Topological linear compactness for Grothendieck categories. Theorem of Tychonoff. Applications to coalgebras

Publ. Mat. 50 (2006), 57–70 TOPOLOGICAL LINEAR COMPACTNESS FOR GROTHENDIECK CATEGORIES. THEOREM OF TYCHONOFF. APPLICATIONS TO COALGEBRAS P. Enache, C. N˘ ast˘ asescu and B. Torrecillas Abstract We show the Tychonoff’s theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra. Introduction The classical Morita theory duality of module categories (cf. [17] and [24]) was extended to Grothendieck categories by Colby and Fuller [6] and a weaker form by ´ Anh and Wiegandt [4]. Many authors have considered this theory in general Grothendieck categories (cf. [12] and [20]) and also in particular Grothendieck categories (e.g. graded modules aver graded rings [16], modules over rings with enough idempotents [3], comodules over a coalgebra [13]). Linear compact objects with respect to a topology play a crucial role in the analysis of Morita dualities in Grothendieck categories (see [4], [12]). In this paper we study linear compact objects in Grothendieck categories. The first section is notational and moreover it contains some preliminary results. Section 2 is devoted to characterizing linear compact objects in semiartinian Grothendieck categories (Theorem 2.3). In Section 3 we consider the Tychonoff’s theorem. This theorem is a classical result for linear compact modules over unitary rings, which does not remain true for general Grothendieck categories. We obtain a 2000 Mathematics Subject Classification. 16W30, 18E15, 16D90. Key words. Tychonoff’s theorem, coalgebras, quasi-finite objects, semiartinian Grothendieck categories, Morita duality. 58 P. Enache, C. N˘ ast˘ asescu, B. Torrecillas positive answer for Grothendieck categories having a set of small projective generators. This result applies for right semiperfect coalgebras, graded rings and rings with local units. In Section 4 we study the linear compactness of the dual algebra of a coalgebra. For this algebras the discrete linear compactness is equivalent to the noetherian property, and when the coalgebra is almost connected, the topological linear compactness is equivalent to the almost noetherian property (Theorem 4.8). Furthermore, we show that if this algebra has a right Morita duality it has a self-duality. 1. Notation and preliminary results for categories Let Cbe an abelian category with direct products. Recall that a topology τon an object X∈ C is a filter base (Xi)i∈Iof subobjects of X. For a subobject Yof Xthe τ-closure of Yis given by Y=∩I(Y+Xi), thus Yis τ-closed if Y=∩I(Y+Xi). Yis open if it belongs to the filter generated by (Xi)i∈I, i.e. Xi⊆Yfor some i(see [4]). Let Aand Bbe two abelian categories with direct products and let F:A → B be an exact functor which commutes with direct products, then i) If (X, τ) is a topological object with (Xi) the filter base, then (F(X), τ′) is a topological object with filter base (F(Xi)). This is clear because Fcommutes with intersections. ii) If Y⊆Xis a τ-closed, then F(Y)⊆F(X) is τ′-closed. Indeed, we note that, since Fis exact, Fcommutes with finite sums and, moreover, since Fpreserve direct products Fcommutes with arbitrary intersections. Thus F(Y) = F(∩i∈I(Y+Xi)) = ∩i∈IF(Y+Xi) = F(Y). iii) Let (Xi, τi) be a family of objects of A. We can construct the topological product object (Qi∈IXi, τ), where τis the product (Tychonoff) topology. Throughout this paper, we shall always assume that QiXiis endowed with the product topology. Definition 1.1. A topological object Xis linearly compact if for every filter base (Xi) of closed subobjects of X, the canonical morphism X−→ lim ←− X/Xi is an epimorphism. Xis discrete linearly compact if Xis linearly compact with respect to the topology defined by the filter base {0}. Theorem of Tychonoff 59 The next result shows the behavior of linearly compact objects under localization. Lemma 1.2. Let Tbe any localizing subcategory of the Grothendieck category A, and let T:A → A/Tbe the localizing functor. Assume that Tcommutes with direct products. If (X, τ)is