Stable étale realization and étale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A¹-homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero.On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.     

Nonstandard étale cohomology

A lot of good properties of étale cohomology only hold for torsion coefficients. We use ultraproducts respectively enlargement construction to define a cohomology theory that inherits the important properties of étale cohomology while allowing greater flexibility with the coefficients. In particular, choosing coefficients ^∗Z/P^∗Z (for P an infinite prime and ^∗Z the enlargement of Z) gives a Weil cohomology, and choosing ^∗Z/l^h∗Z (for l a finite prime and h an infinite number) allows comparison with ordinary l-adic cohomology. More generally, for every N∈^∗Z, we get a category of ^∗Z/N^∗Z-constructible sheaves with good properties.     

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids.Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence.We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E₁-term consists entirely of vector bundle valued cohomology groups.Our theory works for differentiable, holomorphic and algebraic stacks.     

Deformation spaces of one-dimensional formal modules and their cohomology

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over equipped with a level-m-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the ?-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces Mm.     

Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

Cohomology of Crystalline Smooth Sheaves

We define and study a candidate for 'arithmetic' cohomology theory with values in crystalline p-adic smooth sheaves. We show that it injects into arithmetic étale cohomology and prove a duality result.     

The Weil-��tale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.     

Nearby cycles for log smooth families

We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.     

Duality in the cohomology of crystalline local systems

Let k be a perfect field of a positive characteristic p, K-the fraction field of the ring of Witt vectors W(k) Let X be a smooth and proper scheme over W(k). We present a candidate for a cohomology theory with coefficients in crystalline local systems: p -adic étale local systems on X_K characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber X _k. We show that this cohomology satysfies a duality theorem.     

Oriented cohomology and motivic decompositions of relative cellular spaces

For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL.     

Étale wild kernels of exceptional number fields

We clarify the relationship between higher étale wild kernels of a number field at the prime 2 and the Galois-coinvariants of Tate-twisted class groups in the 2-cyclotomic tower of the field. We also determine the relationship between the étale wild kernel and the group of infinitely divisible elements of H²(F,Z₂(j+1)){2}.     

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g?3 and let M₀ be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M₀ have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan-Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.     

On the zeroth stable A1-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.     

Some finiteness results for étale cohomology

We prove some finiteness theorems for the étale cohomology, Borel-Moore homology and cohomology with proper supports with divisible coefficients of schemes of finite type over a finite or p-adic field. This yields vanishing results for their l-adic cohomology, proving part of a conjecture of Jannsen.     

Topological André-Quillen Cohomology and E∞ André-Quillen Cohomology

We show that the André-Quillen cohomology of an E_∞ simplicial algebra with arbitrary coefficients and the topological André-Quillen cohomology of an E_∞ ring spectrum with Eilenberg-Mac Lane coefficients may be calculated as the André-Quillen cohomology of an associated E_∞ differential graded algebra.     

A polar de Rham theorem

We prove an analogue of the de Rham theorem for polar homology; that the polar homology HP_q(X) of a smooth projective variety X is isomorphic to its H^n,n−q Dolbeault cohomology group. This analogue can be regarded as a geometric complexification where arbitrary (sub)manifolds are replaced by complex (sub)manifolds and de Rham's operator d is replaced by Dolbeault's .     

Torsion algebraic cycles and étale cobordism

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of ?-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

Motivic cohomology of the simplicial motive of a Rost variety

We compute the motivic cohomology groups of the simplicial motive X_θ of a Rost variety for an n-symbol θ in Galois cohomology of a field. As an application we compute the kernel and cokernel of multiplication by θ in Galois cohomology. We also show that the reduced norm map on K₂ of a division algebra of square-free degree is injective.