Motivic cohomology and unramified cohomology of quadrics

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension ≤4 and ≥11. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics. For most of the paper we have to assume that the ground field has characteristic 0, because we use Voevodsky’s motivic cohomology. Received August 18, 1999 / final version received December 10, 1999?Published online April 19, 2000     

Syntomic cohomology as a p-adic absolute Hodge cohomology

The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients. Received: 25 September 2000 / In final form: 23 March 2001 / Published online: 28 February 2002     

Stable local cohomology

For a module having a complete injective resolution, we define a stable version of local cohomology. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this functor behaves like the usual local cohomology functor. Our main result is that when there is only one nonzero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.     

On rack cohomology

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in Carter et al. (J. Pure Appl. Algebra, 157 (2001) 135) are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in Carter et al. (Twisted quandle cohomology theory and cocycle knot invariants, math. GT/0108051). We also give a group-theoretical interpretation of the second cohomology group for racks.     

Commutative semigroup cohomology

Triple cohomology for commutative semigroups is described in concrete terms and related to existing extensions.     

Singular cohomology inL

We prove, that in the world of constructible sets, there does not exist a spaceX withH″ (X,Z) isomorphic to the rational numbers. The proof requires a result about the growth of Ext _z ^Emphasis>/i (-, Z) inside of Gödel’s constructible universeL.     

Coclass and cohomology

Recently, it was proved by Leedham-Green and others that with a finite number of exceptions, every p-group of coclass r is a quotient of one of only a finite number of p-adic uniserial space groups. In this paper we use that structure to demonstrate that there are only finitely many isomorphism classes of cohomology rings of 2-groups of coclass r with coefficients in any fixed field k of characteristic 2. In addition, there is experimental evidence indicating that in many cases successive quotients of the uniserial space groups have isomorphic cohomology rings.     

Comparing cohomology obstructions

We show that three different kinds of cohomologies - Baues-Wirsching cohomology, the (S_∗,O)-cohomology of Dwyer and Kan, and the André-Quillen cohomology of aΠ-algebra  - are isomorphic, under certain assumptions. This is then used to identify the cohomological obstructions in three general approaches to realizability problems: the track category version of Baues and Wirsching, the diagram rectifications of Dwyer, Kan, and Smith, and the Π-algebra realization of Dwyer, Kan, and Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory.     

The injectivity of Frobenius acting on cohomology and local cohomology modules

LetR be a two-dimensional normal graded ring over a field of characteristicp>0. We want to describe the tight closure of (O) in the local cohomology moduleH _R+ ² (R) using the graded module structure ofH _R+ ² (R). For this purpose we explore the condition that the Frobenius mapF: [H _R+ ² (R)]_n→[H _R+ ² (R)]_pninduced on graded pieces ofH _R+ ² (R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorD onX=Proj (R) such thatR=R (X, D)= ⊕ _n≥0H⁰(X O _X (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups Our interest is the casen<0, and in this case, a generalization of Tango's method for integral divisors enables us to show thatF _n is injective ifp is greater than a certain bound given explicitly byX andnD. This result is useful to studyF-rationality ofR. The notion ofF-rational rings in characteristicp>0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulop reduction of a rational signularity in characteristic 0 isF-rational forp≫0. Our result answers to this question affirmatively and also sheds light to behavior ofF-rationality in smallp.     

Morphic cohomology and singular cohomology of motives over the complex numbers

Morphic cohomology and singular cohomology of motives over the complex numbers are defined via the triangulated category of motives. Regarding morphic cohomology as functors defined on the triangulated category of motives, natural transformations of morphic cohomology are studied.