Mostrando recursos 1 - 20 de 40

  1. 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...

  2. 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...

  3. 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...

  4. 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.

  5. 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...

  6. 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.

  7. 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...

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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...

  13. 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.

  14. $K\mkern-2mu$-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.

  15. Motivic complexes over nonperfect fields

    Suslin, Andrei
    We show that the theory of motivic complexes developed by Voevodsky over perfect fields works over nonperfect fields as well provided that we work with sheaves with transfers of [math] -modules ( [math] ). In particular we show that every homotopy invariant sheaf with transfers of [math] -modules is strictly homotopy invariant.

  16. Equivariant vector bundles, their derived category and $K$-theory on affine schemes

    Krishna, Amalendu; Ravi, Charanya
    Let [math] be an affine group scheme over a noetherian commutative ring [math] . We show that every [math] -equivariant vector bundle on an affine toric scheme over [math] with [math] -action is equivariantly extended from [math] for several cases of [math] and  [math] . ¶ We show that, given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant [math] -theories as well as equivariant [math] -theories.

  17. Longitudes in $\mathrm{SL}_2$ representations of link groups and Milnor–Witt $K_2$-groups of fields

    Nosaka, Takefumi
    We describe an arithmetic [math] -valued invariant for longitudes of a link [math] , obtained from an [math] representation of the link group. Furthermore, we show a nontriviality on the elements, and compute the elements for some links. As an application, we develop a method for computing longitudes in [math] representations for link groups, where [math] is the universal covering group of  [math] .

  18. Low-dimensional Milnor–Witt stems over $\mathbb R$

    Dugger, Daniel; Isaksen, Daniel
    We compute some motivic stable homotopy groups over [math] . For [math] , we describe the motivic stable homotopy groups [math] of a completion of the motivic sphere spectrum. These are the first four Milnor–Witt stems. We start with the known [math] groups over [math] and apply the [math] -Bockstein spectral sequence to obtain [math] groups over [math] . This is the input to an Adams spectral sequence, which collapses in our low-dimensional range.

  19. On the vanishing of Hochster's $\theta$ invariant

    Walker, Mark
    Hochster’s theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. ¶ Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster’s theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in...

  20. Chern classes and compatible power operations in inertial K-theory

    Edidin, Dan; Jarvis, Tyler; Kimura, Takashi
    Let [math] be a smooth Deligne–Mumford quotient stack. In a previous paper we constructed a class of exotic products called inertial products on [math] , the Grothendieck group of vector bundles on the inertia stack [math] . In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When [math] is diagonalizable these give rise to an augmented [math] -ring structure on inertial K-theory. ¶ One well-known inertial product is the virtual product. Our results show that for toric Deligne–Mumford stacks there is a [math] -ring structure on inertial K-theory. As an example, we...

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