The homotopy category of pseudofunctors and translation cohomology

We develop the obstruction theory of the 2-category of abelian track categories, pseudofunctors and pseudonatural transformations by using the cohomology of categories. The obstructions are defined in Baues-Wirsching cohomology groups. We introduce translation cohomology to classify endomorphisms in the 2-category of abelian track categories. In a sequel to this paper we will show, under certain conditions which are satisfied by all homotopy categories, that a translation cohomology class determines the exact triangles of a triangulated category.     

Grothendieck standard conjectures,morphic cohomology and Hodge index theorem

Using morphic cohomology, we produce a sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. A refinement of Hodge structures is given, and with the assumption of morphic conjectures, we prove a Hodge index theorem. We answer a question of Friedlander and Lawson by assuming the Grothendieck standard conjecture B and prove that the topological filtration from morphic cohomology is equal to the Grothendieck arithmetic filtration for some cases.     

Modules over motivic cohomology

For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of motives. The proof makes use of some highly structured models for motivic stable homotopy theory, motivic Spanier-Whitehead duality, the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper. Working with rational coefficients, we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory.     

On the cohomology rings of small categories

Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.     

Decompositions of categories over posets and cohomology of categories

Let ? be a small category. We present some results which describe cohomology groups and homotopy colimits of functors defined over ? using cohomology groups and homotopy colimits over certain categories associated to functors from ? to posets. Received: 3 May 1999     

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).     

Rational isomorphisms between K-theories and cohomology theories

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the Segre map) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the topological filtration on singular cohomology groups in K-theoretic terms.     

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.     

Hochschild cohomology of ring objects in monoidal categories

We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.     

Clifford semigroups as functors and their cohomology

We prove that the category of Clifford semigroups and prehomomorphisms CSP\mathcal{CSP} is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we find a very close relationship between the cohomology groups of a commutative inverse monoid and the cohomology groups of the colimit group of the functor giving the monoid.     

Intrinsic characterization of Alexander-Spanier cohomology groups of compactifications

In this paper we construct a uniform Alexander-Spanier cohomology functor from the category of pairs of uniform spaces to the category of abelian groups. We show that this functor satisfies all Eilenberg-Steenrod axioms on the category of pairs of precompact uniform spaces, is precompact uniform shape invariant and intrinsically, in terms of uniform structures, describes the Alexander-Spanier cohomology groups of compactifications of completely regular spaces.     

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     

Discrete morse theory and the cohomology ring

In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].

     

Quantum superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.     

Singular Hochschild cohomology via the singularity category

We show that the singular Hochschild cohomology (= Tate–Hochschild cohomology) of an algebra A is isomorphic, as a graded algebra, to the Hochschild cohomology of the differential graded enhancement of the singularity category of A. The existence of such an isomorphism is suggested by recent work by Zhengfang Wang.     

On the sectional category of subgroup inclusions and Adamson cohomology theory

The sectional category of a subgroup inclusion $H?G$ can be defined as the sectional category of the corresponding map between Eilenberg–MacLane spaces. We extend a characterization of topological complexity of aspherical spaces given by Farber, Grant, Lupton and Oprea to the context of sectional category of subgroup inclusions and investigate it by means of Adamson cohomology theory.     

Four-cocycles in commutative semigroup cohomology

Semigroup Forum - The symmetric cohomology group of a commutative semigroup is defined in dimension 4 and is shown to equal its triple cohomology group.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

Resolution of unbounded complexes in Grothendieck categories

As Spaltenstein showed, the category of unbounded complexes of sheaves on a topological space has enough K-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives.