Exact Functor Research Articles

A representability theorem for some huge abelian categories

We define quasi-locally presentable categories as big unions of a chain of coreflective subcategories that are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi-locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi-locally presentable, and we obtain a new proof of the Brown representability theorem. Examples of functors that are not representable are also given.

Functors for unitary representations of classical real groups and affine Hecke algebras

We define exact functors from categories of Harish–Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with “real infinitesimal character”) into the corresponding p-adic spherical unitary dual.

The linear dual of the derived category of a scheme

Let X → S be a projective morphism of schemes. We study the category D(X/S) * of S-linear exact functors D(X) → D(S), and we study the Fourier transform D(X) → D(X/S) * .

1-Dimensional representations and parabolic induction for W-algebras

A W-algebra is an associative algebra constructed from a reductive Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of W-algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan–Goodwin–Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type E8. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor.

Green functions via hyperbolic localization

Let G be a reductive algebraic group, with nilpotent cone \mathcal N and flag variety \mathcal B . We construct an exact functor from perverse sheaves on \mathcal N to locally constant sheaves on \mathcal B , and we use it to study Ext-groups and stalks of simple perverse sheaves on \mathcal N in terms of the cohomology of \mathcal B .

Algebraic K-theory and abstract homotopy theory

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer–Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.

Derived Functors Related to Wall-Crossing

The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic.

The Axiom of Infinity and Transformations j: V → V

AbstractWe suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.

The Eilenberg–Watts theorem over schemes

We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg–Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002) [6] are noncommutative P 1 -bundles in the sense of M. Van den Bergh [10].

On Mirabolic -modules

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D,G) where D is a sheaf of twisted differential operators on X, we form a left ideal D.g in D generated by the Lie algebra g, of G. Then, D/D.g is a holonomic D-module and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/D.g is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case where the group G=SL(n) acts diagonally on X = F \times F \times P^{n-1}, a triple product where F is the flag manifold for SL(n) and P^{n-1} is the projective space. We further relate D-modules on F \times F \times P^{n-1} to D-modules on the space SL(n) \times P^{n-1} via a pair, CH, HC, of adjoint functors, analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules.

On the derived DG functors

Assume that abelian categories A, B over a field admit countable direct limits and that these limits are exact. Let F : D dg(A)→ D + dg(B) be a DG quasi-functor such that the functor Ho(F) : D+(A)→ D+(B) carries D≥0(A) to D≥0(B) and such that, for every i > 0, the functor HiF : A → B is effaceable. We prove that F is canonically isomorphic to the right derived DG functor RH0(F). We also prove a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories Db dg(A), D + dg(A) as the Ext groups in the abelian category of left exact functors A → IndA . The proofs are based on a description of Drinfeld’s category of quasi-functors as the derived category of a category of sheaves.

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.

Cohomological stratification of diagram algebras

The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.

A combinatorial approach to functorial quantum sl k knot invariants

This paper contains a categorification of the ${\frak sl}(k)$ link invariant using parabolic singular blocks of category ${\cal{O}}$. Our approach is intended to be as elementary as possible, providing essentially combinatorial arguments for the main results of Sussan. The justification that our combinatorial arguments and steps are correct uses non-combinatorial geometric and representation theoretic results (e.g., the Kazhdan-Lusztig and Soergel's theorems). We take these results as granted and use them like axioms (called {\it Facts\/} in the text). We first construct an exact functor valued invariant of webs or ``special'' trivalent graphs labelled with $1, 2, k-1, k$ satisfying the MOY relations. Afterwards we extend it to the ${\frak sl}(k)$-invariant of links by passing to the derived categories. The approach of Khovanov using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the ${\frak sl}(k)$-homology, based on foams and the Kapustin-Li formula, from Mackaay, Stosic, and Vaz. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

Coinduction functor and simple comodules

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to group-graded modules is also given.

Invariance de laK-Théorie par équivalences dérivées

RésuméCes notes ont pour but de démontrer que tout foncteur exact à droite entre catégories de Waldhausen raisonnables, induisant une équivalence au niveau des catégories homotopiques, définit une équivalence d'homotopie entre les spectres deK-théorie correspondants. Cela généralise un résultat bien connu de Thomason et Trobaugh. Les ingrédients de démonstration sont une généralisation du théorème d'approximation de Waldhausen et une caractérisation combinatoire simple des équivalences dérivées. On étudie par ailleurs la localisation simpliciale des catégories de Waldhausen. On démontre qu'un foncteur (homotopiquement) exact à droite induit une équivalence de catégories homotopiques si et seulement s'il induit une équivalence au niveau des localisations simpliciales. Cela permet de faire le lien avec laK-théorie des catégories simpliciales introduite par Toën et Vezzosi.

Landweber exact formal group laws and smooth cohomology theories

The main aim of this paper is the construction of a smooth (sometimes called differential) extension \hat{MU} of the cohomology theory complex cobordism MU, using cycles for \hat{MU}(M) which are essentially proper maps W\to M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\to M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.

Construction fonctorielle de catégories de Frobenius

Let A , B be exact categories with A karoubian and M be an exact functor. Under suitable adjunction hypotheses for M, we are able to show that the direct factors of the objects of A of the form MY with Y ∈ B make up a Frobenius category which allows us to define an M-stable category for A only by quotienting. In addition, we propose a construction of an M-stable category for A , B triangulated categories and M a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck (1987) which links the two notions of M-stable category. To cite this article: V. Beck, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 × 3 -lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.

Geometric criteria for Landweber exactness

The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard's theorem and Cartier's classification of p-typical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BP-algebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem, and we use it to give a proof of the original theorem.