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Search Results: 1 - 10 of 224956 matches for " Kay Rülling "
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Additive Chow groups with higher modulus and the generalized de Rham-Witt complex
Kay Rülling
Mathematics , 2005,
Abstract: Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by TH^n(k,n;m), n,m >0. They prove that TH^n(k,n;1) is isomorphic to the group of absolute Kaehler differentials of degree n-1 over k. In this paper we generalize their result and show that TH^n(k,n;m) is isomorphic to W_m\Omega^{n-1}_k, the group of degree n-1 elements in the generalized de Rham-Witt of length m over k. This complex was defined by Hesselholt and Madsen and generalizes the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie. Before we can prove this theorem we have to generalize some classical results to the de Rham-Witt complex. We give a construction of the generalized de Rham-Witt complex for \bbmath{Z}_(p)-algebras analogous to the construction in the p-typical case. We construct a trace for finite field extensions L / k and if K is the function field of a smooth projective curve C over k and P in C is a point we define a residue map Res_P: W_m\Omega^1_K --> W_m(k), which satisfies a "sum-of-residues-equal-zero" theorem.
K-groups of reciprocity functors
Florian Ivorra,Kay Rülling
Mathematics , 2012,
Abstract: In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get close to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or K\"ahler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.
K-Groups of reciprocity functors for G_a and abelian varieties
Kay Rülling,Takao Yamazaki
Mathematics , 2013,
Abstract: We prove that the $K$-group of reciprocity functors, defined by F. Ivorra and the first author, vanishes over a perfect field as soon as one of the reciprocity functors is $\mathbb{G}_a$ and one is an abelian variety.
Vanishing of the higher direct images of the structure sheaf
Andre Chatzistamatiou,Kay Rülling
Mathematics , 2014, DOI: 10.1112/S0010437X15007435
Abstract: We prove that the higher direct images of the structure sheaf under a birational and projective morphism between excellent and regular schemes vanish.
Higher direct images of the structure sheaf in positive characteristic
Andre Chatzistamatiou,Kay Rülling
Mathematics , 2009,
Abstract: We prove vanishing of the higher direct images of the structure (and the canonical) sheaf for a proper birational morphism with source a smooth variety and target the quotient of a smooth variety by a finite group of order prime to the characteristic of the ground field. We also show that for smooth projective varieties the cohomology of the structure sheaf is a birational invariant. These results are well-known in characteristic zero.
Introductory course on $\ell$-adic sheaves and their ramification theory on curves
Lars Kindler,Kay Rülling
Mathematics , 2014,
Abstract: These are the (preliminary) notes accompanying 13 lectures given by the authors at the Clay Mathematics Institute Summer School 2014 in Madrid. The notes give an introduction into the theory of $\ell$-adic sheaves with emphasis on their ramification theory on curves.
Suslin homology of relative curves with modulus
Kay Rülling,Takao Yamazaki
Mathematics , 2015,
Abstract: We compute the Suslin homology of relative curves with modulus. This result may be regarded as a modulus version of the computation of motives for curves, due to Suslin and Voevodsky.
Hodge-Witt cohomology and Witt-rational singularities
Andre Chatzistamatiou,Kay Rülling
Mathematics , 2011,
Abstract: We prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme is a birational invariant.
Higher Chow groups with modulus and relative Milnor K-theory
Kay Rülling,Shuji Saito
Mathematics , 2015,
Abstract: Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism for all i greater or equal the dimension of X between the motivic Nisnevich cohomology of (X,D) in bidegree (i+r,r) and the i-th Nisnevich cohomology of the r-th relative Minor K-sheaf of (X,D). This generalizes the well-known isomorphism in the case D=0. We use this to prove a certain Zariski descent property for the motivic cohomology of the pair (\A^1_k, (m+1){0}).
Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$
Pierre Berthelot,Hélène Esnault,Kay Rülling
Mathematics , 2010,
Abstract: Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometrically connected and of Hodge type $\geq 1$ in positive degrees. Then we show that the number of $k'$-rational points of $X$ satisfies the congruence $|X(k')| \equiv 1$ mod $|k'|$. Thanks to \cite{BBE07}, we deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\sO_{X_k,\Q})$, for $q > 0$. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjective projective morphism $f : Y \to X$.
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