TOPOLOGICAL LINEAR COMPACTNESS FOR GROTHENDIECK CATEGORIES . THEOREM OF TYCHONOFF . APPLICATIONS TO COALGEBRAS

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 Ánh 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 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.

Notation and preliminary results for categories
Let C be an abelian category with direct products.Recall that a topology τ on an object Y is open if it belongs to the filter generated by (X i ) i∈I , i.e.X i ⊆ Y for some i (see [4]).
Let A and B be 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 (X i ) the filter base, then (F (X), τ ′ ) is a topological object with filter base (F (X i )).This is clear because we note that, since F is exact, F commutes with finite sums and, moreover, since F preserve direct products F commutes with arbitrary intersections.Thus iii) Let (X i , τ i ) be a family of objects of A. We can construct the topological product object ( i∈I X i , τ ), where τ is the product (Tychonoff) topology.Throughout this paper, we shall always assume that i X i is endowed with the product topology.
Definition 1.1.A topological object X is linearly compact if for every filter base (X i ) of closed subobjects of X, the canonical morphism X is discrete linearly compact if X is linearly compact with respect to the topology defined by the filter base {0}.
The next result shows the behavior of linearly compact objects under localization.
Lemma 1.2.Let T be any localizing subcategory of the Grothendieck category A, and let T : A → A/T be the localizing functor.Assume that T commutes with direct products.If (X, τ ) is linearly compact then (T (X), τ ′ ) is linearly compact, where τ ′ is the linear topology induced by τ via functor T .
Proof: If τ is the linear topology given by the filter base (X i ) i∈I , then τ ′ is given by the filter base (T (X where S is the right adjoint to T and ψ X : X → ST (X) is the unit of the adjoint situation.Thus, since Y is closed.Hence if we have a filter base of closed subobjects (Y j ) j of T (X), there exists a filter base (Z j ) j of closed subobjects of X such that is exact.Since T is exact and it commutes with direct products then T commutes also with direct projective limits so the following sequence Let A be a Grothendieck category and P ∈ A be a projective object.We consider the class It is easy to see that C P is a TTF class.Following [10], we can consider the quotient category A/C P and the canonical functor TP / / A/C P .

SP o o
Since C P is a TTF class it is well-known that T P commutes with direct products.Hence, by Lemma 1.2, if M is linearly compact then T P (M ) is also linearly compact.
Moreover, we recall that if P is small, then, by [5, Theorem 1.2], A/C P ∼ = R-Mod where R = End A (P ).
Following [23] the ring R is said to be a ring with local units if for every finite subset X of R there exists an idempotent e ∈ R such that X is contained in the ring eRe.For a ring R with local units we denote by R-MOD the category of all left R-modules M with 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.H a Hopf algebra with a nonzero integral) its associated Doi-Hopf module category is isomorphic to the category of R-MOD for certain ring R with local units (see [8]).It is well-knonw (see [23]) that {Re | e 2 = 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 A be a Grothendieck category with a set (P i ) i∈I of small projective generators.Then there exists a ring R with local units such that A is equivalent to the category R-MOD.

Quasi-finite objects
Recall that an object X satisfies AB − 5 * if for any subobjet Y and an inverse family of subobjects {X i } i∈I of X, then 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 T be any localizing subcategory of the semiartinian Grothendieck category A, and let T : A → A/T be the localizing functor.
i) If X is a simple object of A/T , then there exists a simple object Let {S i | i ∈ I} be a complete set of representatives of the isomorphism types of simple objects of A. For every i ∈ I, let T i : A → A/T E(Si) denote the localization functor, where Proposition 2.2.Let A be a semiartinian Grothendieck category.An object A is strictly quasi-finite if and only if T i (A) is artinian for any i.
Proof: Let A be strictly quasi-finite, then by Lemma 2.1 T i (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 T i (A) implies that T i (A)/B is quasi-finite for any subobject B of T i (A).Therefore T i (A)/B is finitely cogenerated and T i (A) is artinian.
Conversely, assume that T i (A) is artinian for every i ∈ I. Since the functors T i are exact, it is enough to prove that A is quasi-finite.Let S be a simple object and consider the unique i ∈ I such that S ∼ = S i .If A i denotes the largest subobject of A such that A i ∈ T E(Si) , then we have an exact sequence Then Hom(S i , A i ) = 0 and Hom(S i , A) ≤ Hom(S i , A/A i ).Thus, we can assume that A i = 0, that is, A is T E(Si) torsionfree.The socle soc(A) is then a direct sum of copies of S i and, since T i (A) is artinian, we have that T i (soc(A)) ≤ T i (A) is artinian as well.Therefore, soc(A) consists of a direct sum of finitely many copies of S i ∼ = S.
Theorem 2.3.Let A be a semiartinian Grothendieck category.Assume that every simple object of A has a projective cover.The following properties of an object M are equivalent: Proof: It remains to show that 3) ⇒ 1).Let {S i : 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 Let T i : A → A/T E(Si) denote the localization functor.By [5, Theorem 3.3] every T E(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], T i has a left adjoint functor H i .Hence T i commutes with projective limits.Let M p λ −→ M i −→ 0 be an inverse system of epimorphisms.Then is an epimorphism for any i.Since T i is exact and is an epimorphism.

The theorem of Tychonoff for Grothendieck categories
Theorem 3.1.Let A be a Grothendieck category with a set of small projective generators and let (X k , τ k ) k∈Λ be a family of topological objects.If (X k , τ k ) is linearly compact for every k ∈ Λ, then ( k∈Λ X k , τ ) is linearly compact.
Proof: By Proposition 1.4 we have that A ∼ = R-MOD where R is a ring with local units.Now we observe that if (M i ) i∈I is a family of objects in R-MOD, then the direct product of this family in R-MOD is equal to R i∈I M i = ∪ e i∈I M i , where i∈I M i is the usual cartesian product and e runs 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 ∈ R is 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. Examples.

Right semiperfect coalgebras
If C is a right semiperfect coalgebra then M C is a semiartinian Grothendieck category and simple objects have projective covers.The next result shows that Tychonoff's theorem holds in M C when C is right semiperfect.
Corollary 3.3.Let C be a semiperfect coalgebra and let (M i ) i∈I be a family of linearly compact right comodules, then i∈I M i is linearly compact in M C (here we consider the direct product in the category M C 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 x over a field k of characteristic zero; its structure of coalgebra is given by ∆(x) = x ⊗ 1 + 1 ⊗ x and ǫ(x) = 0.In this case

where t is the classical torsion radical on K[[x]].
We will see that M ′ , the product of U i in M C , is not linearly compact with respect to the product topology.Let Clearly each H i is a closed submodule of M ′ .Consider the set of coset a i = (0, . . ., 1 + x i , 0, . . . ) + H i , then (x)a i ⊆ H i .This set of cosets has the finite intersection property but not the intersection property.

Graded rings
Let R = ⊕ σ∈G R σ be 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).

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.

Linear compactness of the dual algebra of a coalgebra
Let C be a coalgebra and let C * = Hom k (C, k) be its dual convolution algebra.Denote by C M the category of left C-comodules.
A right Morita duality is an additive contravariant category-equivalence between two categories of Mod-R and 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) E R and S E are injective cogenerators and (b) the right and left multiplication induces isomorphisms End R (E) ∼ = S and End S (E) ∼ = R.If S = R, then we say that R has 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 * .iii) ⇒ i) Let X i be an inverse family of left C * -submodules of M * .Since M * is left noetherian, then the X i 's are finitely generated.Now by [1,Proposition 4.1] and and the Y i will form a direct system.From the exact sequence 0 B. J. Müller [18] showed that for commutative rings the existence of Morita dualities implies the existence of self dualities.

