Mostrando recursos 1 - 20 de 46

  1. Stable $\mathbb{A}^1$-connectivity over Dedekind schemes

    Schmidt, Johannes; Strunk, Florian
    We show that [math] -localization decreases the stable connectivity by at most one over a Dedekind scheme with infinite residue fields. For the proof, we establish a version of Gabber’s geometric presentation lemma over a henselian discrete valuation ring with infinite residue field.

  2. Connectedness of cup products for polynomial representations of $\mathrm{GL}_n$ and applications

    Touzé, Antoine
    We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, such as the cohomological behavior of the Schur functor.

  3. A fixed point theorem on noncompact manifolds

    Hochs, Peter; Wang, Hang
    We generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using [math] -theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah–Segal–Singer’s result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant [math] -theory and...

  4. Algebraic $K$-theory of quotient stacks

    Krishna, Amalendu; Ravi, Charanya
    We prove some fundamental results like localization, excision, Nisnevich descent, and the regular blow-up formula for the algebraic [math] -theory of certain stack quotients of schemes with affine group scheme actions. We show that the homotopy [math] -theory of such stacks is homotopy invariant. This implies a similar homotopy invariance property of the algebraic [math] -theory with coefficients.

  5. Algebraic $K$-theory and a semifinite Fuglede–Kadison determinant

    Hochs, Peter; Kaad, Jens; Schemaitat, André
    In this paper we apply algebraic [math] -theory techniques to construct a Fuglede–Kadison type determinant for a semifinite von Neumann algebra equipped with a fixed trace. Our construction is based on the approach to determinants for Banach algebras developed by Skandalis and de la Harpe. This approach can be extended to the semifinite case since the first topological [math] -group of the trace ideal in a semifinite von Neumann algebra is trivial. Along the way we also improve the methods of Skandalis and de la Harpe by considering relative [math] -groups with respect to an ideal instead of the usual...

  6. An explicit basis for the rational higher Chow groups of abelian number fields

    Kerr, Matthew; Yang, Yu
    We review and simplify A. Beĭlinson’s construction of a basis for the motivic cohomology of a point over a cyclotomic field, then promote the basis elements to higher Chow cycles and evaluate the KLM regulator map on them.

  7. Cohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5

    Cao, Yang
    Dans cet article, où le corps de base est un corps de caractéristique zéro quelconque, pour [math] une variété géométriquement cellulaire, on étudie le quotient du troisième groupe de cohomologie non ramifiée [math] par sa partie constante. Pour [math] une compactification lisse d’un torseur universel sur une surface géométriquement rationnelle, on montre que ce quotient est fini. Pour [math] une surface de del Pezzo de degré [math] , on montre que ce quotient est trivial, sauf si [math] est une surface de del Pezzo de degré 8 d’un type particulier. ¶ We consider geometrically cellular varieties [math] over an arbitrary field...

  8. Equivariant noncommutative motives

    Tabuada, Gonçalo
    Given a finite group [math] , we develop a theory of [math] -equivariant noncommutative motives. This theory provides a well-adapted framework for the study of [math] -schemes, Picard groups of schemes, [math] -algebras, [math] -cocycles, [math] -equivariant algebraic [math] -theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin’s theory. As a first application, we extend Panin’s computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfect complexes of a [math] -scheme [math] admits a full exceptional collection of...

  9. Abstract tilting theory for quivers and related categories

    Groth, Moritz; Šťovíček, Jan
    We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. ¶ Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However, even over a field our main result leads to new derived equivalences, for example, for not necessarily finite or acyclic quivers. ¶ Our...

  10. Suslin's moving lemma with modulus

    Kai, Wataru; Miyazaki, Hiroyasu
    The moving lemma of Suslin (also known as the generic equidimensionality theorem) states that a cycle on [math] meeting all faces properly can be moved so that it becomes equidimensional over [math] . This leads to an isomorphism of motivic Borel–Moore homology and higher Chow groups. ¶ In this short paper we formulate and prove a variant of this. It leads to a modulus version of the isomorphism, in an appropriate pro setting.

