphs.
Chachay Victor
In this paper, we give an enriched count for the 10 lines on a smooth del Pezzo surface S of degree 5 as the quadratic degree of the Euler class of a locally free sheaf. To do this, we see S as a section of the quintic del Pezzo threefold V 5 and the lines on S as a sub-scheme of the P 2 of lines on V 5 . We then use some oriented Schubert calculus and Bott residue formula with N SL 2 -action to compute the degree of the Euler class.
Zhenxing Di, Liping Li, Li Liang +1
We develop a unified approach, based on projectively coresolved Gorenstein flat (PGF) objects, for constructing flat model structures on diagram categories. Specifically, we show that PGF objects in such categories are fully determined by their objectwise components, which in turn enables us to establish hereditary abelian model structures whose trivial cofibrant objects are precisely the flat objects. As an application, we reobtain flat model structures on Q-shaped derived categories, thereby providing a common framework that subsumes classical constructions for chain complexes. Moreover, we obtain an explicit description of the cofibrant objects in these models.