Theory and Applications of Categories, Vol. 6, No. 2, 1999, pp. 25–32.

A BICATEGORICAL APPROACH TO STATIC MODULES

Dedicated to J. Lambek on the occasion of his 75th birthday

RENATO BETTI

ABSTRACT. The purpose of this paper is to indicate some bicategorical properties of ring theory. In this interaction, static modules are analyzed.

Introduction

Categorical generalizations of ring theory usually start from the observation that rings are (one-object) categories enriched in the monoidal closed category Ab of abelian groups. In this setting, R-S-bimodules are profunctors R +^{}S, i.e. functors R^{op} × S \overline{}Ab,

and their composition R +^{P}^{}S +^{Q}^{}T is provided by the tensor product Q ⊗_{S} P over the ring S. So, one has a distributive bicategory Mod, whose objects are the (noncommutative) rings, whose arrows are the profunctors and whose 2-cells are the module morphisms.

To say that Mod is a distributive bicategory is to say that it admits local colimits which distribute over composition on both sides. Moreover, Mod is biclosed (see e.g. [6]), in the sense that it admits right Kan extensions and right liftings: in other words, compositions with an arrow P : R +^{}S are functors:

SMod \overline{− ⊗_{S} P_{}}_{R}Mod
Mod_{R} \overline{P ⊗_{R} −_{}}Mod_{S}

having right adjoints (here _{S}Mod denotes the category of left S-modules and similarly Mod_{S} is the category of right S-modules).

In particular the right Kan extension of the left E-module N along the E-R-bimodule P is the R-module hom^{E}(P, N) of E-module morphisms P ^{}N. Analogously, the right lifting of the right R-module M along the E-R-bimodule P is the right E-module hom_{R}(P, M) of R-module morphisms P ^{}M.

The main observation which relates Mod with a relevant property of modules was first formulated by Lawvere [6]: finitely generated projective modules are exactly those profunctors which, when regarded as arrows in Mod, have a right adjoint. To prove this fact one should consider the dual module P^{∗} = hom_{R}(P, R) of P. It is easy to see that the adjointness P −| P^{∗} has the evaluation x ⊗ f ^{}f(x) as counit P ⊗_{Z} P^{∗} ^{}R and the assignment of a so-called dual basis Z ^{}P^{∗} ⊗_{R} P as unit.

Research partially supported by the Italian MURST.

Received by the editors 1999 February 15 and, in revised form, 1999 September 8.

Published on 1999 November 30.

1991 Mathematics Subject Classification: 18D05, 18E10, 13E05.

Key words and phrases: Bicategory, module, Clifford theory.

\begin{matrix}c \\ ◯ \end{matrix} Renato Betti 1999. Permission to copy for private use granted.

In this paper we want to concentrate on the fact that categorical notions often are directly connected to the notions of ring theory, in general allowing one to formulate problems and prove results in a more conceptual way. Here we consider properties relative to static and e-static modules (with respect to a given module P) i.e. relative to those modules V for which the evaluation P ⊗_{E} hom_{R}(P, V) ^{}V is an isomorphism (or an epimorphism). The calculus of adjoints in a bicategory and its general properties are put in use to study properties of static and e-static modules including aspects of Morita theory and an extension of the Dade-Cline version of Clifford theory to one in general ring theory (see e.g. Cline [3], Dade [4], Alperin [1], Nauman [8]).

1. Static and e-static modules

Notations and the formulation of the problem are from Lambek [5] which regards static modules with respect to a projective module P.

Given a bimodule P : E +^{}R, consider the adjoint pair of functors:

Mod_{R} [figure: hom_{R}(P, −); \underline{P ⊗_{E} −}] Mod_{E}  (1)

(P ⊗_{E} − −| hom_{R}(P, −)).

By general properties of adjoint pairs, these restrict to an equivalence between the full subcategory (F ix ϵ) of Mod_{R} consisting of the modules U whose counit

ϵ_{U} : U ⊗_{E} hom_{R}(P, U) ^{}U

is an isomorphism and the full subcategory (F ix η) of Mod_{E} consisting of those modules V whose unit η_{V} : V ^{}hom_{R}(P, P ⊗_{E} V) is an isomorphism.

