SOME REMARKS
ON MODULAR QFD LATTICES
MIHAI IOSIF
A modular latticeLwith 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for anyx∈L. Following the line of research from [1], we give a condition for a lattice to be QFD, in relation to the poset of its subdirectly irreducible elements. We also study the subdirectly irreducible elements for a class of lattices larger than the class of compactly generated lattices.
Finally, we give an application to Grothendieck categories.
AMS 2000 Subject Classification: 06C05, 06B35, 16P60, 16S90, 18E15.
Key words: modular lattice, Goldie dimension, QFD lattice, Grothendieck cate- gory.
INTRODUCTION
In [1] we investigated some conditions for an upper continuous modular lattice to be QFD. The starting point was a condition that we calledCamillo’s Conditionsince it had been considered for modules in [4]. Camillo’s Condition seemed suitable for compactly generated lattices, and we extended it to more general classes of lattices, in order to apply it to Grothendieck categories.
This paper follows the line of research from [1], in connection to other conditions that we studied in [2]. These conditions involve the poset of the subdirectly irreducible elementsof a lattice. We characterize in these terms the compactly generated lattices among a more general class of lattices that we callalmost compactly generated. This class already appeared in [1], related to Camillo’s Condition in a natural way. From this perspective, we give a charac- terization of upper continuous modular lattices that are compactly generated and QFD. We also study the poset of subdirectly irreducible elements for an interval in an almost compactly generated lattice; we obtain an inequality in- volving the codepth and some information about the dual Krull dimension of this poset.
In the last section we give an application to Grothendieck categories.
Thus, we obtain a result that improves in some way [1, Theorem 6.1].
REV. ROUMAINE MATH. PURES APPL.,52(2007),3, 341–347
0. PRELIMINARIES
In this section we present ([1] and [2]) the basic terminology and some results for posets and lattices that will be used in this paper.
Throughout this paper we assume that all latticesLhave a least element denoted by 0 and a last element denoted by 1. However, partially ordered sets P for short posets, are not necessarily assumed to have 0 or 1. For a posetP and elementsabinP we write
b/a:= [a, b] ={x∈P |axb}.
We also write
[a, b[ := {x∈P |ax < b}.
We say that the intervalb/ais simpleifa=band b/a={a, b}.
Recall that a lattice L with 0 and 1 has finite Goldie (or uniform) di- mension if there is no infinite independent subset ofL. The latticeLis said to beQFD (i.e., quotients have finite Goldie dimension) if 1/x has finite Goldie dimension for everyx∈L.
As in [1, Definition 2.7], we say that the latticeL verifies Condition (C) (orCamillo’s Condition) if for every m∈L there exists a compact elementt ofL such thatt≤m and [t, m[ has no maximal element.
For an upper continuous latticeLwe introduced ([1]) the mapϕ:L→L defined by
ϕ(x) =
{k |k∈x/0, k compact inL}, x∈L.
We list here, as in [1, Lemma 3.3], the basic properties of ϕ.
(1) ϕ is an opener operator on L; that is, it is an order preserving map such thatϕ(x)≤x and ϕ(ϕ(x)) =ϕ(x) for all x∈L.
(2) ϕ(L) ={x ∈L |x=ϕ(x)}.
(3) ϕ(L) is a complete lattice and, for every A⊆ϕ(L), supϕ(L)A= supLA and infϕ(L)A=ϕ(infLA).
(4) k is a compact element in L if and only if k is a compact element inϕ(L).
(5) ϕ(L) is compactly generated.
We shall use these properties without further notice. Clearly,L=ϕ(L) if and only ifL is compactly generated.
Following [2, Definitions 0.1], we say that a poset P is subdirectly irre- ducible ifP ={0}and the set P\ {0} has a least element; i.e., there exists an element 0=x0 ∈P such thatx0 x for every 0=x∈P.
For an arbitrary poset P, an element s ∈ P is said to be a subdirectly irreducible element of P if the interval 1/sis a subdirectly irreducible poset.
We denote byS(P) the set of all subdirectly irreducible elements of P; i.e., S(P) :={x∈P|1/x is subdirectly irreducible}.
Definition 0.1 ([2, Definition 1.1]). A poset P is said to be rich in sub- directly irreducibles if for every a < binP, the interval b/ahas a subdirectly irreducible factor interval b/c.
For all undefined notation and terminology on lattices, the reader is referred to Birkhoff [3], Crawley and Dilworth [5], or Stenstr¨om [6].
1. ALMOST COMPACTLY GENERATED LATTICES We begin with a definition providing a name for a class of lattices that was studied in [1] in relation to Camillo’s Condition.
Definition 1.2. Let L be an upper continuous lattice. We say that the lattice Lis almost compactly generated if, for everya∈Lwith ϕ(a)< a, the intervala/ϕ(a) is a dense poset.
If a is a compactly generated element of L, then a = ϕ(a) and so the interval a/ϕ(a) degenerates to one point. Hence every compactly generated lattice is almost compactly generated. The real interval [0,1] with the usual order has 0 as its only compact element; thus, [0,1] is almost compactly gen- erated, but it is not compactly generated. This last fact enable us to give a large class of examples.