linearly compact then (T(X), τ′)is linearly compact, where τ′is the linear topology induced by τvia functor T. Proof: If τis the linear topology given by the filter base (Xi)i∈I, then τ′ is given by the filter base (T(Xi))i∈I. If Y⊆T(X) is a closed subobject relative to τ′, we consider Z=ψ−1 X(S(Y)) where Sis the right adjoint to Tand ψX:X→ST (X) is the unit of the adjoint situation. Thus, T(Z) = T(S(Y)) = Y. Moreover, T(Z) = ∩i∈I(T(Z) + T(Xi)) = ∩i∈I(Y+T(Xi)) = Y since Yis closed. Hence if we have a filter base of closed subobjects (Yj)j of T(X), there exists a filter base (Zj)jof closed subobjects of Xsuch that T(Zj) = Yj. Now X−→ lim ←− X/Zj−→ 0 is exact. Since Tis exact and it commutes with direct products then Tcommutes also with direct projective limits so the following sequence T(X)−→ T(lim ←− X/Zj)∼ =lim ←− T(X)/Yj−→ 0 is exact. Thus T(X) is linearly compact in A/T. Example 1.3. Let Abe a Grothendieck category and P∈ A be a projective object. We consider the class CP={A∈ A | HomA(P, A) = 0}. It is easy to see that CPis a TTF class. Following [10], we can consider the quotient category A/CPand the canonical functor A TP//A/CP. SP oo Since CPis a TTF class it is well-known that TPcommutes with direct products. Hence, by Lemma 1.2, if Mis linearly compact then TP(M) is also linearly compact. Moreover, we recall that if Pis small, then, by [5, Theorem 1.2], A/CP∼ =RMod where R= EndA(P). 60 P. Enache, C. N˘ ast˘ asescu, B. Torrecillas Following [23] the ring Ris said to be a ring with local units if for every finite subset Xof Rthere exists an idempotent e∈Rsuch that Xis contained in the ring eRe. For a ring Rwith local units we denote by R-MOD the category of all left R-modules Mwith the property that RM =M. Clearly the category R-MOD is a Grothendieck category. Recently, the interest for this class of rings has been motivated by the fact that for certain H-comodule algebras (e.g. Ha Hopf algebra with a nonzero integral) its associated Doi-Hopf module category is isomorphic to the category of R-MOD for certain ring Rwith local units (see [8]). It is well-knonw (see [23]) that {Re |e2=e}is a generating set of finitely generated projective modules in R-MOD. The following result connects rings with local units with Grothendieck categories. Proposition 1.4. Let Abe a Grothendieck category with a set (Pi)i∈I of small projective generators. Then there exists a ring Rwith local units such that Ais equivalent to the category R-MOD. Proof: See [5, Theorem 3.1] and [2]. 2. Quasi-finite objects Recall that an object Xsatisfies AB −5∗if for any subobjet Yand an inverse family of subobjects {Xi}i∈Iof X, then Y+∩i∈IXi=∩i∈I(Y+Xi). As in the case of R-modules, it is easy to show that any discrete linearly compact object satisfies AB −5∗and that the class of linearly compact objects is a Serre subcategory. A semiartinian object is called quasi-finite if its socle has finite generated homogeneous component. A semiartinian object is strictly quasi-finite if any quotient is quasi-finite. The next lemma can be shown using the same argument as [13, Lemma 2.7 and Proposition 2.8]. Lemma 2.1. Let Tbe any localizing subcategory of the semiartinian Grothendieck category A, and let T:A → A/Tbe the localizing functor. i) If Xis a simple object of A/T, then there exists a simple object Y∈ A such that T(Y)∼ =X. ii) If Mis a strictly quasi-finite object in A, then T(M)is strictly quasi-finite. Theorem of Tychonoff 61 Let {Si|i∈I}be a complete set of representatives of the isomorphism types of simple objects of A. For every i∈I, let Ti:A → A/TE(Si)denote the localization functor, where TE(Si)={X∈ A | Hom(X, E(Si)) = 0}. Proposition 2.2. Let Abe a semiartinian Grothendieck category. An object Ais strictly quasi-finite if and only if Ti(A)is artinian for any i. Proof: Let Abe strictly quasi-finite, then by Lemma 2.1 Ti(A) is strictly quasi-finite in the quotient category A/T(E(Si). This category has an unique isomorphic type of simple module, hence strictly quasi-finite of Ti(A) implies that Ti(A)/B is quasi-finite for any subobject