  11. Localization, Whitehead groups and the Atiyah conjecture

    Lück, Wolfgang; Linnell, Peter
    Let [math] be the [math] -group of square matrices over [math] which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let [math] be the division closure of [math] in the algebra [math] of operators affiliated to the group von Neumann algebra. Let [math] be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let [math] be a torsionfree group which belongs to [math] . Then we prove that [math] is isomorphic to [math] . Furthermore we show that [math] is a...

  12. Hochschild homology, lax codescent, and duplicial structure

    Garner, Richard; Lack, Stephen; Slevin, Paul
    We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Böhm and Ştefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.

  13. A plethora of inertial products

    Edidin, Dan; Jarvis, Tyler; Kimura, Takashi
    For a smooth Deligne–Mumford stack [math] , we describe a large number of inertial products on [math] and [math] and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on [math] and an inertial product on [math] and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle [math] on [math] defines two new inertial pairs. We recover, as special cases, the orbifold products considered by Chen and Ruan (2004), Abramovich, Graber and Vistoli (2002), Fantechi and Göttsche (2003), Jarvis, Kaufmann...

  14. The joint spectral flow and localization of the indices of elliptic operators

    Kubota, Yosuke
    We introduce the notion of the joint spectral flow, which is a generalization of the spectral flow, by using Segal’s model of the connective [math] -theory spectrum. We apply it for some localization results of indices motivated by Witten’s deformation of Dirac operators, and rephrase some analytic techniques in terms of topology.

  15. On some negative motivic homology groups

    Kohrita, Tohru
    For an arbitrary separated scheme [math] of finite type over a finite field [math] and a negative integer [math] , we prove, under the assumption of resolution of singularities, that [math] is canonically isomorphic to [math] if [math] or [math] , and [math] vanishes if [math] and [math] . As the group [math] is explicitly known, this gives a explicit calculation of motivic homology of degree [math] and weight [math] or [math] of an arbitrary scheme over a finite field.

  16. On the Deligne–Beilinson cohomology sheaves

    Barbieri-Viale, Luca
    We prove that the Deligne–Beilinson cohomology sheaves [math] are torsion-free as a consequence of the Bloch–Kato conjectures as proven by Rost and Voevodsky. This implies that [math] if [math] is unirational. For a surface [math] with [math] we show that the Albanese kernel, identified with [math] , can be characterized using the integral part of the sheaves associated to the Hodge filtration.

  17. Stable operations and cooperations in derived Witt theory with rational coefficients

    Ananyevskiy, Alexey
    The algebras of stable operations and cooperations in derived Witt theory with rational coefficients are computed and an additive description of cooperations in derived Witt theory is given. The answer is parallel to the well-known case of K-theory of real vector bundles in topology. In particular, we show that stable operations in derived Witt theory with rational coefficients are given by the values on the powers of the Bott element.

  18. Rational mixed Tate motivic graphs

    Agarwala, Susama; Patashnick, Owen
    We study the combinatorics of a subcomplex of the Bloch–Kriz cycle complex that was used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined by Gangl, Goncharov and Levin). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely combinatorial criterion for admissibility. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs...

  19. Chow groups of some generically twisted flag varieties

    Karpenko, Nikita
    We classify the split simple affine algebraic groups [math] of types A and C over a field with the property that the Chow group of the quotient variety [math] is torsion-free, where [math] is a special parabolic subgroup (e.g., a Borel subgroup) and [math] is a generic [math] -torsor (over a field extension of the base field). Examples of [math] include the adjoint groups of type A. Examples of [math] include the Severi–Brauer varieties of generic central simple algebras.

  20. $K\mkern-2mu$-theory of derivators revisited

    Muro, Fernando; Raptis, Georgios
    We define a [math] -theory for pointed right derivators and show that it agrees with Waldhausen [math] -theory in the case where the derivator arises from a good Waldhausen category. This [math] -theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator [math] -theory, as originally defined, is the best approximation to Waldhausen [math] -theory by a functor that is invariant under equivalences of derivators.

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