Proposition1.3. LetF be a compactly generated lattice andDan upper continuous dense lattice. ConsiderL=F×D with the componentwise order.
Then L is almost compactly generated and ϕ(L) = F × {0} F, where 0 denotes the first element of D.
Proof. Notice first that since F is compactly generated, it is upper con- tinuous by [6, Chapter 3, Proposition 5.3]. D is also upper continuous, so L is an upper continuous lattice.
Next, we show that (x, α)∈L is a join of compact elements ofL if and only if α = 0. Indeed, if α >0, then α =
{β |β ∈D, β < α}, since D is complete and dense. Thus, (x, α) =
{(x, β) |β < α in D}, so (x, α) is not compact inL. Since a compact element ofLmust have the second coordinate 0, we obtain that (x, α) is not a join of compact elements of L, for α >0.
However, if c is compact in F and (c,0) =
i∈I(xi, βi), then all βi are zero andc=
i∈Ixi inF, which implies that there exists a finite subsetJ of I such thatc=
i∈Jxi inF. Consequently, (c,0) =
i∈J(xi, βi). Hence (c,0) is compact in L, if c is compact in F. Now, each x ∈F is a join of compact elements ofF, so (x,0) is a join of compact elements ofL.
It follows that ϕ(x, α) = (x,0). Moreover, the interval (x, α)/ϕ(x, α) of Lis isomorphic to the interval [0, α] of D, and so it is a dense poset (even if degenerates to a point).
Remark 1.4. IfLis an almost compactly generated modular lattice, then the lattice ϕ(L) is upper continuous and modular.
Indeed, since ϕ(L) is compactly generated, it is upper continuous, as above. Now, the modularity follows by [1, Lemma 4.4].
The following result characterizes compactly generated lattices among almost compactly generated lattices. We shall use it to characterize the upper continuous modular lattices that are compactly generated and QFD.
Lemma1.5. LetLbe an upper continuous modular lattice. The following assertions are equivalent.
(1) L is compactly generated.
(2) L is almost compactly generated and is rich in subdirectly irreducibles.
Proof. (1)⇒(2): SinceLis compactly generated,Lis rich in subdirectly irreducibles by [2, Corrolary 1.4].
(2)⇒(1): Leta∈L.Ifa=ϕ(a),thena > ϕ(a),and the intervala/ϕ(a) is dense, by hypothesis. SinceLis rich in subdirectly irreducibles,a/ϕ(a) has a subdirectly irreducible factor interval. Thus,a/ϕ(a) has a simple interval, which is a contradiction. Hence a=ϕ(a),for all a∈L.Therefore, L=ϕ(L) is compactly generated.
Proposition 1.6. Let L be an upper continuous modular lattice. The following assertions are equivalent.
(1) L is compactly generated and QFD.
(2) L verifies Condition (C)and is rich in subdirectly irreducibles.
Proof. (1)⇒ (2): Since L is compactly generated and QFD, it follows from [1, Theorem 2.8] that L verifies Condition (C). Moreover, Lbeing com- pactly generated, is rich in subdirectly irreducibles by Lemma 1.5.
(2)⇒(1): SinceLis an upper continuous lattice verifying Condition (C), L is almost compactly generated, by [1, Lemma 3.5]. Now, using Lemma 1.5, one has that L is compactly generated. Since L is also modular and verifies Condition (C), L is QFD by [1, Theorem 2.8].
2. SUBDIRECTLY IRREDUCIBLE ELEMENTS
In this section we study the action of ϕ on subdirectly irreducible ele- ments of an interval of an almost compactly generated modular lattice. We begin with a technical result.
Lemma 2.7. Let L be an almost compactly generated modular lattice.
Then
ϕ(u∨x) =u∨ϕ(x) for allx∈L and u∈ϕ(L).
Proof. We have u∨x u, that implies ϕ(u ∨x) ϕ(u) = u, and u∨x x, that implies ϕ(u ∨x) ϕ(x). So, ϕ(u∨x) u∨ϕ(x). Since x ϕ(x), we have also x∧(u∨ϕ(x) = (x∧u)∨ϕ(x) by modularity law.
Hence, using again modularity, we obtain
(u∨x)/(u∨ϕ(x)) = (u∨ϕ(x)∨x)/(u∨ϕ(x)) x/(x∧(u∨ϕ(x))) = x/((x∧u)∨ϕ(x)).
The last interval is a sub-interval of x/ϕ(x). Since L is almost compactly generated, x/ϕ(x) is a dense interval. Thus (u∨x)/(u∨ϕ(x)) is also a dense interval (even if degenerates to a single point).
Now, if ϕ(u∨x) =u∨ϕ(x),then ϕ(u∨x) > u∨ϕ(x) in ϕ(L),and so there exists a compact element c of L such that c u∨x and c u∨ϕ(x). Thus, we have in L the isomorphism
(c∨u∨ϕ(x))/(u∨ϕ(x)) c/(c∧(u∨ϕ(x))).