In case P is a right R-module and E = hom_{R}(P, P) is its ring of endomorphisms, the modules in (F ix ϵ) are said to be static with respect to P. If ϵ_{U} is an epimorphism, U is said to be e-static and (Epi ϵ) denotes the full subcategory of e-static modules. In [7], McMaster proves that e-static modules relative to a projective P are exactly those modules which are cotorsionfree with respect to P.

Observe that the restriction of the adjunction (1) to the subcategory (Epi ϵ) gives an adjunction:

[figure: (Epi ϵ); (Mono η)]

where (Mono η) is the full subcategory of Mod_{E} constituted by the modules U with a monomorphic unit η_{U} : U ^{}hom_{R}(P, P ⊗_{E} U).

When P is finitely generated and projective then hom_{R}(P, −) ≅ P^{∗} ⊗_{R} −, hence V is a static module if and only if

P ⊗_{E} P^{∗} ⊗_{R} V ≅ V

i.e. V equifies the trace ideal τ : P ⊗_{E} P^{∗} → R.

1.1. Theorem. If P is finitely generated and projective, then it is the equifier of its trace ideal.

Proof. The module P equifies the trace ideal τ : P ⊗_{E} P^{∗} ^{}R because P^{∗} ⊗_{R} P ≅ 1_{E}. Moreover, any static module U factors uniquely up to isomorphism in the form U ≅ P ⊗_{E} hom_{R}(P, U) because hom_{R}(P, R) ⊗_{R} U ≅ hom_{R}(P, U).

If moreover P is a generator of Mod_{R}, then the trace ideal τ is an isomorphism and any module in Mod_{R} is static. This is one of the main results of Morita theory, namely the assertion that in this case Mod_{R} and Mod_{E} are equivalent categories (i.e. R and E are Morita equivalent rings).

In general, for a bimodule P : E +^{}R with E = hom_{R}(P, P), one has:

1.2. Theorem. Any finitely generated projective E-module V is in (F ix η).

Proof. It is enough to remind the reader that, in any bicategory with right liftings, the functor hom_{R}(P, −) preserves composition with right adjoints.

In the case of Mod, if V has a right adjoint, for any W in Mod_{E} one has the following sequence of bijections, natural in W:

W ^{}hom_{R}(P, P) ⊗_{E} V
−
P ⊗_{E} W ^{}P ⊗_{E} V
−
W ^{}hom_{R}(P, P ⊗_{E} V)

From the previous theorem, one has that the subcategory of finitely generated Emodules is contained in (F ix η). Nauman [8] particularly considers this subcategory and, through his result ([8], theorem 3.7) one concludes:

1.3. Corollary. A right R-module U weakly divides P in Mod_{R} (i.e. it is a direct summand of finitely many copies of P) if and only if it is of the form P ⊗_{E} V for a finitely generated and projective E-module V.

Let ( )′ denote the functor which associates to any R-module V the image of the evaluation:

ϵ_{V} :P ⊗_{E} hom_{R}(P, V) \overline{e_{V} ^{}}V′ m_{V}^{}V  (2)

(here e_{V} is epi and m_{V} is mono).

1.4. Theorem. The functor ( )′ lands in (Epi ϵ) and it is the right adjoint to the embedding (Epi ϵ)^{⊂} Mod_{R}.

Proof. First one proves that V′ is in (Epi ϵ). Consider the diagram:

[figure: P ⊗_{E} hom_{R}(P, V′); ϵ_{V′}; V′; P ⊗_{E} hom_{R}(P, V); m′; ϵ_{V}; V; m_{V}]

Here, m′ denotes P ⊗_{E}hom_{R}(P, m_{V}). Now, hom_{R}(P, m_{V}) is mono because hom_{R}(P, −) is a left exact functor. Moreover, under the bijection of P ⊗_{E} − −| hom_{R}(P, −), the evaluation (2) corresponds to the composite:

[figure: hom_{R}(P, V); ẽ_{V}; hom_{R}(P, V′); hom_{R}(P, m_{V}); hom_{R}(P, V)]

which is the identity of hom_{R}(P, V). Hence hom_{R}(P, m_{V}) is also epi. As a consequence m′ is an isomorphism and from

e_{V} · (P ⊗_{E} hom_{R}(P, m_{V})) = ϵ_{V′}

one has that also ϵ_{V′} is epi.

