Mackey Functors and Control of Fusion

This paper presents an algebraic approach to Mislin's theoremthat control of p-fusion is equivalent to inducing a mod-p cohomologyisomorphism. There are consequences for the cohomology of p-permutationmodules. 2000 Mathematics Subject Classification 20J06 (primary),20C05 (secondary).     

Stratifications and Mackey Functors I: Functors For a Single Group

In the context of Mackey functors we introduce a category whichis analogous to the category of modules for a quasi-hereditaryalgebra which have a filtration by standard objects. Many ofthe constructions which work for quasi-hereditary algebras canbe done in this new context. In particular, we construct ananalogue of the ‘Ringel dual’, which turns out hereto be a standardly stratified algebra. The Mackey functors whichplay the role of the standard objects are constructed in thesame way as functors which have been used previously in parametrizingthe simple Mackey functors, but instead of using simple modulesin their construction (as was done before) we use p-permutationmodules. These Mackey functors are obtained as adjoints of theoperations of forming the Brauer quotient and its dual. Thefiltrations which have these Mackey functors as their factorsare closely related to the filtrations whose terms are the sumof induction maps from specified subgroups, or are the commonkernel of restriction maps to these subgroups. These latterfiltrations appear in Conlon's decomposition theorems for theGreen ring, as well as in other places, where they arise quitenaturally. 2000 Mathematics Subject Classification: primary 20C20; secondary20J05, 19A22, 16G70, 16E60.     

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.     

On the Projective Dimensions of Mackey Functors

We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over ?.     

函子的广义逆

本文建立了函子的广义逆．得到：（１）函子的广义逆从不同于视矩阵为态射的另一角度，包含了“矩阵的广义逆”．（２）函子的广义道与态射的广义逆互不蕴含，但却有异曲同工之处．     

Functors of log Artin rings

This paper gives a generalization of the theory of functors of Artin rings in the framework of log geometry. In the final section we apply it to the log smooth deformation theory. Received: 4 August 1997 / Revised version: 13 January 1998     

Derived Functors of Graded Theories

The homotopy category of one point unions of spheres and the homotopy category of Eilenberg–MacLane spaces are examples of graded theories. Spaces yield models of these theories which can be described in terms of iterated comma categories given by functors. The associated derived functors play an important role in spectral sequences computing homotopy groups. Using cross-effect methods, the derived functors are studied and the existence of a vanishing line is proven.     

On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C ^* (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C ^* with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T?=?(T, m _T , e _T ) is a monad and G?=?(G, δ _G , ε _G ) is a comonad on ${\mathbb{A}}$ , additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ^???, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.     

The p-Blocks of the Mackey Algebra

Let p be a prime number. This paper describes the primitive idempotents and prime spectrum of the crossed Burnside algebra of a finite group over a p-local ring. The main application is a formula for the block idempotents of the p-local Mackey algebra of the group, in terms of the corresponding bloks of the group algebra.     

Derived Functors of Inverse Limits Revisited

We prove, correct and extend several results of an earlier paperof ours (using and recalling several of our later papers) aboutthe derived functors of projective limit in abelian categories.In particular we prove that if C is an abelian category satisfyingthe Grothendieck axioms AB3 and AB4* and having a set of generatorsthen the first derived functor of projective limit vanisheson so-called Mittag-Leffler sequences in C. The recent examplesgiven by Deligne and Neeman show that the condition that thecategory has a set of generators is necessary. The conditionAB4* is also necessary, and indeed we give for each integerm 1 an example of a Grothendieck category C_m and a Mittag-Lefflersequence in C_m for which the derived functors of its projectivelimit vanish in all positive degrees except m. This leads toa systematic study of derived functors of infinite productsin Grothendieck categories. Several explicit examples of theapplications of these functors are also studied.     

Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.     

On Quantization Functors of Lie Bialgebras

We show that the set of quantization functors of Lie bialgrebras has the structure of a torsor. Then we show that the Etingof–Kazhdan map is a morphism of torsors. We compute the infinitesimal of this map. As a corollary, we show that the quantization functors of finite-dimensional Lie bialgebras are independent of the choice of an associator.     

Torsion Functors of Local Cohomology Modules

Through a study of torsion functors of local cohomology modules we improve some non-finiteness results on the top non-zero local cohomology modules with respect to an ideal.     

Normal Functors of Degree ≥ 3

Mathematical Notes -     

Divisibility and Factorization of Kernel Functors

Abstract

In this article we explore the semigroup of kernel functors associated to a generalized discrete valuation ring. We will show that, among other interesting features, this semigroup is linearly ordered under divisibility, and we will present a factorization of kernel functors in terms of irreducible elements. The asymmetry of the divisibility condition will also be considered. In a way, these results are closely related to (and in the spirit of) the corresponding results obtained by Brungs for the lattice of ideals of this type of rings.