arXiv subjects
Publications and source records attributed to Uwe Jannsen.
We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.
Suslin proved that for an extension K/k of algebraically closed fields the induced maps K_m(k)[n] --> K_m(K)[n] and K_m(k)/n ---> K_m(K)/n for the higher K-groups are isomorphisms, where A[n] is the subgroup of n-torsion in an abelien group, and A/n = A/nA, by definition. In this paper we generalize this to other functors and other field extensions.
In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open subvariety $U$.
For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field. He proved these conjectures for low dimensions. We prove Kato's conjecture over number fields. In particular this gives a Hasse principle for function fields F over a number field K, involving the corresponding function fields over all completions of K. We get the same results over global fields K of positive characteristic, for coefficients invertible in K. This was proved earlier by M. Kerz and S. Saito, by another method. Finally we obtain a conjecture of Kato over a finite field, and a generalization to finitely generated fields K, assuming resolution of singularities or that the coefficents are invertible in K. The latter case was again obtained earlier by M. Kerz and S. Saito, by different methods.
We establish a purely algebraic tool for studying the Iwasawa adjoints of some natural Iwasawa modules for $p$-adic Lie group extensions of number fields, by relating them to certain continuous Galois cohomology groups via a spectral sequence.
We develop a new cohomology theory in characteristic p>0, the so called F-gauge cohomology, a cohomology with values in the category of so-called F-gauges, which refines the cristalline cohomology. In this first paper we mainly discuss the theory for smooth projective varieties over perfect fields of charatceristic p. We also compare our theory with other existing structures like the F-zips of Moonen and Wedhorn and the displays of Langer and Zink.
We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to automorphisms or etale or Zariski localizations. We treat the embedded case as well as the non-embedded case, with or without a boundary, and we relate the diferent versions. In the non-embedded case, a boundary is a collection of locally principal closed subschemes. Our main tools are the stratifications by Hilbert-Samuel functions and the characteristic polyhedra introduced by H. Hironaka. In an appendix we show that the standard method used in characteristic zero - the theory of maximal contact - does not work for surfaces in positive characteristic (the counterexamples are hypersurfaces in affine threespace and work over any field of positive characteristic). In this new version, we treat the case of locally noetherian but not necessarily noetherian schemes in an appropriate way. Here one does not have a finite resolution sequence, but still a canonical resolution morphism by glueing. The same techniques allow to treat algebraic spaces and stacks.
In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g., to Hasse principles and the computation of motivic cohomology, and discuss some open questions.
We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi) semistable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application we prove that the reciprocity map introduced for smooth projective varieties over local fields by Bloch, Kato and Saito is an isomorphism after profinite completion, if the variety has good reduction or 'almost good' reduction.
For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective smooth over a finite field. We give an application to finiteness questions for some motivic cohomology groups over finite fields.