Artinian objects in the category of functors between IF_{2}-vector spaces

Geoffrey M.L. Powell

LAGA, UMR 7539, Institut Galilée, Université Paris 13, Villetaneuse, France

The category F(k) associated to a field k is the category of functors V_{k}^{f} → V_{k}, where V_{k} is the category of k-vector spaces and V_{k}^{f} is the full sub-category of finite-dimensional spaces. When the field k is infinite, this category is related to the study of polynomial representations of the general linear groups on k. The case of a finite field is less-well understood but is of interest since it is related to the category of unstable modules over the k-Steenrod algebra, a basic object of consideration in algebraic topology. The calculation of extension groups in F between polynomial functors is equivalent to the calculation of MacLane cohomology with coefficients; more recently, Suslin has established that this is in fact equal to the stable groups calculated over the general linear groups, an affirmative response to the conjecture that topological Hochschild homology is equal to stable K-theory. The talk will concentrate on the structure of objects in the category rather than homological considerations; throughout it will be assumed that k = IF_{2}, the prime field with two elements.

The evaluation functor F ↦ F (IF_{2}^{n}) defines a functor F → M_{n}, where M_{n} denotes the category of left End(IF_{2}^{n})-modules. Restriction defines a functor M_{n} → GL_{n} to the category of GL_{n}IF_{2} = Aut(IF_{2}^{n})-modules. This theory has been developed by Kuhn in the ‘recollement’ framework [K2]; in particular, this provides a construction of the simple functors in F from the simple GL_{n}IF_{2} modules, as n varies. There is a connection with the category of modules over the symmetric groups S_{d}, which arises from studying the polynomial filtration of the category F. This may be described using the functor Hom_{F}(T^{d}, −) : F → mod − S_{d}, where T^{d} is the d^{th} tensor power functor, V ↦ V^{⊗d}; the S_{d} action is induced by the evident (right) action on T^{d}. This provides a natural setting for the correspondence between the representation theory of the symmetric groups and the general linear groups. The basic material on the categories F will be covered in the talks by Kuhn and Schwartz.

A functor is said to be finite if it has a finite composition series and locally finite if it is the colimit of its finite sub-objects. The full sub-category of locally finite objects is denoted by F_{ω}; the injective functors in F belong to the category F_{ω}, which has enough injectives. The non-constant indecomposable injectives are of interest because they are not finite objects. The injective envelope of the functors Λ^{1} is habitually written \overline{I} and is a uniserial object, with socle series Λ^{1}, Λ^{2}, …, Λ^{n}, …. However, for all other non-constant indecomposable injectives, the structure is far richer and there is little hope of giving an explicit description of the full lattice of sub-objects. Nevertheless, Lionel Schwartz has proposed the following conjecture, in conjunction with Kuhn and Lannes:

The Artinian Conjecture The indecomposable injectives in F are Artinian objects in F (that is: every descending sequence of sub-objects stabilizes).

The talk will discuss this conjecture, along with related questions. A reduction is provided by introducing a co-Weyl object, J_{S} ∈ F, associated to the simple functor S; this may be defined as the functor R_{n}S(IF_{2}^{n}), where R_{n} is the right adjoint to the functor F → M_{n} and n is the least integer such that S(IF_{2}^{n}) ≠ 0. A concrete description is that J_{S} is the largest sub-functor J ⊂ I_{S} of the injective envelope of S for which J(IF_{2}^{n}) = S(IF_{2}^{n}). The fundamental importance of these objects is shown by the following result, analogous to a property of highest weight categories.

Theorem 1 [P2] Each indecomposable injective in F admits a finite filtration with each filtration quotient isomorphic to some co-Weyl object.

It follows immediately that the Artinian conjecture reduces to the affirmation that each object J_{S} is Artinian.

The essential tool in the study of global properties of the lattice of subobjects of a functor is the shift functor Δ̃ : F → F, defined by Δ̃F (V) = F (V ⊕IF_{2}); this gives a generalization to the category Fof restriction arguments in representation theory. The functor Δ̃ is the right adjoint to the functor −⊗P, where P is the projective (not locally finite) functor V ↦ IF_{2}[V]; this admits a natural filtration by powers of the augmentation ideal and hence quotients P → q_{n}P. The right adjoints to the functors − ⊗ q_{n}P are written as [p_{n}Δ̃] and there are inclusions of endo-functors of F, [p_{n}Δ̃] ⊂ Δ̃. The quotients ˜∇_{n} = Δ̃/[p_{n−1}Δ̃] restrict to functors F_{ω} → F_{ω} which are not exact but preserve injections and surjections [P]. The following result establishes the most general result known concerning the structure of these functors:

Theorem 2 [P2] Suppose that S, n, J_{S} are as above, then ˜∇_{n+1}J_{S} = 0 and there is an inclusion J_{S} ⊂ ˜∇_{n}J_{S}. Moreover, if F ⊂ J_{S} is a proper sub-functor of J_{S}, then there exists a natural number t such that ( ˜∇_{n})^{k}F = 0.

This result is used to establish the following particular case of the Artinian conjecture:

Theorem 3 [PS] The injective object (not indecomposable) \overline{I} ⊗ \overline{I} is injective in F.

The proof of this theorem requires the study of the functors \overline{I} ⊗ Λ^{n}, as n varies. More generally, one studies the functors \overline{I} ⊗S, where S is a simple functor. One says that a functor is simple modulo finites if every proper sub-object is a finite object. The class of simple modulo finite objects is a generalization of the class of uniserial objects. It is established easily that the functor \overline{I} ⊗ S is Artinian if and only if it admits a finite filtration of which each filtration quotient is simple modulo finites. The approach to the study of \overline{I} ⊗ S relies heavily on the usage of the functors ∇, mimicking the argument of [P1].

References

[K1] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: I, Amer. J. Math. 116 (1993), 327-360.

[K2] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: II, K-Theory 8 (1994), 395-426.

[K3] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: III, K-Theory 9 (1995), 273-303.

[PS] G.M.L. POWELL with an Appendix by L. SCHWARTZ, The Artinian conjecture for I ⊗ I, To appear, J. Pure Appl. Alg..

[P] G.M.L. POWELL, Polynomial filtrations and Lannes’ T -functor, K-Theory 13 (1998), 279-304.

[P1] G.M.L. POWELL, The structure of \overline{I} ⊗ Λ^{n} in generic representation theory, J. Alg. 194, (1997), 455-466.

[P2] G.M.L. POWELL, The structure of the indecomposable injectives in generic representation theory, To appear, TAMS