For the adjointness, it is easy to prove that, if U is in (Epi ϵ), the universal property of images provides a natural bijection:

U ^{}V′
−
U ^{}V

given by composition with m_{V}.

A known characterization of static modules is provided by Auslander equivalence [2] (see also Alperin [1]): a module V is static with respect to P if and only if there is a presentation (i.e. an exact sequence):

[figure: Σ_{Y} P; Σ_{X}P; V; 0]

such that the application of the functor hom_{R}(P, −) gives another exact sequence (here X and Y are sets and Σ denotes coproduct). In case P is projective, the presentation is enough, as proved directly by Lambek [5].

By substituting the presentation of V with a condition on its generation by P, e-static modules can be characterized. Say that the R-module V is generated by P if there is an epimorphism Σ_{X}P ^{}V. Moreover, say that the generators are preserved if the application of the functor hom_{R}(P, −) gives another epimorphism hom_{R}(P, Σ_{X}P) ^{}hom_{R}(P, V).

1.5. Theorem. The R-module V is in (Epi ϵ) if and only if it is generated by P and the generators are preserved by hom_{R}(P, −).

Proof. First, observe that (Epi ϵ) is a coreflective subcategory of Mod_{R} and thus it is closed under colimits. Moreover the module P is static because E = hom_{R}(P, P).

Now suppose that V is generated by P and that generators are preserved by hom_{R}(P, −). Tensoring the epimorphism hom_{R}(P, Σ_{X}P) ^{}hom_{R}(P, V) with P gives another epimorphism, as tensoring is a right exact functor. So, one has the commutative diagram:

[figure: P ⊗_{E} hom_{R}(P, Σ_{X}P); P ⊗_{E} hom_{R}(P, V); Σ_{X}P; ϵ_{Σ_{X}P}; V; ϵ_{V}]

where the horizontal arrows are epimorphism and also ϵ_{Σ_{X}P} is an isomorphism because (Epi ϵ) is closed under coproducts. Hence ϵ_{V} is an epimorphism.

Conversely, suppose that ϵ_{V} : P ⊗_{E} hom_{R}(P, V) ^{}V is epi. Now, hom_{R}(P, V) as a right E-module is a quotient of a free E-module, i.e. it is generated in the form Σ_{X}E → hom_{R}(P, V). By tensoring and composing with ϵ_{V}, one has that V is generated by P:

[figure: Σ_{X}P ≅ P ⊗_{E} Σ_{X}E; P ⊗_{E} hom_{R}(P, V); ϵ_{V}; V]

Applying hom_{R}(P, −) we get the commutative diagram:

[figure: Σ_{X}E; α; hom_{R}(P, V); hom_{R}(P, P ⊗_{E} Σ_{X}E); η_{Σ_{X}E}; β; hom_{R}(P, P ⊗_{E} hom_{R}(P, V))); η_{hom_{R}(P,V)}]

where the arrow α is epi. Taking into account the triangular identity:

id = hom_{R}(P, ϵ_{V}) · η_{hom_{R}(P,V)}

one has that:

α = hom_{R}(P, ϵ_{V}) · β · η_{Σ_{X}E}

hence hom_{R}(P, ϵ_{V}) is epi and the generators of V by P are preserved.

2. Application to Clifford theory

We shall now prove a general result about modules obtained by induction and restriction along a ring homomorphism.

In [3], Cline extended the classical Clifford theory of projective representations and stable modules by utilising rings and modules which are graded by a group. His main result can be stated as an equivalence of categories and was reobtained by Dade ([4], theorem 7.4) by using only the functors Hom and ⊗ in a natural way. Restricting it to the essential case of group algebras, the equivalence regards the homomorphism kH ^{}kG induced by a normal subgroup H of G (here G is a finite group, k is a field and kG is the group algebra). In this case, Dade’s result [4] characterizes those kG-modules which are static when restricted to kH. Furthemore, Alperin [1] gives a generalization to non-normal subgroups H.

