Brown representability for the dual

Let T be the homotopy category of all spectra. Brown proved that a homological functor H: T ^{o p} → Ab is representable if it takes coproducts to products. That is, the functors [−,h] may be characterised as the homological functors taking coproducts to products. In this article, we will prove the dual. A covariant functor H:T → Ab which takes products to products is representable; it is of the form [h,−]. Oblatum 10-VII-1997 & 23-VII-1997     

Failure of Brown representability in derived categories

Let be a triangulated category with coproducts, the full subcategory of compact objects in . If is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors are the restrictions of representable functors on , and all natural transformations are the restrictions of morphisms in . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories of rings, and homological functors which are not restrictions of representables.     

The Grothendieck duality theorem via Bousfield's techniques and Brown representability

Grothendieck proved that if is a proper morphism of nice schemes, then has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. 1It follows almost as immediately, by ``smashing' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an easy consequence.

     

Brown representability for triangulated categories with a linear action by a graded ring

Archiv der Mathematik - In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a...     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.     

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander [Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed. V. Franjou is partially supported by the LMJL—Laboratoire de Mathématiques Jean Leray, CNRS: Université de Nantes, école Centrale de Nantes, and acknowledges the hospitality and support of CRM Barcelona where the final corrections on this paper were implemented.     

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper, we study unimodular amenable groups. The first part of the paper is devoted to results on the existence of uniform families of ε-quasi tilings for these groups. First we extend constructions of Ornstein and Weiss by quantitative estimates for the covering properties of the corresponding decompositions. Then we apply the methods developed to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable amenable group into some Banach space. This result extends significantly and complements related results found in the literature. Further, using the Lindenstrauss ergodic theorem, we link our results to classical ergodic theory. We conclude with two important applications: uniform approximation of the integrated density of states on amenable Cayley graphs and almost-sure convergence of cluster densities in an amenable bond percolation model.     

Ellipsoidal mixed-integer representability

Representability results for mixed-integer linear systems play a fundamental role in optimization since they give geometric characterizations of the feasible sets that can be formulated by mixed-integer linear programming. We consider a natural extension of mixed-integer linear systems obtained by adding just one ellipsoidal inequality. The set of points that can be described, possibly using additional variables, by these systems are called ellipsoidal mixed-integer representable. In this work, we give geometric conditions that characterize ellipsoidal mixed-integer representable sets.     

Projective functors for algebraic groups

We investigate the regular subquotient category introduced by Soergel in [W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (1–3) (2000) 311–335]. A detailed study of projective functors enables one to relate those categories for semi-simple algebraic group G and its subgroup schemata G_rT. As an application we derive some information about the characters of tilting modules for G.     

Estimation of parameters for Hilbert space-valued partially observable stochastic processes

We give the asymptotic statistical theory (strong consistency and asymptotic normality) of a modified least-square-estimator for the parameters of a linear time discrete Kalman-filter-system. The method of proof uses a strong law of large numbers for martingale difference and ergodic sequences and a central limit theorem for q-dependent stationary processes.     

Plane functors

Tensor products of functors over the set category are studied in this work. The concept of a plane functor is defined and a criterion under which a functor is planar is given, by means of which the existence of injective envelopes in the functor category over the set category is proved.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 577–586, April, 1976.In conclusion, the author wishes to express his deep appreciation to A. Kh. Livshits for his deep attention to the work.     

Closure properties for fitting functors

Fitting functors have been introduced as a generalization of injectors and radicals for Fitting classes of finite (soluble) groups. This paper provides definitions and properties of certain types of Fitting functors which resemble injectors and radicals for Fitting classes with additional closure properties, most prominently subgroup- or quotient-closed Fitting classes and Fischer classes.     

Enright functors for Kac-Moody superalgebras

A simple generalization of the Enright functor associated with a non-isotropic simple root of Kac-Moody superalgebras is introduced. Two applications for a Kac-Moody superalgebra without isotropic simple root are given: the uniqueness (up to scalar) of homomorphisms between Verma modules and the Malikov-Feigin-Fuks type singular vector formula. The braid relations of the Enright functors are also discussed.     

Finite representability of operators

We introduce operator local supportability as a new type of operator finite representability that generalizes Bellenot finite representability. We prove that local supportability and local representability are mutually independent. New examples of both types of finite representability are given. For instance, for every operator T, we prove that is locally supportable in . We also prove that, given an operator T with range in , T∗ is locally representable in .