Since c u∨ϕ(x), we have c ∧(u ∨ϕ(x)) < c. Moreover, c is compact in L, so the interval c/(c∧(u∨ϕ(x))) is a non-degenerated compact lattice and, consequently, it contains a simple interval by Krull’s Lemma. Thus, (c∨u∨ϕ(x))/(u∨ϕ(x)) also contains a simple interval. On the other hand, (c∨u∨ϕ(x))/(u∨ϕ(x)) is a sub-interval in the dense interval (u∨x)/(u∨ϕ(x)), and so it is also a dense interval, which is a contradiction.
Proposition 2.8. Let L be an almost compactly generated modular lat- tice, and a < b in L. Then x → ϕ(x) is a strictly increasing map from S(b/a) ⊆L toS(ϕ(b)/ϕ(a))⊆ϕ(L).
Proof. We show first that for a subdirectly irreducible element x of the interval b/a of L, the element ϕ(x) is subdirectly irreducible in the interval ϕ(b)/ϕ(a) of ϕ(L).Let sbe the unique atom ofb/x .Then the interval s/xis simple inL,and soϕ(s)/ϕ(x) is a simple interval inϕ(L) by [1, Lemma 3.7]. If u∈ϕ(L) such thatu∈ϕ(b)/ϕ(a) andu > ϕ(x),thenuxand, consequently, u∨x > x inb/a . Since x∈ S(b/a), we have u∨xs, and so u∨x =u∨s.
By Lemma 2.7, we obtain that u=u∨ϕ(x) =ϕ(u∨x) =ϕ(u∨s) =u∨ϕ(s). Thus,uϕ(s),that ends the first part of the proof.
Now, let us show that the map is strictly increasing. To see this, let x < y be two subdirect irreducible elements in b/a .Letsbe the unique atom in b/x .Thus, y > sby subdirect irreducibility. We have ϕ(x) ϕ(s) ϕ(y) and, as above, ϕ(x)=ϕ(s).Thus, ϕ(x)< ϕ(y) and we are done.
As in [2], for a posetP we shall denote byλ(P) the so calledcodepth of P; i.e., the least ordinal that does not embed inP.
Corollary2.9. LetLbe an almost compactly generated modular lattice, anda < b in L.Then
λ(S(b/a))λ(S(ϕ(b)/ϕ(a))),
where b/ais an interval of L and ϕ(b)/ϕ(a) is an interval of ϕ(L).
For a posetP we denote by k0(P) thedual Krull dimension of P. Proposition 2.10. Let L be an almost compactly generated modular lattice. If k0(S(ϕ(L))) exists and is countable, then for every two elements a < b in L the dual Krull dimension k0(S(b/a)) exists and is countable.
Proof. ϕ(L) is an upper continuous modular lattice by Remark 1.4. Let a < b in L and consider the interval ϕ(b)/ϕ(a) of ϕ(L). Since k0(S(ϕ(L))) exists and is countable,k0(S(ϕ(b)/ϕ(a))) exists and is countable by [2, Corol- lary 1.12]. By Proposition 2.8, we have a strictly increasing map fromS(b/a) inLtoS(ϕ(b)/ϕ(a)) inϕ(L).So, there existsk0(S(b/a)) k0(S(ϕ(b)/ϕ(a))) and we are done.
3. APPLICATION TO GROTHENDIECK CATEGORIES In this section we shall denote by G a Grothendieck category, that is, an Abelian category with exact direct limits and with a generator. For any object X ∈ G, we shall denote by L(X) the lattice of all subobjects of X.
This lattice is upper continuous and modular (see e.g., Stenstr¨om [6, Chapter 4, Proposition 5.3, and Chapter 5, Section 1]. For all undefined notation and terminology on Abelian categories the reader is referred to Stenstr¨om [6].
We say that an objectX∈ G has finite Goldie dimension (resp. is QFD) if the lattice L(X) has finite Goldie dimension (resp. is QFD).
Following [2, Definition 2.15], we say that an objectX ofG issubdirectly irreducible if the latticeL(X) is subdirectly irreducible. Consequently, a sub- object Y of an object X ∈ G is said to be subdirectly irreducible in X if the quotient objectX/Y is subdirectly irreducible.
If we specialize Proposition 1.6 for the lattice L(X), we immediately obtain
Proposition 3.11. Let G be a Grothendieck category and let X be an object ofG. The following statements are equivalent.
(1) X is finitely generated and QFD.
(2) X satisfies the conditions
(a) for every subobject Y of X there exists a finitely generated subob- jectT ofY such that the quotient objectY/T ofGhas no maximal proper subobject;
(b) for every two subobjectsZ < Y of X there exists a subobjectS of X such that Z S < Y and S is subdirectly irreducible in Y.
Notice that Proposition 3.11 holds under a more general hypothesis than in [1, Theorem 6.1], which needs for the Grothendieck categoryGto belocally finitely generated(i.e., to have a family of finitely generated generators).
Acknowledgements.The author would like to thank Professor T. Albu for his valu- able suggestions.
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Received 18 June 2006 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest 1, Romania miosif@gta.math.unibuc.ro
