Motives and modules over motivic cohomology

In this Note we summarize the main results and techniques in our homotopical algebraic approach to motives. A major part of this work relies on highly structured models for motivic stable homotopy theory. For any noetherian and separated base scheme of finite Krull dimension these frameworks give rise to a homotopy theoretic meaningful study of modules over motivic cohomology. When the base scheme is $\mathrm{Spec}(k)$, for k a field of characteristic zero, the corresponding homotopy category is equivalent to Voevodsky's big category of motives. To cite this article: O. Röndigs, P.A. Østvær, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Tate cohomology lowers chromatic Bousfield classes

Let be a finite group. We use recent results of J. P. C. Greenlees and H. Sadofsky to show that the Tate homology of local spectra with respect to produces local spectra. We also show that the Bousfield class of the Tate homology of (for finite) is the same as that of . To be precise, recall that Tate homology is a functor from -spectra to -spectra. To produce a functor from spectra to spectra, we look at a spectrum as a naive -spectrum on which acts trivially, apply Tate homology, and take -fixed points. This composite is the functor we shall actually study, and we'll prove that when is finite. When , the symmetric group on letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald's functor ).

     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

Galois module structure of Tate modules

Dedicated to Professor Peter Roquette Received 1 February 1996; in final form 26 April 1996     

Tate cohomology and periodic localization of polynomial functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S–module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n)_* is independent of choices. Goodwillies general theory says that to any homotopy functor F from S–modules to S–modules, there is an associated tower under F, {P_dF}, such that FP_dF is the universal arrow to a d–excisive functor. Our first main theorem says that P_dFP_d-1F always admits a homotopy section after localization with respect to T(n)_* (and so also after localization with respect to Morava K–theory K(n)_*). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second main theorem which is equivalent to the following: for any finite group G, the Tate spectrum is weakly contractible. This strengthens and extends previous theorems of Greenlees–Sadofsky, Hovey–Sadofsky, and Mahowald–Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. Mathematics Subject Classification (2000) 55P65, 55N22, 55P60, 55P91     

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ:D₊→D⁺ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D⁺. As a consequence, we obtain an exact sequence of Mackey functors

     

Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov–Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category ??????(X) of quasi‐coherent sheaves on X is such a category and so has these features. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cofiniteness and finiteness of local cohomology modules over regular local rings

Let (R,m) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAss _R (R/I) = Assh _R (_I). It is shown that the R- module H^ht(I) _I (R) is I-cofinite if and only if cd(I,R) = ht(I). Also we present a sufficient condition under which this condition the R-module H ⁱ _I (R) is finitely generated if and only if it vanishes.     

A ∞-modules over A ∞-algebras and Hochschild cohomology for modules over algebras

For each left graded module M over a graded algebra A, a Hochschild cochain complex S*(A, M) whose homology is responsible for the existence of nontrivial structures of A _∞-modules over A _∞-algebras on the given module is constructed.     

Artinianness of local cohomology modules

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.     

The associated primes of local cohomology modules over rings of small dimension

Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H ⁱ _I (M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H ^{d − 1} _I (M) has finite support for arbitrary R, I, and M. Received: 31 October 2000 / Revised version: 8 January 2001     

Jacquet modules and Dirac cohomology

We show that Dirac cohomology of the Jacquet module of a Harish-Chandra module is a Harish-Chandra module for the corresponding Levi subgroup. We obtain an explicit formula of Dirac cohomology of the Jacquet module for most of the principal series, based on our determination of Dirac cohomology of irreducible generalized Verma modules with regular infinitesimal characters.     

Associated primes for cohomology modules

Let G be a finite group acting linearly on a finite dimensional vector space V defined over a field k of characteristic p, where p is assumed to divide the group order. Let R := S(V *) be the symmetric algebra of the dual on which G acts naturally by algebra automorphisms. We study the R^G-modules Hⁱ(G, R) for i > 0. In particular we give a formula which describes the annihilator of a general element of Hⁱ(G, R) in terms of the relative transfer ideals of R^G, and consequently prove that the associated primes of these cohomology modules are equal to the radicals of certain relative transfer ideals. Received: 5 June 2008     

Faltings' theorem for the annihilation of local cohomology modules over a Gorenstein ring

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.