Proposition 4 . 1 .Proposition 4 . 3 .
Let C be 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 J denotes the Jacobson radical of C * .Thus C * has only a finite number of isomorphic types of simple modules.If follows that C C * has finite generated socle.Since the injective envelope of each simple right C * -module is reflexive, then C C * , as finite cogenerated, is reflexive.For any right C * -submodule X of 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 C C * is quasi-injective, we can apply [1, Corollary 4.4] and C * is left noetherian.Theorem 4.2.Let C be 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 C is injective.By [11, Theorem 2], C as right C * -module is linearly compact.Hence C C * has a finitely generated socle, then C is an almost connected coalgebra.Hence C is an injective cogenerator of C * -Mod.By a similar argument, C is also an injective cogenerator of Mod-C * .Since C satisfies that End C * (C C * ) = End C * ( C * C) = C * , then C * C C * gives a sef-duality of C * .The other implication follows from Proposition 4.1 and its left side version.Let M be a left C-comodule (then M is 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) M is artinian as a left C * -comodule.Proof: i) ⇒ iii) Let Xbe a right C * -submodule of M and 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 = ⊕ I S i , then Y * = I S * i , but Y * linearly compact implies that I is finite.Hence M/X is finitely cogenerated and therefore M is artinian as right C * -comodule.iii) ⇔ ii) We can apply [1, Corollary 4.3].We note that C is quasiinjective as right C * -module, Hom C * (M, C) ∼ = Hom k (M, k) = M * and any submodule of M is closed.

Corollary 4 . 4 .
C * is left linearly compact for the discrete topology if and only if C * is left noetherian.

Corollary 4 . 5 .
Let C be a coalgebra.If C * has a right Morita dualiy then C * has a self-Morita duality.Proof: If C * has a right Morita duality, then C * is left noetherian by Proposition 4.1.Moreover, by [18, Theorem 1] C * C * is discrete linearly compact, then Corollary 4.4 implies that C * is right noetherian.By Theorem 4.2, C * has self-Morita duality.Let R be a k-algebra.A left R-module M is called almost noetherian if every cofinite submodule of M is finitely generated.Submodules, quotients, and extensions of almost noetherian modules are almost noetherian.The algebra R is left almost noetherian if its almost noetherian as a left module.We call R almost noetherian if it is left and right almost noetherian.See [7, Section 1.1] for further details on almost noetherian modules and algebra.The coradical filtration on C induces a filtration on C * ,

Proposition 4 . 6 .
Let C be a coalgebra and R the graded dual ring of gr(C).Denote by gr(C * ) the graded ring associated to the filtration (1).Then i) gr(C * ) ∼ = R as graded rings.ii)C * is complete with respect to (1).Lemma 4.7.C * is linearly compact with respect to (1) if and only if C * i is linearly compact for the discrete topology for all i.Proof: It follows from[22, Theorem 28.15]  and the fact thatC * /C ⊥ i ∼ = C *i and C * is complete with respect to(1).It is well-known that the category of right C-comodules M C is isomorphic to Rat(C * -Mod), the subcategory of C * -Mod consisting of all rational left C * -modules.Rat(C * -Mod) is closed under submodules, quotients and arbitrary direct sums.In the sense of[21], it is a hereditary pretorsion class in C * -Mod.See loc.cit.for further detail on torsion classes.The left exact preradical associated to Rat(C * -Mod) is the rational functor Rat C (−) :C * -Mod → C * -Mod.Given M ∈ C * -Mod, Rat C (M )is the sum of all rational modules contained in M .The left linear topology F C on C corresponding to Rat(C * -Mod) is the family of all closed (in the weak-* topology) and cofinite left ideals of C * .If I ∈ F C there is a finite-dimensional left coideal W of C such that I = W ⊥ .Recall that a coalgebra C is called almost connected if its coradical C 0 is finite dimensional.

Theorem 4 . 8 .
If C * is linearly compact for the topology (1), then C * is almost noetherian.Moreover, if C is almost connected both conditions are equivalent.Proof: Assume that C * is linearly compact for the topology(1).SinceC * /C ⊥ 0 ∼ = C * 0 is discrete linearly compact it follows that C is almost connected.Then since C ⊥ i is open, it follows that C * /C ⊥ i is linearly compact for the discrete topology.But C * /C ⊥ i ∼ = C * i , hence C * i is linearly compact.Now the exact sequence 0 −→ C 1 −→ C 0 −→ C 0 /C 1 −→ 0 induces 0 −→ (C 1 /C 0 ) * −→ C * 1 −→ C * 0 −→ 0. Thus (C 1 /C 0 ) * is linearly compact.Assume that C 1 /C 0 = ⊕ I S i , then (C 1 /C 0 ) * ∼ = I S * i .It follows that I is finite because (C 1 /C 0 ) * is linearly compact.Hence C 1 /C 0 is finite dimensional, and C 1 is finite dimensional.Using induction we obtain that C i is finite dimensional for all i.By [7, Theorem 2.8], C * is almost noetherian.