Bof Ti(A). Therefore Ti(A)/B is finitely cogenerated and Ti(A) is artinian. Conversely, assume that Ti(A) is artinian for every i∈I. Since the functors Tiare exact, it is enough to prove that Ais quasi-finite. Let S be a simple object and consider the unique i∈Isuch that S∼ =Si. If Ai denotes the largest subobject of Asuch that Ai∈ TE(Si), then we have an exact sequence 0−→ Ai−→ A−→ A/Ai−→ 0. Then Hom(Si, Ai) = 0 and Hom(Si, A)≤Hom(Si, A/Ai). Thus, we can assume that Ai= 0, that is, Ais TE(Si)torsionfree. The socle soc(A) is then a direct sum of copies of Siand, since Ti(A) is artinian, we have that Ti(soc(A)) ≤Ti(A) is artinian as well. Therefore, soc(A) consists of a direct sum of finitely many copies of Si∼ =S. By [19, Proposition 2.5], if Ais a semiartinian object in Asatisfying AB −5∗, then Ais strictly quasi-finite. Theorem 2.3. Let Abe a semiartinian Grothendieck category. Assume that every simple object of Ahas a projective cover. The following properties of an object Mare equivalent: 1) Mis discrete linearly compact; 2) Mis AB −5∗; 3) Mis strictly quasi-finite. Proof: It remains to show that 3) ⇒1). Let {Si:i∈I}be a complete set of representatives of the isomorphism types of simple objects of A. For every i∈I, we may consider the localizing subcategory TE(Si)={X∈ A | Hom(X, E(Si) = 0}. 62 P. Enache, C. N˘ ast˘ asescu, B. Torrecillas Let Ti:A → A/TE(Si)denote the localization functor. By [5, Theorem 3.3] every TE(Si)is a TTF-class. Since the projective cover of the simple modules form a family of finitely generated projective generators by [5, Theorem 3.2], Tihas a left adjoint functor Hi. Hence Ticommutes with projective limits. Let Mpλ −→ Mi−→ 0 be an inverse system of epimorphisms. Then Ti(pλ): Ti(M)−→ Ti(Mλ) are epimorphisms. Since Mis strictly quasi-finite Ti(M) is an artinian object. Hence lim ←− Ti(pλ): M−→ lim ←− Ti(Mλ) is an epimorphism. Thus Ti(lim ←− pλ) is an epimorphism for any i. Since Tiis exact and ∩Ker Ti= 0, then lim ←− pλ:M−→ lim ←− Mλ is an epimorphism. 3. The theorem of Tychonoff for Grothendieck categories Theorem 3.1. Let Abe a Grothendieck category with a set of small projective generators and let (Xk, τk)k∈Λbe a family of topological objects. If (Xk, τk)is linearly compact for every k∈Λ, then (Qk∈ΛXk, τ) is linearly compact. Proof: By Proposition 1.4 we have that A∼ =RMOD where Ris a ring with local units. Now we observe that if (Mi)i∈Iis a family of objects in RMOD, then the direct product of this family in RMOD is equal to RQi∈IMi=∪eQi∈IMi, where Qi∈IMiis the usual cartesian product and eruns over all the idempotents of R. Moreover, the same statement holds for inverse limits. These two observations join with the definition of linearly compact object reduce the problem to the case of unitary rings of the form eRe, where e∈Ris idempotent. But for unitary rings the result is true (see [22, Theorem 28.7]. Remark 3.2.An alternative proof of the preceding result and the classical result for linearly compact modules over unitary rings can be given using Lemma 1.2, Proposition 1.4 and Example 1.3. Theorem of Tychonoff 63 Examples. 1. Right semiperfect coalgebras If Cis a right semiperfect coalgebra then MCis a semiartinian Grothendieck category and simple objects have projective covers. The next result shows that Tychonoff’s theorem holds in MCwhen Cis right semiperfect. Corollary 3.3. Let Cbe a semiperfect coalgebra and let (Mi)i∈Ibe a family of linearly compact right comodules, then Qi∈IMiis linearly compact in MC(here we consider the direct product in the category MC with the product topology). In general the result is not true for the category of comodules, as we can show in the following Counterexample 3.4. Let C=k[x] be the Hopf algebra of the polynomials in one indeterminate xover a field kof characteristic zero; its structure of coalgebra is given by ∆(x) = x⊗1 + 1 ⊗xand ǫ(x) = 0. In this case C∗=K[[x]] and MCis isomorphic to the subcategory of torsion K[[x]]-modules. Take Ui=K[[x]]/(xi) for i= 1,2,... . Then the Uiare artinian objects, so Uiare (discrete) linearly compact in MC and C∗-Mod. Consider M′=t ∞ Y i=1 Ui!