In general, the equivalence is relative to a given ring homomorphism f : R ^{}S. This gives rise to a bimodule f_{∗} : R +^{}S and to a bimodule f^{∗} : S +^{}R. These bimodules are given by S itself, and it is easy to prove that they are adjoint arrows in Mod. Precisely: f_{∗} −| f^{∗}.

Induction and restriction functors along f are given by the compositions with f_{∗} and f^{∗} respectively. These are adjoint functors (f_{∗} ⊗_{R} − −| f^{∗} ⊗_{S} −):

Mod_{R} [figure: f_{∗} ⊗_{R} −; \underline{f^{∗} ⊗_{S} −}] Mod_{S}

If P : E +^{}R is such that E = hom_{R}(P, P), consider the module P′ = f_{∗} ⊗_{R} P induced by P through f as a module F +^{}S, where F = hom_{S}(P′, P′). One has another ring homomorphism i : E ^{}F, easily described in Mod by the canonical morphism of right liftings

hom_{R}(P, P) ^{}hom_{S}(P′, P′)

The generalization of Cline’s [3] and Dade’s [4] works on Clifford theory referred to above regards those S-modules U whose restriction f^{∗} ⊗_{S} U is static with respect to P. In [8], Nauman expresses it in a purely ring theoretical way:

2.1. Theorem. [Nauman [8], theorem 5.5] The restrictions of the additive functors hom_{S}(P′, −) and P′ ⊗_{F} − form an equivalence between the full additive subcategory of Mod_{S} having as objects all S-modules U such that f^{∗} ⊗_{S} U weakly divides P in Mod_{R} and the full additive subcategory of Mod_{F} whose objects are the F -modules V such that i^{∗} ⊗ V is finitely generated and projective.

By the universal property of adjoint pairs one proves that:

P′ ⊗_{F} i_{∗} ≅ f_{∗} ⊗_{R} P  (3)

Adjointness is essential also in proving the following:

2.2. Theorem. The restriction of the right S-module U along f is in (F ix ϵ) if and only if the restriction of hom_{S}(P′, U) along i is in (F ix η). In symbols:

f^{∗} ⊗_{S} U ∈ (F ix ϵ) ⟺ i^{∗} ⊗_{F} hom_{S}(P′, U) ∈ (F ix η)

Proof. For any right S-module U:

f^{∗} ⊗_{S} U ∈ (F ix ϵ) ⟺ hom_{R}(P, f^{∗} ⊗_{S} U) ∈ (F ix η)

Consider the following diagram. By (3) the square of left adjoints commutes up to isomorphisms, and thus so does the square of right adjoints:

Mod_{R} [figure: hom_{R}(P, −); \underline{P ⊗_{E} −}] Mod_{E}

[figure: Mod_{S}; f_{∗} ⊗_{R} −; f^{∗} ⊗_{S} −; hom_{S}(P′, −); \underline{P′ ⊗_{F} −}; Mod_{F}; i_{∗} ⊗_{E} −; i^{∗} ⊗_{F} −]

Hence:

hom_{R}(P, f^{∗} ⊗_{S} U) ≅ i^{∗} ⊗_{F} hom_{S}(P′, U)

Acknowledgment. I would like to thank the anonymous referee for helpful suggestions and corrections.

References

[1] J.L. Alperin, Static modules and non-normal Clifford theory, J. Austral. Math. Soc. (Series A) 49 (1990), 347-353.

[2] M. Auslander, Representation of Artin algebras, Comm. in Algebra 1 (1974), 177-268.

[3] E. Cline, Stable Clifford theory, J. of Algebra 22 (1972), 350-364.

[4] E.C. Dade, Group-graded rings and modules, Math. Z. (1980), 241-262.

[5] J. Lambek, Remarks on colocalization and equivalence, Comm. in Algebra 10 (1983), 1145-1153.

[6] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. e Fis. Milano 43 (1973), 135-166.

[7] R.J. McMaster, Cotorsion theories and colocalization, Can. J. Math. 27 (1975), 618-628.

[8] S.K. Nauman, Static modules and stable Clifford theory, J. of Algebra 128 (1990), 497-509.

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