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.     

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 ?.     

$U({\mathfrak {so}}(7,{\mathbb C}))$的向量表示的范畴化

The aim of this paper is to categorify the n-th tensor power of the vector representation of U (so(7, C)). The main tools are certain singular blocks and projective functors of the BGG category of the complex Lie algebra gl n .     

$D_4$型李代数包络代数旋模的范畴化

本文我们定义复数域$C$上一般线性李代数${\rm gl}_n$ BGG 范畴的若干子范畴及其上的投射函子,利用这些子范畴和投射函子范畴化了$D_4$型李代数包络代数旋模的$n$-次张量积.     

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.     

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.     

Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension

We study finitely generated modules M over a ring R, noetherianon both sides. If M has finite Gorenstein dimension G-dim_RMin the sense of Auslander and Bridger, then it determines twoother cohomology theories besides the one given by the absolutecohomology functors . Relative cohomology functors are defined for all non-negative integers n; they treat the modules of Gorensteindimension 0 as projectives and vanish for n > G-dim_RM. Tatecohomology functors are defined for all integers n; all groups vanish if M or N has finite projective dimension. Comparisonmorphisms and link these functors. We give a self-contained treatmentof modules of finite G-dimension, establish basic propertiesof relative and Tate cohomology, and embed the comparison morphismsinto a canonical long exact sequence . We show that these results provide efficient tools for computingold and new numerical invariants of modules over commutativelocal rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.     

Projective Limits of Finite-Dimensional Lie Groups

For a topological group G we define N to be the set of all normalsubgroups modulo which G is a finite-dimensional Lie group.Call G a pro-Lie group if, firstly, G is complete, secondly,N is a filter basis, and thirdly, every identity neighborhoodof G contains some member of N. It is easy to see that everypro-Lie group G is a projective limit of the projective systemof all quotients of G modulo subgroups from N. The converseimplication emerges as a difficult proposition, but it is shownhere that any projective limit of finite-dimensional Lie groupsis a pro-Lie group. It is also shown that a closed subgroupof a pro-Lie group is a pro-Lie group, and that for any closednormal subgroup N of a pro-Lie group G, for any one parametersubgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The categoryof all pro-Lie groups and continuous group homomorphisms betweenthem is closed under the formation of all limits in the categoryof topological groups and the Lie algebra functor on the categoryof pro-Lie groups preserves all limits and quotients. 2000 MathematicsSubject Classification 22E65, 22D05, 22E20, 22A05, 54B35.     

Uno's Invariant Conjecture for the Finite Symplectic Group Sp4(q) in the Defining Characteristic

The purpose of this article is to study a categorification of the n-th tensor power of the spin representation of U(𝔰𝔬(7, ?)) by using certain subcategories and projective functors of the Bernstein–Gelfand–Gelfand (BGG) category of the complex Lie algebra 𝔤𝔩 _n .     

Über projektive Moduln und Endlichkeitshindernisse bei Transformationsgruppen

We study projective modules in the category of functors from homogeneous spaces into abelian groups. Such functors have been considered by Bredon [1]. We show that protective functors are determined by a set of ordinary projective modules over suitable group rings. The general notions are applied to give a quick proof for the product formula of the finiteness obstruction for transformation groups. These finiteness obstructions are straightforward extensions of the Swan-Wall obstructions (see e. g. Quinn [7]). They are important in the study of homotopy representations (tom Dieck — Petrie [3], [4]). This work is also related to Rothenberg [8].     

On the Exceptional Principal Bundles Over a Surface

Let G be a simple simply connected complex Lie group. Some criteriaare given for the nonexistence of exceptional principal G-bundlesover a complex projective surface. As an application, it isshown that there are no exceptional G-bundles over a surfacewhose arithmetic genus is zero or one. It is also shown thatthere are no stable exceptional G-bundles over an abelian surface.2000 Mathematics Subject Classification 32L20, 14J60.     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].     

Tensoring with Infinite-Dimensional Modules in $\mathcal {O}_0$

We show that the principal block O₀\mathcal {O}_0 of the BGG category O\mathcal {O} for a semisimple Lie algebra \frak g\frak g acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category O\mathcal {O}. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on O₀\mathcal {O}_0. Furthermore, all this generalises to parabolic subcategories of O₀\mathcal {O}_0. As an example, we present some explicit computations for the algebra \fraksl₃\frak{sl}_3.     

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)     

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof–Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are “compatible with duality”, i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof–Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld–Jimbo-type quantizations. To the memory of Paulette Libermann (1919–2007)     

Essential norms and weak compactness of integration operators

The aim of this paper is to present a general method to construct projective resolutions of globally defined Mackey functors over a field of characteristic zero and apply it to obtain explicit resolutions for inflation functors. Our method is a special case of Bouc’s method in (Proc. Symp. Pure Math. 63 (1998), 31–84) and uses global Mackey functors to construct the projective resolutions.     

Extension groups between simple Mackey functors

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.     

Free group functors

We construct group functors whose Lie algebras are free.     

Derived categories of sheaves on singular schemes with an application to reconstruction

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.     

Hecke correspondences for smooth moduli spaces of sheaves

Publications mathématiques de l'IHÉS - We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and...