⊆M= ∞ Y i=1 Ui, where t is the classical torsion radical on K[[x]]. We will see that M′, the product of Uiin MC, is not linearly compact with respect to the product topology. Let Hi= ((x)/(xi)×K[[x]]/(xi+1)× · · · )∩M′. Clearly each Hiis a closed submodule of M′. Consider the set of coset ai= (0,...,1 + xi,0,...) + Hi, then (x)ai⊆Hi. This set of cosets has the finite intersection property but not the intersection property. 2. Graded rings Let R=⊕σ∈GRσbe a graded ring. Since {R(σ), σ ∈G}is a family of small projective generators in R-gr, then the products of linearly compact graded modules are linearly compact (see [16] for details about linearly compact graded modules). 64 P. Enache, C. N˘ ast˘ asescu, B. Torrecillas 3. Rings with local units Linearly compact modules for this category has been considered in [3]. From the proof of Theorem 3.1 it follows that the Tychonoff’s theorem is valid for modules over ring with local units. 4. Linear compactness of the dual algebra of a coalgebra Let Cbe a coalgebra and let C∗= Homk(C, k) be its dual convolution algebra. Denote by CMthe category of left C-comodules. A right Morita duality is an additive contravariant category-equivalence between two categories of Mod-Rand S-Mod which are both closed under submodules and factor modules and contains all finitely generated modules. This is equivalent to say that there exists a (S, R)-bimodule such that (a) ERand SEare injective cogenerators and (b) the right and left multiplication induces isomorphisms EndR(E)∼ =Sand EndS(E)∼ =R. If S=R, then we say that Rhas a right Morita selfduality. It is well-known the connection between Morita duality theory and the notion of linear compactness (see [18]). First, we examine the Morita duality theory for C∗. Proposition 4.1. Let Cbe a coalgebra. If C∗possesses a right Morita duality then C∗is left noetherian. Proof: If C∗has a right Morita duality, then C∗is right linearly compact [18, Theorem 1]. Hence C∗/J is right linearly compact, therefore it has finite Goldie dimension, then it is semisimple, where Jdenotes the Jacobson radical of C∗. Thus C∗has only a finite number of isomorphic types of simple modules. If follows that CC∗has finite generated socle. Since the injective envelope of each simple right C∗-module is reflexive, then CC∗, as finite cogenerated, is reflexive. For any right C∗-submodule Xof C, we have that (C/X)C∗is reflexive. Hence C/X has a finite generated socle and it is finitely cogenerated. Therefore C as right C∗-module is artinian. As CC∗is quasi-injective, we can apply [1, Corollary 4.4] and C∗is left noetherian. Theorem 4.2. Let Cbe a coalgebra. C∗has a Morita self-duality if and only if C∗is both sides noetherian. Proof: Assume that C∗is left noetherian, then by [7, Theorem 3.2] the class of rational left C∗-modules is closed under injective envelopes. Hence the left C∗-module Cis injective. By [11, Theorem 2], Cas right C∗-module is linearly compact. Hence CC∗has a finitely generated socle, Theorem of Tychonoff 65 then Cis an almost connected coalgebra. Hence Cis an injective cogenerator of C∗-Mod. By a similar argument, Cis also an injective cogenerator of Mod-C∗. Since Csatisfies that EndC∗(CC∗) = EndC∗(C∗C) = C∗, then C∗CC∗gives a sef-duality of C∗. The other implication follows from Proposition 4.1 and its left side version. Proposition 4.3. Let Mbe a left C-comodule (then Mis also a right C∗-module). The following conditions are equivalent: i) C∗M∗is discrete linearly compact; ii) M∗is a left noetherian C∗-module; iii) Mis artinian as a left C∗-comodule. Proof: i) ⇒iii) Let Xbe a right C∗-submodule of Mand consider Y= soc(M/X). We have the following commutative diagram of right C∗-modules 0  Y  0//X//M//M/X //0 By taking dual we obtain a commutative diagram of left C∗-modules 0//(M/X)∗  //M∗//X∗//0 Y∗  0 Hence Y∗is left linearly compact. If Y=⊕ISi, then Y∗=QIS∗ i, but Y∗linearly compact implies that Iis finite. Hence M/X is finitely cogenerated and therefore Mis artinian as right C∗-comodule.